Parallel curve

<h2 id="parallel-curve-of-a-parametrically-given-curve">Parallel curve of a parametrically given curve</h2>
If there is a regular parametric representation 
 
 
 
 
 
 
 x
 →
 
 
 
 =
 (
 x
 (
 t
 )
 ,
 y
 (
 t
 )
 )
 
 
 {\displaystyle {\vec {x}}=(x(t),y(t))}
 
 of the given curve available, the second definition of a parallel curve (s. above) leads to the following parametric representation of the parallel curve with distance 
 
 
 
 
 |
 
 d
 
 |
 
 
 
 {\displaystyle |d|}
 
:

x
 →
 
 
 
 
 d
 
 
 (
 t
 )
 =
 
 
 
 x
 →
 
 
 
 (
 t
 )
 +
 d
 
 
 
 n
 →
 
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {x}}_{d}(t)={\vec {x}}(t)+d{\vec {n}}(t)}
 
 with the unit normal 
 
 
 
 
 
 
 n
 →
 
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {n}}(t)}
 
.
In cartesian coordinates:

x
          
            d
          
        
        (
        t
        )
        =
        x
        (
        t
        )
        +
        
          
            
              d
              
              
                y
                ′
              
              (
              t
              )
            
            
              
                x
                ′
              
              (
              t
              
                )
                
                  2
                
              
              +
              
                y
                ′
              
              (
              t
              
                )
                
                  2
                
              
            
          
        
      
    
    {\displaystyle x_{d}(t)=x(t)+{\frac {d\;y'(t)}{\sqrt {x'(t)^{2}+y'(t)^{2}}}}}

y
          
            d
          
        
        (
        t
        )
        =
        y
        (
        t
        )
        −
        
          
            
              d
              
              
                x
                ′
              
              (
              t
              )
            
            
              
                x
                ′
              
              (
              t
              
                )
                
                  2
                
              
              +
              
                y
                ′
              
              (
              t
              
                )
                
                  2
                
              
            
          
        
         
        .
      
    
    {\displaystyle y_{d}(t)=y(t)-{\frac {d\;x'(t)}{\sqrt {x'(t)^{2}+y'(t)^{2}}}}\ .}

The distance parameter 
 
 
 
 d
 
 
 {\displaystyle d}
 
 may be negative. In this case, one gets a parallel curve on the opposite side of the curve (see diagram on the parallel curves of a circle). One can easily check that a parallel curve of a line is a parallel line in the common sense, and the parallel curve of a circle is a concentric circle.

<h3>Geometric properties:<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a></h3>
<ul><li>
 
 
 
 
 
 
 
 x
 →
 
 
 
 
 d
 
 ′
 
 (
 t
 )
 ∥
 
 
 
 
 x
 →
 
 
 
 ′
 
 (
 t
 )
 ,
 
 
 
 {\displaystyle {\vec {x}}'_{d}(t)\parallel {\vec {x}}'(t),\quad }
 
 that means: the tangent vectors for a fixed parameter are parallel.</li>
<li>
 
 
 
 
 k
 
 d
 
 
 (
 t
 )
 =
 
 
 
 k
 (
 t
 )
 
 
 1
 +
 d
 k
 (
 t
 )
 
 
 
 ,
 
 
 
 {\displaystyle k_{d}(t)={\frac {k(t)}{1+dk(t)}},\quad }
 
 with 
 
 
 
 k
 (
 t
 )
 
 
 {\displaystyle k(t)}
 
 the <a href="/facts/Curvature/RFCAEsj8">curvature</a> of the given curve and 
 
 
 
 
 k
 
 d
 
 
 (
 t
 )
 
 
 {\displaystyle k_{d}(t)}
 
 the curvature of the parallel curve for parameter 
 
 
 
 t
 
 
 {\displaystyle t}
 
.</li>
<li>
 
 
 
 
 R
 
 d
 
 
 (
 t
 )
 =
 R
 (
 t
 )
 +
 d
 ,
 
 
 
 {\displaystyle R_{d}(t)=R(t)+d,\quad }
 
 with 
 
 
 
 R
 (
 t
 )
 
 
 {\displaystyle R(t)}
 
 the <a href="/facts/Curvature/RFCAEsj8">radius of curvature</a> of the given curve and 
 
 
 
 
 R
 
 d
 
 
 (
 t
 )
 
 
 {\displaystyle R_{d}(t)}
 
 the radius of curvature of the parallel curve for parameter 
 
 
 
 t
 
 
 {\displaystyle t}
 
.</li>
<li>When they exist, the <a href="/facts/Osculating_circle/mDs0CS3x">osculating circles</a> to parallel curves at corresponding points are concentric. <a class="footnote-ref" id="fnref:18" href="#fn:18">18</a></li>
<li>As for <a href="/facts/Parallel_(geometry)/16r5yq7D">parallel lines</a>, a normal line to a curve is also normal to its parallels.</li>
<li>When parallel curves are constructed they will have <a href="/facts/Cusp_(singularity)/QH3fEGry">cusps</a> when the distance from the curve matches the radius of <a href="/facts/Curvature/RFCAEsj8">curvature</a>. These are the points where the curve touches the <a href="/facts/Evolute/TmNQdwFY">evolute</a>.</li>
<li>If the progenitor curve is a boundary of a planar set and its parallel curve is without self-intersections, then the latter is the boundary of the <a href="/facts/Minkowski_sum/dEQfoJ1A">Minkowski sum</a> of the planar set and the disk of the given radius.</li></ul>
If the given curve is polynomial (meaning that 
 
 
 
 x
 (
 t
 )
 
 
 {\displaystyle x(t)}
 
 and 
 
 
 
 y
 (
 t
 )
 
 
 {\displaystyle y(t)}
 
 are polynomials), then the parallel curves are usually not polynomial. In CAD area this is a drawback, because CAD systems use polynomials or rational curves. In order to get at least rational curves, the square root of the representation of the parallel curve has to be solvable. Such curves are called pythagorean hodograph curves and were investigated by R.T. Farouki.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>

<h2 id="parallel-curves-of-an-implicit-curve">Parallel curves of an implicit curve</h2>

Generally the analytic representation of a parallel curve of an <a href="/facts/Implicit_curve/pMFUYp7e">implicit curve</a> is not possible. Only for the simple cases of lines and circles the parallel curves can be described easily.
For example:

Line 
 
 
 
 
 f
 (
 x
 ,
 y
 )
 =
 x
 +
 y
 −
 1
 =
 0
 
 
 
 {\displaystyle \;f(x,y)=x+y-1=0\;}
 
 → distance function: 
 
 
 
 
 h
 (
 x
 ,
 y
 )
 =
 
 
 
 x
 +
 y
 −
 1
 
 
 2
 
 
 
 =
 d
 
 
 
 {\displaystyle \;h(x,y)={\frac {x+y-1}{\sqrt {2}}}=d\;}
 
 (Hesse normalform)
Circle 
 
 
 
 
 f
 (
 x
 ,
 y
 )
 =
 
 x
 
 2
 
 
 +
 
 y
 
 2
 
 
 −
 1
 =
 0
 
 
 
 {\displaystyle \;f(x,y)=x^{2}+y^{2}-1=0\;}
 
 → distance function: 
 
 
 
 
 h
 (
 x
 ,
 y
 )
 =
 
 
 
 x
 
 2
 
 
 +
 
 y
 
 2
 
 
 
 
 −
 1
 =
 d
 
 .
 
