Trilinear interpolation

<h2 id="related-methods">Related methods</h2>
<p>Trilinear interpolation is the extension of <a href="/facts/Linear_interpolation/galkvRGy">linear interpolation</a>, which operates in spaces with <a href="/facts/Dimension/0ktmUyac">dimension</a> 
  
    
      
        D
        =
        1
      
    
    {\displaystyle D=1}
  
, and <a href="/facts/Bilinear_interpolation/yoke6SzF">bilinear interpolation</a>, which operates with dimension 
  
    
      
        D
        =
        2
      
    
    {\displaystyle D=2}
  
, to dimension 
  
    
      
        D
        =
        3
      
    
    {\displaystyle D=3}
  
. These interpolation schemes all use polynomials of order 1, giving an accuracy of order 2, and it requires 
  
    
      
        
          2
          
            D
          
        
        =
        8
      
    
    {\displaystyle 2^{D}=8}
  
 adjacent pre-defined values surrounding the interpolation point. There are several ways to arrive at trilinear interpolation, which is equivalent to 3-dimensional <a href="/facts/Tensor/pNjFo5tV">tensor</a> <a href="/facts/B-spline/MLTCWCEd">B-spline</a> interpolation of order 1, and the trilinear interpolation operator is also a tensor product of 3 linear interpolation operators.
</p><p>For an arbitrary, <a href="/facts/Unstructured_grid/2CxbbbFx">unstructured mesh</a> (as used in <a href="/facts/Finite_element/eUcfP1vn">finite element</a> analysis), other methods of interpolation must be used; if all the mesh elements are <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a> (3D <a href="/facts/Simplex/0FCfNRN8">simplices</a>), then <a href="/facts/Barycentric_coordinates_(mathematics)/ymlRaudU">barycentric coordinates</a> provide a straightforward procedure.
</p>
<h2 id="formulation">Formulation</h2>

<p>On a periodic and cubic lattice, let 
  
    
      
        
          x
          
            d
          
        
      
    
    {\displaystyle x_{\text{d}}}
  
, 
  
    
      
        
          y
          
            d
          
        
      
    
    {\displaystyle y_{\text{d}}}
  
, and 
  
    
      
        
          z
          
            d
          
        
      
    
    {\displaystyle z_{\text{d}}}
  
 
be the differences between each of 
  
    
      
        x
      
    
    {\displaystyle x}
  
, 
  
    
      
        y
      
    
    {\displaystyle y}
  
, 
  
    
      
        z
      
    
    {\displaystyle z}
  
 and the smaller coordinate related, that is:
</p>

x
                  
                    d
                  
                
                =
                
                  
                    
                      x
                      −
                      
                        x
                        
                          0
                        
                      
                    
                    
                      
                        x
                        
                          1
                        
                      
                      −
                      
                        x
                        
                          0
                        
                      
                    
                  
                
              
            
            
              
                
                  y
                  
                    d
                  
                
                =
                
                  
                    
                      y
                      −
                      
                        y
                        
                          0
                        
                      
                    
                    
                      
                        y
                        
                          1
                        
                      
                      −
                      
                        y
                        
                          0
                        
                      
                    
                  
                
              
            
            
              
                
                  z
                  
                    d
                  
                
                =
                
                  
                    
                      z
                      −
                      
                        z
                        
                          0
                        
                      
                    
                    
                      
                        z
                        
                          1
                        
                      
                      −
                      
                        z
                        
                          0
                        
                      
                    
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}x_{\text{d}}={\frac {x-x_{0}}{x_{1}-x_{0}}}\\y_{\text{d}}={\frac {y-y_{0}}{y_{1}-y_{0}}}\\z_{\text{d}}={\frac {z-z_{0}}{z_{1}-z_{0}}}\end{aligned}}}

<p>where 
  
    
      
        
          x
          
            0
          
        
      
    
    {\displaystyle x_{0}}
  
 indicates the lattice point below 
  
    
      
        x
      
    
    {\displaystyle x}
  
, and 
  
    
      
        
          x
          
            1
          
        
      
    
    {\displaystyle x_{1}}
  
 indicates the lattice point above 
  
    
      
        x
      
    
    {\displaystyle x}
  
 and similarly for

y
          
            0
          
        
        ,
        
          y
          
            1
          
        
        ,
        
          z
          
            0
          
        
      
    
    {\displaystyle y_{0},y_{1},z_{0}}
  
 and 
  
    
      
        
          z
          
            1
          
        
      
    
    {\displaystyle z_{1}}
  
.
</p><p>First one interpolates along 
  
    
      
        x
      
    
    {\displaystyle x}
  
 (imagine one is "pushing" the face of the cube defined by 
  
    
      
        
          C
          
            0
            j
            k
          
        
      
    
    {\displaystyle C_{0jk}}
  
 to the opposing face, defined by 
  
    
      
        
          C
          
            1
            j
            k
          
        
      
    
    {\displaystyle C_{1jk}}
  
), giving:
</p>

c
                  
                    00
                  
                
              
              
                
                =
                
                  c
                  
                    000
                  
                
                (
                1
                −
                
                  x
                  
                    d
                  
                
                )
                +
                
                  c
                  
                    100
                  
                
                
                  x
                  
                    d
                  
                
              
            
            
              
                
                  c
                  
                    01
                  
                
              
              
                
                =
                
                  c
                  
                    001
                  
                
                (
                1
                −
                
                  x
                  
                    d
                  
                
                )
                +
                
                  c
                  
                    101
                  
                
                
                  x
                  
                    d
                  
                
              
            
            
              
                
                  c
                  
                    10
                  
                
              
              
                
                =
                
                  c
                  
                    010
                  
                
                (
                1
                −
                
                  x
                  
                    d
                  
                
                )
                +
                
                  c
                  
                    110
                  
                
                
                  x
                  
                    d
                  
                
              
            
            
              
                
                  c
                  
                    11
                  
                
              
              
                
                =
                
                  c
                  
                    011
                  
                
                (
                1
                −
                
                  x
                  
                    d
                  
                
                )
                +
                
                  c
                  
                    111
                  
                
                
                  x
                  
                    d
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}c_{00}&=c_{000}(1-x_{\text{d}})+c_{100}x_{\text{d}}\\c_{01}&=c_{001}(1-x_{\text{d}})+c_{101}x_{\text{d}}\\c_{10}&=c_{010}(1-x_{\text{d}})+c_{110}x_{\text{d}}\\c_{11}&=c_{011}(1-x_{\text{d}})+c_{111}x_{\text{d}}\end{aligned}}}

<p>Where 
  
    
      
        
          c
          
            000
          
        
      
    
    {\displaystyle c_{000}}
  
 means the function value of 
  
    
      
        (
        
          x
          
            0
          
        
        ,
        
          y
          
            0
          
        
        ,
        
          z
          
            0
          
        
        )
        .
      
    
    {\displaystyle (x_{0},y_{0},z_{0}).}
  
 Then one interpolates these values (along 
  
    
      
        y
      
    
    {\displaystyle y}
  
, "pushing" from 
  
    
      
        
          C
          
            i
            0
            k
          
        
      
    
    {\displaystyle C_{i0k}}
  
 to 
  
    
      
        
          C
          
            i
            1
            k
          
        
      
    
    {\displaystyle C_{i1k}}
  
), giving:
</p>

c
                  
                    0
                  
                
              
              
                
                =
                
                  c
                  
                    00
                  
                
                (
                1
                −
                
                  y
                  
                    d
                  
                
                )
                +
                
                  c
                  
                    10
                  
                
                
                  y
                  
                    d
                  
                
              
            
            
              
                
                  c
                  
                    1
                  
                
              
              
                
                =
                
                  c
                  
                    01
                  
                
                (
                1
                −
                
                  y
                  
                    d
                  
                
                )
                +
                
                  c
                  
                    11
                  
                
                
                  y
                  
                    d
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}c_{0}&=c_{00}(1-y_{\text{d}})+c_{10}y_{\text{d}}\\c_{1}&=c_{01}(1-y_{\text{d}})+c_{11}y_{\text{d}}\end{aligned}}}

<p>Finally one interpolates these values along 
  
    
      
        z
      
    
    {\displaystyle z}
  
 (walking through a line):
</p>

c
        =
        
          c
          
            0
          
        
        (
        1
        −
        
          z
          
            d
          
        
        )
        +
        
          c
          
            1
          
        
        
          z
          
            d
          
        
        .
      
    
    {\displaystyle c=c_{0}(1-z_{\text{d}})+c_{1}z_{\text{d}}.}

<p>This gives us a predicted value for the point.
</p><p>The result of trilinear interpolation is independent of the order of the interpolation steps along the three axes: any other order, for instance along 
  
    
      
        x
      
    
    {\displaystyle x}
  
, then along 
  
    
      
        y
      
    
    {\displaystyle y}
  
, and finally along 
  
    
      
        z
      
    
    {\displaystyle z}
  
, produces the same value.
</p>
<h3>Algorithm visualization</h3>

<p>The above operations can be visualized as follows: First we find the eight corners of a cube that surround our point of interest. These corners have the values 
  
    
      
        
          c
          
            000
          
        
      
    
    {\displaystyle c_{000}}
  
, 
  
    
      
        
          c
          
            100
          
        
      
    
    {\displaystyle c_{100}}
  
, 
  
    
      
        
          c
          
            010
          
        
      
    
    {\displaystyle c_{010}}
  
, 
  
    
      
        
          c
          
            110
          
        
      
    
    {\displaystyle c_{110}}
  
, 
  
    
      
        
          c
          
            001
          
        
      