 
 {\displaystyle \;h(x,y)={\sqrt {x^{2}+y^{2}}}-1=d\;.}

In general, presuming certain conditions, one can prove the existence of an <a href="/facts/Oriented_distance_function/lqgd3TAv">oriented distance function</a> 
 
 
 
 h
 (
 x
 ,
 y
 )
 
 
 {\displaystyle h(x,y)}
 
. In practice one has to treat it numerically.<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a> Considering parallel curves the following is true:

<ul><li>The parallel curve for distance d is the <a href="/facts/Level_set/JUmNbdNh">level set</a> 
 
 
 
 h
 (
 x
 ,
 y
 )
 =
 d
 
 
 {\displaystyle h(x,y)=d}
 
 of the corresponding oriented distance function 
 
 
 
 h
 
 
 {\displaystyle h}
 
.</li></ul>
<h3>Properties of the distance function:<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a><a class="footnote-ref" id="fnref:22" href="#fn:22">22</a></h3>
<ul><li>
 
 
 
 
 |
 
 grad
 ⁡
 h
 (
 
 
 
 x
 →
 
 
 
 )
 
 |
 
 =
 1
 
 ,
 
 
 {\displaystyle |\operatorname {grad} h({\vec {x}})|=1\;,}
 
</li>
<li>
 
 
 
 h
 (
 
 
 
 x
 →
 
 
 
 +
 d
 grad
 ⁡
 h
 (
 
 
 
 x
 →
 
 
 
 )
 )
 =
 h
 (
 
 
 
 x
 →
 
 
 
 )
 +
 d
 
 ,
 
 
 {\displaystyle h({\vec {x}}+d\operatorname {grad} h({\vec {x}}))=h({\vec {x}})+d\;,}
 
</li>
<li>
 
 
 
 grad
 ⁡
 h
 (
 
 
 
 x
 →
 
 
 
 +
 d
 grad
 ⁡
 h
 (
 
 
 
 x
 →
 
 
 
 )
 )
 =
 grad
 ⁡
 h
 (
 
 
 
 x
 →
 
 
 
 )
 
 .
 
 
 {\displaystyle \operatorname {grad} h({\vec {x}}+d\operatorname {grad} h({\vec {x}}))=\operatorname {grad} h({\vec {x}})\;.}
 
</li></ul>
Example:
The diagram shows parallel curves of the implicit curve with equation 
 
 
 
 
 f
 (
 x
 ,
 y
 )
 =
 
 x
 
 4
 
 
 +
 
 y
 
 4
 
 
 −
 1
 =
 0
 
 .
 
 
 {\displaystyle \;f(x,y)=x^{4}+y^{4}-1=0\;.}
 
 
Remark:
The curves 
 
 
 
 
 f
 (
 x
 ,
 y
 )
 =
 
 x
 
 4
 
 
 +
 
 y
 
 4
 
 
 −
 1
 =
 d
 
 
 
 {\displaystyle \;f(x,y)=x^{4}+y^{4}-1=d\;}
 
 are not parallel curves, because 
 
 
 
 
 
 |
 
 grad
 ⁡
 f
 (
 x
 ,
 y
 )
 
 |
 
 =
 1
 
 
 
 {\displaystyle \;|\operatorname {grad} f(x,y)|=1\;}
 
 is not true in the area of interest.

<h2 id="further-examples">Further examples</h2>

<ul><li>The <a href="/facts/Involute/Ql8vqIHr">involutes</a> of a given curve are a set of parallel curves. For example: the involutes of a circle are parallel spirals (see diagram).</li></ul>
And:<a class="footnote-ref" id="fnref:23" href="#fn:23">23</a>

<ul><li>A <a href="/facts/Parabola/Cweo3wgw">parabola</a> has as (two-sided) offsets <a href="/facts/Rational_curve/LnWsPmDP">rational curves</a> of degree 6.</li>
<li>A <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a> or an <a href="/facts/Ellipse/1wUDnGxY">ellipse</a> has as (two-sided) offsets an <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curve</a> of degree 8.</li>
<li>A <a href="/facts/B%25C3%25A9zier_curve/zJx3Uj60">Bézier curve</a> of degree n has as (two-sided) offsets <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curves</a> of degree 4n − 2. In particular, a cubic Bézier curve has as (two-sided) offsets algebraic curves of degree 10.</li></ul>
<h2 id="parallel-curve-to-a-curve-with-a-corner">Parallel curve to a curve with a corner</h2>

When determining the cutting path of part with a sharp corner for <a href="/facts/Machining/Yrjfj6IV">machining</a>, you must define the parallel (offset) curve to a given curve that has a discontinuous normal at the corner. Even though the given curve is not smooth at the sharp corner, its parallel curve may be smooth with a continuous normal, or it may have <a href="/facts/Cusp_(singularity)/QH3fEGry">cusps</a> when the distance from the curve matches the radius of <a href="/facts/Curvature/RFCAEsj8">curvature</a> at the sharp corner.

<h3>Normal fans</h3>
As described above, the parametric representation of a parallel curve, 
 
 
 
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {x}}_{d}(t)}
 
, to a given curver, 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {x}}(t)}
 
, with distance 
 
 
 
 
 |
 
 d
 
 |
 
 
 
 {\displaystyle |d|}
 
 is:

x
 →
 
 
 
 
 d
 
 
 (
 t
 )
 =
 
 
 
 x
 →
 
 
 
 (
 t
 )
 +
 d
 
 
 
 n
 →
 
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {x}}_{d}(t)={\vec {x}}(t)+d{\vec {n}}(t)}
 
 with the unit normal 
 
 
 
 
 
 
 n
 →
 
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {n}}(t)}
 
.
At a sharp corner (
 
 
 
 t
 =
 
 t
 
 c
 
 
 
 
 {\displaystyle t=t_{c}}
 
), the normal to 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 
 t
 
 c
 
 
 )
 
 
 {\displaystyle {\vec {x}}(t_{c})}
 
 given by 
 
 
 
 
 
 
 n
 →
 
 
 
 (
 
 t
 
 c
 
 
 )
 
 
 {\displaystyle {\vec {n}}(t_{c})}
 
 is discontinuous, meaning the <a href="/facts/One-sided_limit/zLfr8gc5">one-sided limit</a> of the normal from the left 
 
 
 
 
 
 
 n
 →
 
 
 
 (
 
 t
 
 c
 
 
 −
 
 
 )
 
 
 {\displaystyle {\vec {n}}(t_{c}^{-})}
 
 is unequal to the limit from the right 
 
 
 
 
 
 
 n
 →
 
 
 
 (
 
 t
 
 c
 
 
 +
 
 
 )
 
 
 {\displaystyle {\vec {n}}(t_{c}^{+})}
 
. Mathematically,

n
              →
            
          
        
        (
        
          t
          
            c
          
          
            −
          
        
        )
        =
        
          lim
          
            t
            →
            
              t
              
                c
              
              
                −
              
            
          
        
        
          
            
              n
              →
            
          
        
        (
        t
        )
        ≠
        
          
            
              n
              →
            
          
        
        (
        
          t
          
            c
          
          
            +
          
        
        )
        =
        
          lim
          
            t
            →
            
              t
              
                c
              
              
                +
              
            
          
        
        
          
            
              n
              →
            
          
        
        (
        t
        )
      
    
    {\displaystyle {\vec {n}}(t_{c}^{-})=\lim _{t\to t_{c}^{-}}{\vec {n}}(t)\neq {\vec {n}}(t_{c}^{+})=\lim _{t\to t_{c}^{+}}{\vec {n}}(t)}
  
.

However, we can define a normal fan<a class="footnote-ref" id="fnref:24" href="#fn:24">24</a> 
 
 
 
 
 
 
 
 n
 →
 
 
 
 
 f
 
 
 (
 α
 )
 
 
 {\displaystyle {\vec {n}}_{f}(\alpha )}
 
 that provides an <a href="/facts/Interpolation/8PyetX4f">interpolant</a> between 
 
 
 
 
 
 
 n
 →
 
 
 
 (
 
 t
 
 c
 
 
 −
 
 
 )
 
 
 {\displaystyle {\vec {n}}(t_{c}^{-})}
 
 and 
 
 
 
 
 
 
 n
 →
 
 
 
 (
 
 t
 
 c
 
 
 +
 
 
 )
 
 
 {\displaystyle {\vec {n}}(t_{c}^{+})}
 
, and use 
 
 
 
 
 
 
 
 n
 →
 
 
 
 
 f
 
 
 (
 α
 )
 
 
 {\displaystyle {\vec {n}}_{f}(\alpha )}
 
 in place of 
 
 
 
 
 
 
 n
 →
 
 
 
 (
 
 t
 
 c
 
 
 )
 
 
 {\displaystyle {\vec {n}}(t_{c})}
 
 at the sharp corner:

n
 →
 
 
 
 
 f
 
 
 (
 α
 )
 =
 
 
 
 (
 1
 −
 α
 )
 
 
 
 n
 →
 
 
 
 (
 
 t
 
 c
 
 
 −
 
 
 )
 +
 α
 
 
 
 n
 →
 
 
 
 (
 
 t
 
 c
 
 
 +
 
 
 )
 
 
 ‖
 (
 1
 −
 α
 )
 
 
 
 n
 →
 
 
 
 (
 
 t
 
 c
 
 
 −
 
 
 )
 +
 α
 
 
 
 n
 →
 
 
 
 (
 
 t
 
 c
 
 
 +
 
 
 )
 ‖
 
 
 
 ,
 
 
 
 {\displaystyle {\vec {n}}_{f}(\alpha )={\frac {(1-\alpha ){\vec {n}}(t_{c}^{-})+\alpha {\vec {n}}(t_{c}^{+})}{\lVert (1-\alpha ){\vec {n}}(t_{c}^{-})+\alpha {\vec {n}}(t_{c}^{+})\rVert }},\quad }
 
where 
 
 
 
 0
 <
 α
 <
 1
 
 
 {\displaystyle 0<\alpha <1}
 
.
The resulting definition of the parallel curve 
 
 
 
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {x}}_{d}(t)}
 
 provides the desired behavior:

x
 →
 
 
 
 
 d
 
 
 (
 t
 )
 =
 
 
 {
 
 
 
 
 
 
 x
 →
 
 
 
 (
 t
 )
 +
 d
 
 
 
 n
 →
 
 
 
 (
 t
 )
 ,
 
 
 
 if 
 
 t
 <
 
 t
 
 c
 
 
 
  or 
 
 t
 >
 
 t
 
 c
 
 
 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 
 t
 
 c
 
 
 )
 +
 d
 
 
 
 
 n
 →
 
 
 
 
 f
 
 
 (
 α
 )
 ,
 
 
 
 if 
 
 t
 =
 
 t
 
 c
 
 
 
  where 
 
 0
 <
 α
 <
 1
 
 
 
 
 
 
 
 
 {\displaystyle {\vec {x}}_{d}(t)={\begin{cases}{\vec {x}}(t)+d{\vec {n}}(t),&{\text{if }}t<t_{c}{\text{ or }}t>t_{c}\\{\vec {x}}(t_{c})+d{\vec {n}}_{f}(\alpha ),&{\text{if }}t=t_{c}{\text{ where }}0<\alpha <1\end{cases}}}

<h2 id="algorithms">Algorithms</h2>

In general, the parallel curve of a <a href="/facts/B%25C3%25A9zier_curve/zJx3Uj60">Bézier curve</a> is not another Bézier curve, a result proved by Tiller and Hanson in 1984.<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> Thus, in practice, approximation techniques are used. Any desired level of accuracy is possible by repeatedly subdividing the curve, though better techniques require fewer subdivisions to attain the same level of accuracy. A 1997 survey by Elber, Lee and Kim<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> is widely cited, though better techniques have been proposed more recently. A modern technique based on <a href="/facts/Curve_fitting/zrLKw6Xj">curve fitting</a>, with references and comparisons to other algorithms, as well as open source JavaScript source code, was published in a blog post<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a> in September 2022.
Another efficient algorithm for offsetting is the level approach described by
<a href="/facts/Ron_Kimmel/qf0QSHuj">Kimmel</a> and Bruckstein (1993).<a class="footnote-ref" id="fnref:28" href="#fn:28">28</a>

<h2 id="parallel-offset-surfaces">Parallel (offset) surfaces</h2>

Offset surfaces are important in <a href="/facts/Numerically_controlled/B2c7UbaX">numerically controlled</a> <a href="/facts/Machining/Yrjfj6IV">machining</a>, where they describe the shape of the cut made by a ball nose end mill of a three-axis mill.<a class="footnote-ref" id="fnref:29" href="#fn:29">29</a> If there is a regular parametric representation 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 u
 ,
 v
 )
 =
 (
 x
 (
 u
 ,
 v
 )
 ,
 y
 (
 u
 ,
 v
 )
 ,
 z
 (
 u
 ,
 v
 )
 )
 
 
 {\displaystyle {\vec {x}}(u,v)=(x(u,v),y(u,v),z(u,v))}
 
 of the given surface available, the second definition of a parallel curve (see above) generalizes to the following parametric representation of the parallel surface with distance 
 
 
 
 
 |
 
 d
 
 |
 
 
 