    
    {\displaystyle c_{001}}
  
, 
  
    
      
        
          c
          
            101
          
        
      
    
    {\displaystyle c_{101}}
  
, 
  
    
      
        
          c
          
            011
          
        
      
    
    {\displaystyle c_{011}}
  
, 
  
    
      
        
          c
          
            111
          
        
      
    
    {\displaystyle c_{111}}
  
.
</p><p>Next, we perform linear interpolation between 
  
    
      
        
          c
          
            000
          
        
      
    
    {\displaystyle c_{000}}
  
 and 
  
    
      
        
          c
          
            100
          
        
      
    
    {\displaystyle c_{100}}
  
 to find 
  
    
      
        
          c
          
            00
          
        
      
    
    {\displaystyle c_{00}}
  
, 
  
    
      
        
          c
          
            001
          
        
      
    
    {\displaystyle c_{001}}
  
 and 
  
    
      
        
          c
          
            101
          
        
      
    
    {\displaystyle c_{101}}
  
 to find 
  
    
      
        
          c
          
            01
          
        
      
    
    {\displaystyle c_{01}}
  
, 
  
    
      
        
          c
          
            011
          
        
      
    
    {\displaystyle c_{011}}
  
 and 
  
    
      
        
          c
          
            111
          
        
      
    
    {\displaystyle c_{111}}
  
 to find 
  
    
      
        
          c
          
            11
          
        
      
    
    {\displaystyle c_{11}}
  
, 
  
    
      
        
          c
          
            010
          
        
      
    
    {\displaystyle c_{010}}
  
 and 
  
    
      
        
          c
          
            110
          
        
      
    
    {\displaystyle c_{110}}
  
 to find 
  
    
      
        
          c
          
            10
          
        
      
    
    {\displaystyle c_{10}}
  
.
</p><p>Now we do interpolation between 
  
    
      
        
          c
          
            00
          
        
      
    
    {\displaystyle c_{00}}
  
 and 
  
    
      
        
          c
          
            10
          
        
      
    
    {\displaystyle c_{10}}
  
 to find 
  
    
      
        
          c
          
            0
          
        
      
    
    {\displaystyle c_{0}}
  
, 
  
    
      
        
          c
          
            01
          
        
      
    
    {\displaystyle c_{01}}
  
 and 
  
    
      
        
          c
          
            11
          
        
      
    
    {\displaystyle c_{11}}
  
 to find 
  
    
      
        
          c
          
            1
          
        
      
    
    {\displaystyle c_{1}}
  
. Finally, we calculate the value 
  
    
      
        c
      
    
    {\displaystyle c}
  
 via linear interpolation of 
  
    
      
        
          c
          
            0
          
        
      
    
    {\displaystyle c_{0}}
  
 and 
  
    
      
        
          c
          
            1
          
        
      
    
    {\displaystyle c_{1}}

</p><p>In practice, a trilinear interpolation is identical to two <a href="/facts/Bilinear_interpolation/yoke6SzF">bilinear interpolation</a> combined with a linear interpolation:
</p>

c
        ≈
        l
        
          (
          
            b
            (
            
              c
              
                000
              
            
            ,
            
              c
              
                010
              
            
            ,
            
              c
              
                100
              
            
            ,
            
              c
              
                110
              
            
            )
            ,
            
            b
            (
            
              c
              
                001
              
            
            ,
            
              c
              
                011
              
            
            ,
            
              c
              
                101
              
            
            ,
            
              c
              
                111
              
            
            )
          
          )
        
      
    
    {\displaystyle c\approx l\left(b(c_{000},c_{010},c_{100},c_{110}),\,b(c_{001},c_{011},c_{101},c_{111})\right)}

<h3>Alternative algorithm</h3>
<p>An alternative way to write the solution to the interpolation problem is
</p>

f
        (
        x
        ,
        y
        ,
        z
        )
        ≈
        
          a
          
            0
          
        
        +
        
          a
          
            1
          
        
        x
        +
        
          a
          
            2
          
        
        y
        +
        
          a
          
            3
          
        
        z
        +
        
          a
          
            4
          
        
        x
        y
        +
        
          a
          
            5
          
        
        x
        z
        +
        
          a
          
            6
          
        
        y
        z
        +
        
          a
          
            7
          
        
        x
        y
        z
      
    
    {\displaystyle f(x,y,z)\approx a_{0}+a_{1}x+a_{2}y+a_{3}z+a_{4}xy+a_{5}xz+a_{6}yz+a_{7}xyz}

<p>where the coefficients are found by solving the linear system
</p>

[
                    
                      
                        
                          1
                        
                        
                          
                            x
                            
                              0
                            
                          
                        
                        
                          
                            y
                            
                              0
                            
                          
                        
                        
                          
                            z
                            
                              0
                            
                          
                        
                        
                          
                            x
                            
                              0
                            
                          
                          
                            y
                            
                              0
                            
                          
                        
                        
                          
                            x
                            
                              0
                            
                          
                          
                            z
                            
                              0
                            
                          
                        
                        
                          
                            y
                            
                              0
                            
                          
                          
                            z
                            
                              0
                            
                          
                        
                        
                          
                            x
                            
                              0
                            
                          
                          
                            y
                            
                              0
                            
                          
                          
                            z
                            
                              0
                            
                          
                        
                      
                      
                        
                          1
                        
                        
                          
                            x
                            
                              1
                            
                          
                        
                        
                          
                            y
                            
                              0
                            
                          
                        
                        
                          
                            z
                            
                              0
                            
                          
                        
                        
                          
                            x
                            
                              1
                            
                          
                          
                            y
                            
                              0
                            
                          
                        
                        
                          
                            x
                            
                              1
                            
                          
                          
                            z
                            
                              0
                            
                          
                        
                        
                          
                            y
                            
                              0
                            
                          
                          
                            z
                            
                              0
                            
                          
                        
                        
                          
                            x
                            
                              1
                            
                          
                          
                            y
                            
                              0
                            
                          
                          
                            z
                            
                              0
                            
                          
                        
                      
                      
                        
                          1
                        
                        
                          
                            x
                            
                              0
                            
                          
                        
                        
                          
                            y
                            
                              1
                            
                          
                        
                        
                          
                            z
                            
                              0
                            
                          
                        
                        
                          
                            x
                            
                              0
                            
                          
                          
                            y
                            
                              1
                            
                          
                        
                        
                          
                            x
                            
                              0
                            
                          
                          
                            z
                            
                              0
                            
                          
                        
                        
                          
                            y
                            
                              1
                            
                          
                          
                            z
                            
                              0
                            
                          
                        
                        
                          
                            x
                            
                              0
                            
                          
                          
                            y
                            
                              1
                            
                          
                          
                            z
                            
                              0
                            
                          
                        
                      
                      
                        
                          1
                        
                        
                          
                            x
                            
                              1
                            
                          
                        
                        
                          
                            y
                            
                              1
                            
                          
                        
                        
                          
                            z
                            
                              0
                            
                          
                        
                        
                          
                            x
                            
                              1
                            
                          
                          
                            y
                            
                              1
                            
                          
                        
                        
                          
                            x
                            
                              1
                            
                          
                          
                            z
                            
                              0
                            
                          
                        
                        
                          
                            y
                            
                              1
                            
                          
                          
                            z
                            
                              0
                            
                          
                        
                        
                          
                            x
                            
                              1
                            
                          
                          
                            y
                            
                              1
                            
                          
                          
                            z
                            
                              0
                            
                          
                        
                      
                      
                        
                          1
                        
                        
                          
                            x
                            
                              0
                            
                          
                        
                        
                          
                            y
                            
                              0
                            
                          
                        
                        
                          
                            z
                            
                              1
                            
                          
                        
                        
                          
                            x
                            
                              0
                            
                          
                          
                            y
                            
                              0
                            
                          
                        
                        
                          
                            x
                            
                              0
                            
                          
                          
                            z
                            
                              1
                            
                          
                        
                        
                          
                            y
                            
                              0
                            
                          
                          
                            z
                            
                              1
                            
                          
                        
                        
                          
                            x
                            
                              0
                            
                          
                          
                            y
                            
                              0
                            
                          
                          
                            z
                            
                              1
                            
                          
                        
                      
                      
                        
                          1
                        
                        
                          
                            x
                            
                              1
                            
                          
                        
                        
                          
                            y
                            
                              0
                            
                          
                        
                        
                          
                            z
                            
                              1
                            
                          
                        
                        
                          
                            x
                            
                              1
                            
                          
                          
                            y
                            
                              0
                            
                          
                        
                        
                          
                            x
                            
                              1
                            
                          
                          
                            z
                            
                              1
                            
                          
                        
                        
                          
                            y
                            
                              0
                            
                          
                          
                            z
                            
                              1
                            
                          
                        
                        
                          
                            x
                            
                              1
                            
                          
                          
                            y
                            
                              0
                            
                          
                          
                            z
                            
                              1
                            
                          
                        
                      
                      
                        
                          1
                        
                        
                          
                            x
                            
                              0
                            
                          
                        
                        
                          
                            y
                            
                              1
                            
                          
                        
                        
                          
                            z
                            
                              1
                            
                          
                        
                        
                          
                            x
                            
                              0
                            
                          
                          
                            y
                            
                              1
                            
                          
                        
                        
                          
                            x
                            
                              0
                            
                          
                          
                            z
                            
                              1
                            
                          
                        
                        
                          
                            y
                            
                              1
                            
                          
                          