 {\displaystyle |d|}
 
:

x
 →
 
 
 
 
 d
 
 
 (
 u
 ,
 v
 )
 =
 
 
 
 x
 →
 
 
 
 (
 u
 ,
 v
 )
 +
 d
 
 
 
 n
 →
 
 
 
 (
 u
 ,
 v
 )
 
 
 {\displaystyle {\vec {x}}_{d}(u,v)={\vec {x}}(u,v)+d{\vec {n}}(u,v)}
 
 with the unit normal 
 
 
 
 
 
 
 
 n
 →
 
 
 
 
 d
 
 
 (
 u
 ,
 v
 )
 =
 
 
 
 
 
 
 ∂
 
 
 
 x
 →
 
 
 
 
 
 ∂
 u
 
 
 
 ×
 
 
 
 ∂
 
 
 
 x
 →
 
 
 
 
 
 ∂
 v
 
 
 
 
 
 
 |
 
 
 
 
 
 ∂
 
 
 
 x
 →
 
 
 
 
 
 ∂
 u
 
 
 
 ×
 
 
 
 ∂
 
 
 
 x
 →
 
 
 
 
 
 ∂
 v
 
 
 
 
 
 |
 
 
 
 
 
 
 {\displaystyle {\vec {n}}_{d}(u,v)={{{\partial {\vec {x}} \over \partial u}\times {\partial {\vec {x}} \over \partial v}} \over {|{{\partial {\vec {x}} \over \partial u}\times {\partial {\vec {x}} \over \partial v}}|}}}
 
.
Distance parameter 
 
 
 
 d
 
 
 {\displaystyle d}
 
 may be negative, too. In this case one gets a parallel surface on the opposite side of the surface (see similar diagram on the parallel curves of a circle). One easily checks: a parallel surface of a plane is a parallel plane in the common sense and the parallel surface of a sphere is a concentric sphere.

<h3>Geometric properties:<a class="footnote-ref" id="fnref:30" href="#fn:30">30</a></h3>
<ul><li>
 
 
 
 
 
 
 ∂
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 
 
 ∂
 u
 
 
 
 ∥
 
 
 
 ∂
 
 
 
 x
 →
 
 
 
 
 
 ∂
 u
 
 
 
 ,
 
 
 
 
 ∂
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 
 
 ∂
 v
 
 
 
 ∥
 
 
 
 ∂
 
 
 
 x
 →
 
 
 
 
 
 ∂
 v
 
 
 
 ,
 
 
 
 {\displaystyle {\partial {\vec {x}}_{d} \over \partial u}\parallel {\partial {\vec {x}} \over \partial u},\quad {\partial {\vec {x}}_{d} \over \partial v}\parallel {\partial {\vec {x}} \over \partial v},\quad }
 
 that means: the tangent vectors for fixed parameters are parallel.</li>
<li>
 
 
 
 
 
 
 
 n
 →
 
 
 
 
 d
 
 
 (
 u
 ,
 v
 )
 =
 ±
 
 
 
 n
 →
 
 
 
 (
 u
 ,
 v
 )
 ,
 
 
 
 {\displaystyle {\vec {n}}_{d}(u,v)=\pm {\vec {n}}(u,v),\quad }
 
 that means: the normal vectors for fixed parameters match direction.</li>
<li>
 
 
 
 
 S
 
 d
 
 
 =
 (
 1
 +
 d
 S
 
 )
 
 −
 1
 
 
 S
 ,
 
 
 
 {\displaystyle S_{d}=(1+dS)^{-1}S,\quad }
 
 where 
 
 
 
 
 S
 
 d
 
 
 
 
 {\displaystyle S_{d}}
 
 and 
 
 
 
 S
 
 
 {\displaystyle S}
 
 are the <a href="/facts/Shape_operator/cslVnmd6">shape operators</a> for 
 
 
 
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 
 
 {\displaystyle {\vec {x}}_{d}}
 
 and 
 
 
 
 
 
 
 x
 →
 
 
 
 
 
 {\displaystyle {\vec {x}}}
 
, respectively.</li></ul>
The principal curvatures are the <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a> of the <a href="/facts/Shape_operator/cslVnmd6">shape operator</a>, the principal curvature directions are its <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvectors</a>, the <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> is its <a href="/facts/Determinant/eGYqJ2TF">determinant</a>, and the mean curvature is half its <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a>.
<ul><li>
 
 
 
 
 S
 
 d
 
 
 −
 1
 
 
 =
 
 S
 
 −
 1
 
 
 +
 d
 I
 ,
 
 
 
 {\displaystyle S_{d}^{-1}=S^{-1}+dI,\quad }
 
 where 
 
 
 
 
 S
 
 d
 
 
 −
 1
 
 
 
 
 {\displaystyle S_{d}^{-1}}
 
 and 
 
 
 
 
 S
 
 −
 1
 
 
 
 
 {\displaystyle S^{-1}}
 
 are the inverses of the <a href="/facts/Shape_operator/cslVnmd6">shape operators</a> for 
 
 
 
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 
 
 {\displaystyle {\vec {x}}_{d}}
 
 and 
 
 
 
 
 
 
 x
 →
 
 
 
 
 
 {\displaystyle {\vec {x}}}
 
, respectively.</li></ul>
The principal radii of curvature are the <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a> of the inverse of the <a href="/facts/Shape_operator/cslVnmd6">shape operator</a>, the principal curvature directions are its <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvectors</a>, the reciprocal of the <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> is its <a href="/facts/Determinant/eGYqJ2TF">determinant</a>, and the mean radius of curvature is half its <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a>.
Note the similarity to the geometric properties of parallel curves.

<h2 id="generalizations">Generalizations</h2>
The problem generalizes fairly obviously to higher dimensions e.g. to offset surfaces, and slightly less trivially to <a href="/facts/Pipe_surface/QGo3IH6u">pipe surfaces</a>.<a class="footnote-ref" id="fnref:31" href="#fn:31">31</a> Note that the terminology for the higher-dimensional versions varies even more widely than in the planar case, e.g. other authors speak of parallel fibers, ribbons, and tubes.<a class="footnote-ref" id="fnref:32" href="#fn:32">32</a> For curves embedded in 3D surfaces the offset may be taken along a <a href="/facts/Geodesic/fbWR6l80">geodesic</a>.<a class="footnote-ref" id="fnref:33" href="#fn:33">33</a>
Another way to generalize it is (even in 2D) to consider a variable distance, e.g. parametrized by another curve.<a class="footnote-ref" id="fnref:34" href="#fn:34">34</a> One can for example stroke (envelope) with an ellipse instead of circle<a class="footnote-ref" id="fnref:35" href="#fn:35">35</a> as it is possible for example in <a href="/facts/METAFONT/j7vCXw8s">METAFONT</a>.<a class="footnote-ref" id="fnref:36" href="#fn:36">36</a>