                            z
                            
                              1
                            
                          
                        
                        
                          
                            x
                            
                              0
                            
                          
                          
                            y
                            
                              1
                            
                          
                          
                            z
                            
                              1
                            
                          
                        
                      
                      
                        
                          1
                        
                        
                          
                            x
                            
                              1
                            
                          
                        
                        
                          
                            y
                            
                              1
                            
                          
                        
                        
                          
                            z
                            
                              1
                            
                          
                        
                        
                          
                            x
                            
                              1
                            
                          
                          
                            y
                            
                              1
                            
                          
                        
                        
                          
                            x
                            
                              1
                            
                          
                          
                            z
                            
                              1
                            
                          
                        
                        
                          
                            y
                            
                              1
                            
                          
                          
                            z
                            
                              1
                            
                          
                        
                        
                          
                            x
                            
                              1
                            
                          
                          
                            y
                            
                              1
                            
                          
                          
                            z
                            
                              1
                            
                          
                        
                      
                    
                    ]
                  
                
                
                  
                    [
                    
                      
                        
                          
                            a
                            
                              0
                            
                          
                        
                      
                      
                        
                          
                            a
                            
                              1
                            
                          
                        
                      
                      
                        
                          
                            a
                            
                              2
                            
                          
                        
                      
                      
                        
                          
                            a
                            
                              3
                            
                          
                        
                      
                      
                        
                          
                            a
                            
                              4
                            
                          
                        
                      
                      
                        
                          
                            a
                            
                              5
                            
                          
                        
                      
                      
                        
                          
                            a
                            
                              6
                            
                          
                        
                      
                      
                        
                          
                            a
                            
                              7
                            
                          
                        
                      
                    
                    ]
                  
                
                =
                
                  
                    [
                    
                      
                        
                          
                            c
                            
                              000
                            
                          
                        
                      
                      
                        
                          
                            c
                            
                              100
                            
                          
                        
                      
                      
                        
                          
                            c
                            
                              010
                            
                          
                        
                      
                      
                        
                          
                            c
                            
                              110
                            
                          
                        
                      
                      
                        
                          
                            c
                            
                              001
                            
                          
                        
                      
                      
                        
                          
                            c
                            
                              101
                            
                          
                        
                      
                      
                        
                          
                            c
                            
                              011
                            
                          
                        
                      
                      
                        
                          
                            c
                            
                              111
                            
                          
                        
                      
                    
                    ]
                  
                
                ,
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\begin{bmatrix}1&x_{0}&y_{0}&z_{0}&x_{0}y_{0}&x_{0}z_{0}&y_{0}z_{0}&x_{0}y_{0}z_{0}\\1&x_{1}&y_{0}&z_{0}&x_{1}y_{0}&x_{1}z_{0}&y_{0}z_{0}&x_{1}y_{0}z_{0}\\1&x_{0}&y_{1}&z_{0}&x_{0}y_{1}&x_{0}z_{0}&y_{1}z_{0}&x_{0}y_{1}z_{0}\\1&x_{1}&y_{1}&z_{0}&x_{1}y_{1}&x_{1}z_{0}&y_{1}z_{0}&x_{1}y_{1}z_{0}\\1&x_{0}&y_{0}&z_{1}&x_{0}y_{0}&x_{0}z_{1}&y_{0}z_{1}&x_{0}y_{0}z_{1}\\1&x_{1}&y_{0}&z_{1}&x_{1}y_{0}&x_{1}z_{1}&y_{0}z_{1}&x_{1}y_{0}z_{1}\\1&x_{0}&y_{1}&z_{1}&x_{0}y_{1}&x_{0}z_{1}&y_{1}z_{1}&x_{0}y_{1}z_{1}\\1&x_{1}&y_{1}&z_{1}&x_{1}y_{1}&x_{1}z_{1}&y_{1}z_{1}&x_{1}y_{1}z_{1}\end{bmatrix}}{\begin{bmatrix}a_{0}\\a_{1}\\a_{2}\\a_{3}\\a_{4}\\a_{5}\\a_{6}\\a_{7}\end{bmatrix}}={\begin{bmatrix}c_{000}\\c_{100}\\c_{010}\\c_{110}\\c_{001}\\c_{101}\\c_{011}\\c_{111}\end{bmatrix}},\end{aligned}}}

<p>yielding the result
</p>

a
                  
                    0
                  
                
                =

−
                      
                        c
                        
                          000
                        
                      
                      
                        x
                        
                          1
                        
                      
                      
                        y
                        
                          1
                        
                      
                      
                        z
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          001
                        
                      
                      
                        x
                        
                          1
                        
                      
                      
                        y
                        
                          1
                        
                      
                      
                        z
                        
                          0
                        
                      
                      +
                      
                        c
                        
                          010
                        
                      
                      
                        x
                        
                          1
                        
                      
                      
                        y
                        
                          0
                        
                      
                      
                        z
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          011
                        
                      
                      
                        x
                        
                          1
                        
                      
                      
                        y
                        
                          0
                        
                      
                      
                        z
                        
                          0
                        
                      
                    
                    
                      (
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        x
                        
                          1
                        
                      
                      )
                      (
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        y
                        
                          1
                        
                      
                      )
                      (
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        z
                        
                          1
                        
                      
                      )
                    
                  
                
                +

c
                        
                          100
                        
                      
                      
                        x
                        
                          0
                        
                      
                      
                        y
                        
                          1
                        
                      
                      
                        z
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          101
                        
                      
                      
                        x
                        
                          0
                        
                      
                      
                        y
                        
                          1
                        
                      
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        c
                        
                          110
                        
                      
                      
                        x
                        
                          0
                        
                      
                      
                        y
                        
                          0
                        
                      
                      
                        z
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          111
                        
                      
                      
                        x
                        
                          0
                        
                      
                      
                        y
                        
                          0
                        
                      
                      
                        z
                        
                          0
                        
                      
                    
                    
                      (
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        x
                        
                          1
                        
                      
                      )
                      (
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        y
                        
                          1
                        
                      
                      )
                      (
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        z
                        
                          1
                        
                      
                      )
                    
                  
                
                ,
              
            
            
              
                
                  a
                  
                    1
                  
                
                =

c
                        
                          000
                        
                      
                      
                        y
                        
                          1
                        
                      
                      
                        z
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          001
                        
                      
                      
                        y
                        
                          1
                        
                      
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        c
                        
                          010
                        
                      
                      
                        y
                        
                          0
                        
                      
                      
                        z
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          011
                        
                      
                      
                        y
                        
                          0
                        
                      
                      
                        z
                        
                          0
                        
                      
                    
                    
                      (
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        x
                        
                          1
                        
                      
                      )
                      (
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        y
                        
                          1
                        
                      
                      )
                      (
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        z
                        
                          1
                        
                      
                      )
                    
                  
                
                +

−
                      
                        c
                        
                          100
                        
                      
                      
                        y
                        
                          1
                        
                      
                      
                        z
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          101
                        
                      
                      
                        y
                        
                          1
                        
                      
                      
                        z
                        
                          0
                        
                      
                      +
                      
                        c
                        
                          110
                        
                      
                      
                        y
                        
                          0
                        
                      
                      
                        z
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          111
                        
                      
                      
                        y
                        
                          0
                        
                      
                      
                        z
                        
                          0
                        
                      
                    
                    
                      (
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        x
                        
                          1
                        
                      
                      )
                      (
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        y
                        
                          1
                        
                      
                      )
                      (
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        z
                        
                          1
                        
                      
                      )
                    
                  
                
                ,
              
            
            
              
                
                  a
                  
                    2
                  
                
                =

c
                        
                          000
                        
                      
                      
                        x
                        
                          1
                        
                      
                      
                        z
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          001
                        
                      
                      
                        x
                        
                          1
                        
                      
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        c
                        
                          010
                        
                      
                      
                        x
                        
                          1
                        
                      
                      
                        z
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          011
                        
                      
                      
                        x
                        
                          1
                        
                      
                      
                        z
                        
                          0
                        
                      
                    
                    
                      (
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        x
                        
                          1
                        
                      
                      )
                      (
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        y
                        
                          1
                        
                      
                      )
                      (
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        z
                        
                          1
                        
                      
                      )
                    
                  
                
                +

−
                      
                        c
                        
                          100
                        
                      
                      
                        x
                        
                          0
                        
                      
                      
                        z
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          101
                        
                      
                      
                        x
                        
                          0
                        
                      
                      
                        z
                        
                          0
                        
                      
                      +
                      
                        c
                        
                          110
                        
                      
                      
                        x
                        
                          0
                        
                      
                      
                        z
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          111
                        
                      
                      
                        x
                        
                          0
                        
                      
                      
                        z
                        
                          0
                        
                      
                    
                    
                      (
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        x
                        
                          1
                        
                      
                      )
                      (
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        y
                        
                          1
                        
                      
                      )
                      (
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        z
                        
                          1
                        
                      
                      )
                    
                  
                
                ,
              
            
            
              
                
                  a
                  
                    3
                  
                
                =

c
                        
                          000
                        
                      
                      
                        x
                        
                          1
                        
                      
                      
                        y
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          001
                        
                      
                      
                        x
                        
                          1
                        
                      
                      
                        y
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          010
                        
                      
                      
                        x
                        
                          1
                        
                      
                      
                        y
                        
                          0
                        
                      
                      +
                      
                        c
                        
                          011
                        
                      
                      
                        x
                        
                          1
                        
                      
                      
                        y
                        
                          0
                        
                      
                    