More recently <a href="/facts/Adobe_Illustrator/T0L5wGoT">Adobe Illustrator</a> has added somewhat similar facility in version <a href="/facts/CS5/Aew6RW3a">CS5</a>, although the control points for the variable width are visually specified.<a class="footnote-ref" id="fnref:37" href="#fn:37">37</a> In contexts where it's important to distinguish between constant and variable distance offsetting the acronyms CDO and VDO are sometimes used.<a class="footnote-ref" id="fnref:38" href="#fn:38">38</a>

<h3>General offset curves</h3>
Assume you have a regular parametric representation of a curve, 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 t
 )
 =
 (
 x
 (
 t
 )
 ,
 y
 (
 t
 )
 )
 
 
 {\displaystyle {\vec {x}}(t)=(x(t),y(t))}
 
, and you have a second curve that can be parameterized by its unit normal, 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}})}
 
, where the normal of 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 =
 
 
 
 n
 →
 
 
 
 
 
 {\displaystyle {\vec {d}}({\vec {n}})={\vec {n}}}
 
 (this parameterization by normal exists for curves whose curvature is strictly positive or negative, and thus convex, smooth, and not straight). The parametric representation of the general offset curve of 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {x}}(t)}
 
 offset by 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}})}
 
 is:

x
 →
 
 
 
 
 d
 
 
 (
 t
 )
 =
 
 
 
 x
 →
 
 
 
 (
 t
 )
 +
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 (
 t
 )
 )
 ,
 
 
 
 {\displaystyle {\vec {x}}_{d}(t)={\vec {x}}(t)+{\vec {d}}({\vec {n}}(t)),\quad }
 
 where 
 
 
 
 
 
 
 n
 →
 
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {n}}(t)}
 
 is the unit normal of 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {x}}(t)}
 
.
Note that the trival offset, 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 =
 d
 
 
 
 n
 →
 
 
 
 
 
 {\displaystyle {\vec {d}}({\vec {n}})=d{\vec {n}}}
 
, gives you ordinary parallel (aka, offset) curves.

<h4>Geometric properties:<a class="footnote-ref" id="fnref:39" href="#fn:39">39</a></h4>
<ul><li>
 
 
 
 
 
 
 
 x
 →
 
 
 
 
 d
 
 ′
 
 (
 t
 )
 ∥
 
 
 
 
 x
 →
 
 
 
 ′
 
 (
 t
 )
 ,
 
 
 
 {\displaystyle {\vec {x}}'_{d}(t)\parallel {\vec {x}}'(t),\quad }
 
 that means: the tangent vectors for a fixed parameter are parallel.</li>
<li>As for <a href="/facts/Parallel_(geometry)/16r5yq7D">parallel lines</a>, a normal to a curve is also normal to its general offsets.</li>
<li>
 
 
 
 
 k
 
 d
 
 
 (
 t
 )
 =
 
 
 
 
 k
 (
 t
 )
 
 
 1
 +
 
 
 
 
 k
 (
 t
 )
 
 
 
 k
 
 n
 
 
 (
 t
 )
 
 
 
 
 
 
 
 
 ,
 
 
 
 {\displaystyle k_{d}(t)={\dfrac {k(t)}{1+{\dfrac {k(t)}{k_{n}(t)}}}},\quad }
 
 with 
 
 
 
 
 k
 
 d
 
 
 (
 t
 )
 
 
 {\displaystyle k_{d}(t)}
 
 the <a href="/facts/Curvature/RFCAEsj8">curvature</a> of the general offset curve, 
 
 
 
 k
 (
 t
 )
 
 
 {\displaystyle k(t)}
 
 the curvature of 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {x}}(t)}
 
, and 
 
 
 
 
 k
 
 n
 
 
 (
 t
 )
 
 
 {\displaystyle k_{n}(t)}
 
 the curvature of 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 (
 t
 )
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}}(t))}
 
 for parameter 
 
 
 
 t
 
 
 {\displaystyle t}
 
.</li>
<li>
 
 
 
 
 R
 
 d
 
 
 (
 t
 )
 =
 R
 (
 t
 )
 +
 
 R
 
 n
 
 
 (
 t
 )
 ,
 
 
 
 {\displaystyle R_{d}(t)=R(t)+R_{n}(t),\quad }
 
 with 
 
 
 
 
 R
 
 d
 
 
 (
 t
 )
 
 
 {\displaystyle R_{d}(t)}
 
 the <a href="/facts/Curvature/RFCAEsj8">radius of curvature</a> of the general offset curve, 
 
 
 
 R
 (
 t
 )
 
 
 {\displaystyle R(t)}
 
 the radius of curvature of 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {x}}(t)}
 
, and 
 
 
 
 
 R
 
 n
 
 
 (
 t
 )
 
 
 {\displaystyle R_{n}(t)}
 
 the radius of curvature of 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 (
 t
 )
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}}(t))}
 
 for parameter 
 
 
 
 t
 
 
 {\displaystyle t}
 
.</li>
<li>When general offset curves are constructed they will have <a href="/facts/Cusp_(singularity)/QH3fEGry">cusps</a> when the <a href="/facts/Curvature/RFCAEsj8">curvature</a> of the curve matches curvature of the offset. These are the points where the curve touches the <a href="/facts/Evolute/TmNQdwFY">evolute</a>.</li></ul>
<h3>General offset surfaces</h3>
General offset surfaces describe the shape of cuts made by a variety of cutting bits used by three-axis end mills in <a href="/facts/Numerically_controlled/B2c7UbaX">numerically controlled</a> <a href="/facts/Machining/Yrjfj6IV">machining</a>.<a class="footnote-ref" id="fnref:40" href="#fn:40">40</a> Assume you have a regular parametric representation of a surface, 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 u
 ,
 v
 )
 =
 (
 x
 (
 u
 ,
 v
 )
 ,
 y
 (
 u
 ,
 v
 )
 ,
 z
 (
 u
 ,
 v
 )
 )
 
 
 {\displaystyle {\vec {x}}(u,v)=(x(u,v),y(u,v),z(u,v))}
 
, and you have a second surface that can be parameterized by its unit normal, 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}})}
 
, where the normal of 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 =
 
 
 
 n
 →
 
 
 
 
 
 {\displaystyle {\vec {d}}({\vec {n}})={\vec {n}}}
 
 (this parameterization by normal exists for surfaces whose <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> is strictly positive, and thus convex, smooth, and not flat). The parametric representation of the general offset surface of 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 t
 )
 
 
 {\displaystyle {\vec {x}}(t)}
 
 offset by 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}})}
 
 is:

x
 →
 
 
 
 
 d
 
 
 (
 u
 ,
 v
 )
 =
 
 
 
 x
 →
 
 
 
 (
 u
 ,
 v
 )
 +
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 (
 u
 ,
 v
 )
 )
 ,
 
 
 
 {\displaystyle {\vec {x}}_{d}(u,v)={\vec {x}}(u,v)+{\vec {d}}({\vec {n}}(u,v)),\quad }
 
 where 
 
 
 
 
 
 
 n
 →
 
 
 
 (
 u
 ,
 v
 )
 
 
 {\displaystyle {\vec {n}}(u,v)}
 
 is the unit normal of 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 u
 ,
 v
 )
 
 
 {\displaystyle {\vec {x}}(u,v)}
 
.
Note that the trival offset, 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 =
 d
 
 
 
 n
 →
 
 
 
 
 
 {\displaystyle {\vec {d}}({\vec {n}})=d{\vec {n}}}
 
, gives you ordinary parallel (aka, offset) surfaces.

<h4>Geometric properties:<a class="footnote-ref" id="fnref:41" href="#fn:41">41</a></h4>
<ul><li>As for <a href="/facts/Parallel_(geometry)/16r5yq7D">parallel lines</a>, the tangent plane of a surface is parallel to the tangent plane of its general offsets.</li>
<li>As for <a href="/facts/Parallel_(geometry)/16r5yq7D">parallel lines</a>, a normal to a surface is also normal to its general offsets.</li>
<li>
 
 
 
 
 S
 
 d
 
 
 =
 (
 1
 +
 S
 
 S
 
 n
 
 
 −
 1
 
 
 
 )
 
 −
 1
 
 
 S
 ,
 
 
 
 {\displaystyle S_{d}=(1+SS_{n}^{-1})^{-1}S,\quad }
 
 where 
 
 
 
 
 S
 
 d
 
 
 ,
 S
 ,
 
 
 {\displaystyle S_{d},S,}
 
 and 
 
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S_{n}}
 
 are the <a href="/facts/Shape_operator/cslVnmd6">shape operators</a> for 
 
 
 
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 ,
 
 
 
 x
 →
 
 
 
 ,
 
 
 {\displaystyle {\vec {x}}_{d},{\vec {x}},}
 
 and 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}})}
 
, respectively.</li></ul>
The principal curvatures are the <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a> of the <a href="/facts/Shape_operator/cslVnmd6">shape operator</a>, the principal curvature directions are its <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvectors</a>, the <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> is its <a href="/facts/Determinant/eGYqJ2TF">determinant</a>, and the mean curvature is half its <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a>.
<ul><li>
 
 
 
 
 S
 
 d
 
 
 −
 1
 
 
 =
 
 S
 
 −
 1
 
 
 +
 
 S
 
 n
 
 
 −
 1
 
 
 ,
 
 
 
 {\displaystyle S_{d}^{-1}=S^{-1}+S_{n}^{-1},\quad }
 
 where 
 
 
 
 
 S
 
 d
 
 
 −
 1
 
 
 ,
 
 S
 
 −
 1
 
 
 
 
 {\displaystyle S_{d}^{-1},S^{-1}}
 
 and 
 
 
 
 
 S
 
 n
 
 
 −
 1
 
 
 
 
 {\displaystyle S_{n}^{-1}}
 
 are the inverses of the <a href="/facts/Shape_operator/cslVnmd6">shape operators</a> for 
 
 
 
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 ,
 
 
 
 x
 →
 
 
 
 ,
 
 
 {\displaystyle {\vec {x}}_{d},{\vec {x}},}
 
 and 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}})}
 
, respectively.</li></ul>
The principal radii of curvature are the <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a> of the inverse of the <a href="/facts/Shape_operator/cslVnmd6">shape operator</a>, the principal curvature directions are its <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvectors</a>, the reciprocal of the <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> is its <a href="/facts/Determinant/eGYqJ2TF">determinant</a>, and the mean radius of curvature is half its <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a>.
Note the similarity to the geometric properties of general offset curves.

<h3>Derivation of geometric properties for general offsets</h3>
The geometric properties listed above for general offset curves and surfaces can be derived for offsets of arbitrary dimension. Assume you have a regular parametric representation of an n-dimensional surface, 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {x}}({\vec {u}})}
 
, where the dimension of 
 
 
 
 
 
 
 u
 →
 
 
 
 
 
 {\displaystyle {\vec {u}}}
 
 is n-1. Also assume you have a second n-dimensional surface that can be parameterized by its unit normal, 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}})}
 
, where the normal of 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 =
 
 
 
 n
 →
 
 
 
 
 
 {\displaystyle {\vec {d}}({\vec {n}})={\vec {n}}}
 
 (this parameterization by normal exists for surfaces whose <a href="/facts/Gaussian_curvature/ARdhnb7o">Gaussian curvature</a> is strictly positive, and thus convex, smooth, and not flat). The parametric representation of the general offset surface of 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {x}}({\vec {u}})}
 
 offset by 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}})}
 
 is:

x
 →
 
 
 
 
 d
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 =
 
 
 
 x
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 +
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 )
 ,
 
 
 
 {\displaystyle {\vec {x}}_{d}({\vec {u}})={\vec {x}}({\vec {u}})+{\vec {d}}({\vec {n}}({\vec {u}})),\quad }
 
 where 
 
 
 
 
 
 
 n
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {n}}({\vec {u}})}
 
 is the unit normal of 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {x}}({\vec {u}})}
 
. (The trival offset, 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 =
 d
 
 
 
 n
 →
 
 
 
 
 
 {\displaystyle {\vec {d}}({\vec {n}})=d{\vec {n}}}
 
, gives you ordinary parallel surfaces.)
First, notice that the normal of 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 =
 
 
 {\displaystyle {\vec {x}}({\vec {u}})=}
 
 the normal of 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 )
 =
 
 
 
 n
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 ,
 
 
 {\displaystyle {\vec {d}}({\vec {n}}({\vec {u}}))={\vec {n}}({\vec {u}}),}
 
 by definition. Now, we'll apply the differential w.r.t. 
 
 
 
 
 
 
 u
 →
 
 
 
 
 
 {\displaystyle {\vec {u}}}
 
 to 
 
 
 
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 
 
 {\displaystyle {\vec {x}}_{d}}
 
, which gives us its tangent vectors spanning its tangent plane.

∂
        
          
            
              
                x
                →
              
            
          
          
            d
          
        
        (
        
          
            
              u
              →
            
          
        
        )
        =
        ∂
        
          
            
              x
              →
            
          
        
        (
        
          
            
              u
              →
            
          
        
        )
        +
        ∂
        
          
            
              d
              →
            
          
        
        (
        
          
            
              n
              →
            
          
        
        (
        
          
            
              u
              →
            
          
        
        )
        )
      
    
    {\displaystyle \partial {\vec {x}}_{d}({\vec {u}})=\partial {\vec {x}}({\vec {u}})+\partial {\vec {d}}({\vec {n}}({\vec {u}}))}

Notice, the tangent vectors for 
 
 
 
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 
 
 {\displaystyle {\vec {x}}_{d}}
 
 are the sum of tangent vectors for 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {x}}({\vec {u}})}
 
 and its offset 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}})}
 
, which share the same unit normal. Thus, the general offset surface shares the same tangent plane and normal with 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {x}}({\vec {u}})}
 
 and 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}}({\vec {u}}))}
 
. That aligns with the nature of envelopes.
We now consider the <a href="/facts/Weingarten_equations/95HmwPA7">Weingarten equations</a> for the <a href="/facts/Shape_operator/cslVnmd6">shape operator</a>, which can be written as 
 