                    
                      (
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        x
                        
                          1
                        
                      
                      )
                      (
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        y
                        
                          1
                        
                      
                      )
                      (
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        z
                        
                          1
                        
                      
                      )
                    
                  
                
                +

−
                      
                        c
                        
                          100
                        
                      
                      
                        x
                        
                          0
                        
                      
                      
                        y
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          101
                        
                      
                      
                        x
                        
                          0
                        
                      
                      
                        y
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          110
                        
                      
                      
                        x
                        
                          0
                        
                      
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        c
                        
                          111
                        
                      
                      
                        x
                        
                          0
                        
                      
                      
                        y
                        
                          0
                        
                      
                    
                    
                      (
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        x
                        
                          1
                        
                      
                      )
                      (
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        y
                        
                          1
                        
                      
                      )
                      (
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        z
                        
                          1
                        
                      
                      )
                    
                  
                
                ,
              
            
            
              
                
                  a
                  
                    4
                  
                
                =

−
                      
                        c
                        
                          000
                        
                      
                      
                        z
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          001
                        
                      
                      
                        z
                        
                          0
                        
                      
                      +
                      
                        c
                        
                          010
                        
                      
                      
                        z
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          011
                        
                      
                      
                        z
                        
                          0
                        
                      
                      +
                      
                        c
                        
                          100
                        
                      
                      
                        z
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          101
                        
                      
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        c
                        
                          110
                        
                      
                      
                        z
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          111
                        
                      
                      
                        z
                        
                          0
                        
                      
                    
                    
                      (
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        x
                        
                          1
                        
                      
                      )
                      (
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        y
                        
                          1
                        
                      
                      )
                      (
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        z
                        
                          1
                        
                      
                      )
                    
                  
                
                ,
              
            
            
              
                
                  a
                  
                    5
                  
                
                =
              
              
                
                  
                    
                      −
                      
                        c
                        
                          000
                        
                      
                      
                        y
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          001
                        
                      
                      
                        y
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          010
                        
                      
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        c
                        
                          011
                        
                      
                      
                        y
                        
                          0
                        
                      
                      +
                      
                        c
                        
                          100
                        
                      
                      
                        y
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          101
                        
                      
                      
                        y
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          110
                        
                      
                      
                        y
                        
                          0
                        
                      
                      +
                      
                        c
                        
                          111
                        
                      
                      
                        y
                        
                          0
                        
                      
                    
                    
                      (
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        x
                        
                          1
                        
                      
                      )
                      (
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        y
                        
                          1
                        
                      
                      )
                      (
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        z
                        
                          1
                        
                      
                      )
                    
                  
                
                ,
              
            
            
              
                
                  a
                  
                    6
                  
                
                =

−
                      
                        c
                        
                          000
                        
                      
                      
                        x
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          001
                        
                      
                      
                        x
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          010
                        
                      
                      
                        x
                        
                          1
                        
                      
                      −
                      
                        c
                        
                          011
                        
                      
                      
                        x
                        
                          1
                        
                      
                      +
                      
                        c
                        
                          100
                        
                      
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        c
                        
                          101
                        
                      
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        c
                        
                          110
                        
                      
                      
                        x
                        
                          0
                        
                      
                      +
                      
                        c
                        
                          111
                        
                      
                      
                        x
                        
                          0
                        
                      
                    
                    
                      (
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        x
                        
                          1
                        
                      
                      )
                      (
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        y
                        
                          1
                        
                      
                      )
                      (
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        z
                        
                          1
                        
                      
                      )
                    
                  
                
                ,
              
            
            
              
                
                  a
                  
                    7
                  
                
                =

c
                        
                          000
                        
                      
                      −
                      
                        c
                        
                          001
                        
                      
                      −
                      
                        c
                        
                          010
                        
                      
                      +
                      
                        c
                        
                          011
                        
                      
                      −
                      
                        c
                        
                          100
                        
                      
                      +
                      
                        c
                        
                          101
                        
                      
                      +
                      
                        c
                        
                          110
                        
                      
                      −
                      
                        c
                        
                          111
                        
                      
                    
                    
                      (
                      
                        x
                        
                          0
                        
                      
                      −
                      
                        x
                        
                          1
                        
                      
                      )
                      (
                      
                        y
                        
                          0
                        
                      
                      −
                      
                        y
                        
                          1
                        
                      
                      )
                      (
                      
                        z
                        
                          0
                        
                      
                      −
                      
                        z
                        
                          1
                        
                      
                      )
                    