 
 
 ∂
 
 
 
 n
 →
 
 
 
 =
 −
 ∂
 
 
 
 x
 →
 
 
 
 S
 
 
 {\displaystyle \partial {\vec {n}}=-\partial {\vec {x}}S}
 
. If 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is invertable, 
 
 
 
 ∂
 
 
 
 x
 →
 
 
 
 =
 −
 ∂
 
 
 
 n
 →
 
 
 
 
 S
 
 −
 1
 
 
 
 
 {\displaystyle \partial {\vec {x}}=-\partial {\vec {n}}S^{-1}}
 
. Recall that the principal curvatures of a surface are the <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvalues</a> of the shape operator, the principal curvature directions are its <a href="/facts/Eigenvalues_and_eigenvectors/8TjEoT8u">eigenvectors</a>, the Gauss curvature is its <a href="/facts/Determinant/eGYqJ2TF">determinant</a>, and the mean curvature is half its <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a>. The inverse of the shape operator holds these same values for the radii of curvature.
Substituting into the equation for the differential of 
 
 
 
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 
 
 {\displaystyle {\vec {x}}_{d}}
 
, we get:

∂
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 =
 ∂
 
 
 
 x
 →
 
 
 
 −
 ∂
 
 
 
 n
 →
 
 
 
 
 S
 
 n
 
 
 −
 1
 
 
 ,
 
 
 
 {\displaystyle \partial {\vec {x}}_{d}=\partial {\vec {x}}-\partial {\vec {n}}S_{n}^{-1},\quad }
 
 where 
 
 
 
 
 S
 
 n
 
 
 
 
 {\displaystyle S_{n}}
 
 is the shape operator for 
 
 
 
 
 
 
 d
 →
 
 
 
 (
 
 
 
 n
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 )
 
 
 {\displaystyle {\vec {d}}({\vec {n}}({\vec {u}}))}
 
.
Next, we use the <a href="/facts/Weingarten_equations/95HmwPA7">Weingarten equations</a> again to replace 
 
 
 
 ∂
 
 
 
 n
 →
 
 
 
 
 
 {\displaystyle \partial {\vec {n}}}
 
:

∂
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 =
 ∂
 
 
 
 x
 →
 
 
 
 +
 ∂
 
 
 
 x
 →
 
 
 
 S
 
 S
 
 n
 
 
 −
 1
 
 
 ,
 
 
 
 {\displaystyle \partial {\vec {x}}_{d}=\partial {\vec {x}}+\partial {\vec {x}}SS_{n}^{-1},\quad }
 
 where 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is the shape operator for 
 
 
 
 
 
 
 x
 →
 
 
 
 (
 
 
 
 u
 →
 
 
 
 )
 
 
 {\displaystyle {\vec {x}}({\vec {u}})}
 
.
Then, we solve for 
 
 
 
 ∂
 
 
 
 x
 →
 
 
 
 
 
 {\displaystyle \partial {\vec {x}}}
 
 and multiple both sides by 
 
 
 
 −
 S
 
 
 {\displaystyle -S}
 
 to get back to the <a href="/facts/Weingarten_equations/95HmwPA7">Weingarten equations</a>, this time for 
 
 
 
 ∂
 
 
 
 
 x
 →
 
 
 
 
 d
 
 
 
 
 {\displaystyle \partial {\vec {x}}_{d}}
 
:

∂
        
          
            
              
                x
                →
              
            
          
          
            d
          
        
        (
        I
        +
        S
        
          S
          
            n
          
          
            −
            1
          
        
        
          )
          
            −
            1
          
        
        =
        ∂
        
          
            
              x
              →
            
          
        
        ,
      
    
    {\displaystyle \partial {\vec {x}}_{d}(I+SS_{n}^{-1})^{-1}=\partial {\vec {x}},}

−
        ∂
        
          
            
              
                x
                →
              
            
          
          
            d
          
        
        (
        I
        +
        S
        
          S
          
            n
          
          
            −
            1
          
        
        
          )
          
            −
            1
          
        
        S
        =
        −
        ∂
        
          
            
              x
              →
            
          
        
        S
        =
        ∂
        
          
            
              n
              →
            
          
        
        .
      
    
    {\displaystyle -\partial {\vec {x}}_{d}(I+SS_{n}^{-1})^{-1}S=-\partial {\vec {x}}S=\partial {\vec {n}}.}

Thus, 
 
 
 
 
 S
 
 d
 
 
 =
 (
 I
 +
 S
 
 S
 
 n
 
 
 −
 1
 
 
 
 )
 
 −
 1
 
 
 S
 
 
 {\displaystyle S_{d}=(I+SS_{n}^{-1})^{-1}S}
 
, and inverting both sides gives us, 
 
 
 
 
 S
 
 d
 
 
 −
 1
 
 
 =
 
 S
 
 −
 1
 
 
 +
 
 S
 
 n
 
 
 −
 1
 
 
 
 
 {\displaystyle S_{d}^{-1}=S^{-1}+S_{n}^{-1}}
 
.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bump_mapping/c97rLmK6">Bump mapping</a></li>
<li><a href="/facts/Distance_function/RkUptSo8">Distance function</a> and <a href="/facts/Signed_distance_function/lqgd3TAv">signed distance function</a></li>
<li><a href="/facts/Distance_field/IeAfIYua">Distance field</a></li>
<li><a href="/facts/Offset_printing/dADgaujb">Offset printing</a></li>
<li><a href="/facts/Tubular_neighborhood/4Ftas0ry">Tubular neighborhood</a></li></ul>