                  
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}a_{0}={}&{\frac {-c_{000}x_{1}y_{1}z_{1}+c_{001}x_{1}y_{1}z_{0}+c_{010}x_{1}y_{0}z_{1}-c_{011}x_{1}y_{0}z_{0}}{(x_{0}-x_{1})(y_{0}-y_{1})(z_{0}-z_{1})}}+{}\\&{\frac {c_{100}x_{0}y_{1}z_{1}-c_{101}x_{0}y_{1}z_{0}-c_{110}x_{0}y_{0}z_{1}+c_{111}x_{0}y_{0}z_{0}}{(x_{0}-x_{1})(y_{0}-y_{1})(z_{0}-z_{1})}},\\[4pt]a_{1}={}&{\frac {c_{000}y_{1}z_{1}-c_{001}y_{1}z_{0}-c_{010}y_{0}z_{1}+c_{011}y_{0}z_{0}}{(x_{0}-x_{1})(y_{0}-y_{1})(z_{0}-z_{1})}}+{}\\&{\frac {-c_{100}y_{1}z_{1}+c_{101}y_{1}z_{0}+c_{110}y_{0}z_{1}-c_{111}y_{0}z_{0}}{(x_{0}-x_{1})(y_{0}-y_{1})(z_{0}-z_{1})}},\\[4pt]a_{2}={}&{\frac {c_{000}x_{1}z_{1}-c_{001}x_{1}z_{0}-c_{010}x_{1}z_{1}+c_{011}x_{1}z_{0}}{(x_{0}-x_{1})(y_{0}-y_{1})(z_{0}-z_{1})}}+{}\\&{\frac {-c_{100}x_{0}z_{1}+c_{101}x_{0}z_{0}+c_{110}x_{0}z_{1}-c_{111}x_{0}z_{0}}{(x_{0}-x_{1})(y_{0}-y_{1})(z_{0}-z_{1})}},\\[4pt]a_{3}={}&{\frac {c_{000}x_{1}y_{1}-c_{001}x_{1}y_{1}-c_{010}x_{1}y_{0}+c_{011}x_{1}y_{0}}{(x_{0}-x_{1})(y_{0}-y_{1})(z_{0}-z_{1})}}+{}\\&{\frac {-c_{100}x_{0}y_{1}+c_{101}x_{0}y_{1}+c_{110}x_{0}y_{0}-c_{111}x_{0}y_{0}}{(x_{0}-x_{1})(y_{0}-y_{1})(z_{0}-z_{1})}},\\[4pt]a_{4}={}&{\frac {-c_{000}z_{1}+c_{001}z_{0}+c_{010}z_{1}-c_{011}z_{0}+c_{100}z_{1}-c_{101}z_{0}-c_{110}z_{1}+c_{111}z_{0}}{(x_{0}-x_{1})(y_{0}-y_{1})(z_{0}-z_{1})}},\\[4pt]a_{5}=&{\frac {-c_{000}y_{1}+c_{001}y_{1}+c_{010}y_{0}-c_{011}y_{0}+c_{100}y_{1}-c_{101}y_{1}-c_{110}y_{0}+c_{111}y_{0}}{(x_{0}-x_{1})(y_{0}-y_{1})(z_{0}-z_{1})}},\\[4pt]a_{6}={}&{\frac {-c_{000}x_{1}+c_{001}x_{1}+c_{010}x_{1}-c_{011}x_{1}+c_{100}x_{0}-c_{101}x_{0}-c_{110}x_{0}+c_{111}x_{0}}{(x_{0}-x_{1})(y_{0}-y_{1})(z_{0}-z_{1})}},\\[4pt]a_{7}={}&{\frac {c_{000}-c_{001}-c_{010}+c_{011}-c_{100}+c_{101}+c_{110}-c_{111}}{(x_{0}-x_{1})(y_{0}-y_{1})(z_{0}-z_{1})}}.\end{aligned}}}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Linear_interpolation/galkvRGy">Linear interpolation</a></li>
<li><a href="/facts/Bilinear_interpolation/yoke6SzF">Bilinear interpolation</a></li>
<li><a href="/facts/Tricubic_interpolation/U8EwEvLl">Tricubic interpolation</a></li>
<li>Radial interpolation</li>
<li>Tetrahedral interpolation</li>
<li><a href="/facts/Slerp/A40A6390">Spherical linear interpolation</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://www.grc.nasa.gov/WWW/winddocs/utilities/b4wind_guide/trilinear.html">pseudo-code from NASA</a>, describes an iterative inverse trilinear interpolation (given the vertices and the value of C find Xd, Yd and Zd).</li>
<li>Paul Bourke, <a href="http://paulbourke.net/miscellaneous/interpolation/">Interpolation methods</a>, 1999. Contains a very clever and simple method to find trilinear interpolation that is based on binary logic and can be extended to any dimension (Tetralinear, Pentalinear, ...).</li>
<li>Kenwright, Free-Form Tetrahedron Deformation. International Symposium on Visual Computing. Springer International Publishing, 2015 <a href="https://link.springer.com/chapter/10.1007/978-3-319-27863-6_74">[1]</a>.</li></ul>

Trilinear interpolation open-in-new

Trilinear interpolation