<ul><li>Josef Hoschek: Offset curves in the plane. In: CAD. 17 (1985), S. 77–81.</li>
<li>Takashi Maekawa: An overview of offset curves and surfaces. In: CAD. 31 (1999), S. 165–173.</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>Farouki, R. T.; Neff, C. A. (1990). "Analytic properties of plane offset curves". Computer Aided Geometric Design. 7 (1–4): 83–99. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0167-8396%2890%2990023-K">10.1016/0167-8396(90)90023-K</a>.</li>
<li>Piegl, Les A. (1999). "Computing offsets of NURBS curves and surfaces". Computer-Aided Design. 31 (2): 147–156. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.360.2793">10.1.1.360.2793</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2FS0010-4485%2898%2900066-9">10.1016/S0010-4485(98)00066-9</a>.</li>
<li>Porteous, Ian R. (2001). Geometric Differentiation: For the Intelligence of Curves and Surfaces (2nd ed.). Cambridge University Press. pp. 1–25. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-00264-6.</li>
<li>Patrikalakis, Nicholas M.; Maekawa, Takashi (2010) [2002]. Shape Interrogation for Computer Aided Design and Manufacturing. Springer Science & Business Media. Chapter 11. Offset Curves and Surfaces. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-04074-0. <a href="http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node210.html">Free online version</a>.</li>
<li>Anton, François; Emiris, Ioannis Z.; Mourrain, Bernard; <a href="/facts/Monique_Teillaud/IFW7p7JG">Teillaud, Monique</a> (May 2005). "The O set to an Algebraic Curve and an Application to Conics". International Conference on Computational Science and its Applications. Singapore: Springer Verlag. pp. 683–696.</li>
<li>Farouki, Rida T. (2008). Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer Science & Business Media. pp. 141–178. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-73397-3. Pages listed are the general and introductory material.</li>
<li>Au, C. K.; Ma, Y.-S. (2013). "Computation of Offset Curves Using a Distance Function: Addressing a Key Challenge in Cutting Tool Path Generation". In Ma, Y.-S. (ed.). Semantic Modeling and Interoperability in Product and Process Engineering: A Technology for Engineering Informatics. Springer Science & Business Media. pp. 259–273. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4471-5073-2.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://mathworld.wolfram.com/ParallelCurves.html">Parallel curves on MathWorld</a></li>
<li><a href="http://xahlee.org/SpecialPlaneCurves_dir/Parallel_dir/parallel.html">Visual Dictionary of Plane Curves</a> Xah Lee</li>
<li><a href="http://library.imageworks.com/pdfs/imageworks-library-offset-curve-deformation-from-Skeletal-Anima.pdf">http://library.imageworks.com/pdfs/imageworks-library-offset-curve-deformation-from-Skeletal-Anima.pdf</a> application to animation; patented as <a href="https://patents.google.com/patent/US8400455">U.S. patent 8,400,455</a></li>
<li><a href="http://www2.uah.es/fsegundo/Otros/Offset/16-SanSegundoSendraSendra-1532.pdf">http://www2.uah.es/fsegundo/Otros/Offset/16-SanSegundoSendraSendra-1532.pdf</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Willson, Frederick Newton (1898). Theoretical and Practical Graphics. Macmillan. p. 66. ISBN 978-1-113-74312-1. {{cite book}}: ISBN / Date incompatibility (help) <a href="978-1-113-74312-1" target="_blank">978-1-113-74312-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Devadoss, Satyan L.; O'Rourke, Joseph (2011). Discrete and Computational Geometry. Princeton University Press. pp. 128–129. ISBN 978-1-4008-3898-1. <a href="978-1-4008-3898-1" target="_blank">978-1-4008-3898-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Devadoss, Satyan L.; O'Rourke, Joseph (2011). Discrete and Computational Geometry. Princeton University Press. pp. 128–129. ISBN 978-1-4008-3898-1. <a href="978-1-4008-3898-1" target="_blank">978-1-4008-3898-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Sendra, J. Rafael; Winkler, Franz; Pérez Díaz, Sonia (2007). Rational Algebraic Curves: A Computer Algebra Approach. Springer Science & Business Media. p. 10. ISBN 978-3-540-73724-7. <a href="978-3-540-73724-7" target="_blank">978-3-540-73724-7</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Agoston, Max K. (2005). Computer Graphics and Geometric Modelling: Mathematics. Springer Science & Business Media. p. 586. ISBN 978-1-85233-817-6. <a href="978-1-85233-817-6" target="_blank">978-1-85233-817-6</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Vince, John (2006). Geometry for Computer Graphics: Formulae, Examples and Proofs. Springer Science & Business Media. p. 293. ISBN 978-1-84628-116-7. <a href="978-1-84628-116-7" target="_blank">978-1-84628-116-7</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Marsh, Duncan (2006). Applied Geometry for Computer Graphics and CAD (2nd ed.). Springer Science & Business Media. p. 107. ISBN 978-1-84628-109-9. <a href="978-1-84628-109-9" target="_blank">978-1-84628-109-9</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Mark Kilgard (2012-04-10). "CS 354 Vector Graphics & Path Rendering". www.slideshare.net. p. 28. <a href="https://www.slideshare.net/Mark_Kilgard/22pathrender" target="_blank">https://www.slideshare.net/Mark_Kilgard/22pathrender</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Willson, Frederick Newton (1898). Theoretical and Practical Graphics. Macmillan. p. 66. ISBN 978-1-113-74312-1. {{cite book}}: ISBN / Date incompatibility (help) <a href="978-1-113-74312-1" target="_blank">978-1-113-74312-1</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Devadoss, Satyan L.; O'Rourke, Joseph (2011). Discrete and Computational Geometry. Princeton University Press. pp. 128–129. ISBN 978-1-4008-3898-1. <a href="978-1-4008-3898-1" target="_blank">978-1-4008-3898-1</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Sendra, J. Rafael; Winkler, Franz; Pérez Díaz, Sonia (2007). Rational Algebraic Curves: A Computer Algebra Approach. Springer Science & Business Media. p. 10. ISBN 978-3-540-73724-7. <a href="978-3-540-73724-7" target="_blank">978-3-540-73724-7</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Agoston, Max K. (2005). Computer Graphics and Geometric Modelling. Springer Science & Business Media. pp. 638–645. ISBN 978-1-85233-818-3. <a href="978-1-85233-818-3" target="_blank">978-1-85233-818-3</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">http://www.cc.gatech.edu/~jarek/papers/localVolume.pdf, p. 3 <a href="http://www.cc.gatech.edu/~jarek/papers/localVolume.pdf" target="_blank">http://www.cc.gatech.edu/~jarek/papers/localVolume.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Agoston, Max K. (2005). Computer Graphics and Geometric Modelling. Springer Science & Business Media. pp. 638–645. ISBN 978-1-85233-818-3. <a href="978-1-85233-818-3" target="_blank">978-1-85233-818-3</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Faux, I. D.; Pratt, Michael J. (1979). Computational Geometry for Design and Manufacture. Halsted Press. ISBN 978-0-47026-473-7. OCLC 4859052. <a href="978-0-47026-473-7" target="_blank">978-0-47026-473-7</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Brechner, Eric (1990). Envelopes and tool paths for three-axis end milling (PhD). Rensselaer Polytechnic Institute. <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">E. Hartmann: Geometry and Algorithms for COMPUTER AIDED DESIGN. S. 30. <a href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf" target="_blank">http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Fiona O'Neill: Planar Bertrand Curves (with Pictures!). <a href="https://fionasmathblog.com/2022/04/26/planar-bertrand-curves-with-pictures/" target="_blank">https://fionasmathblog.com/2022/04/26/planar-bertrand-curves-with-pictures/</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Rida T. Farouki:
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable (Geometry and Computing). Springer, 2008, ISBN 978-3-540-73397-3. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">E. Hartmann: Geometry and Algorithms for COMPUTER AIDED DESIGN. S. 81, S. 30, 41, 44. <a href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf" target="_blank">http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">E. Hartmann: Geometry and Algorithms for COMPUTER AIDED DESIGN. S. 30. <a href="http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf" target="_blank">http://www.mathematik.tu-darmstadt.de/~ehartmann/cdgen0104.pdf</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Thorpe, John A. (1994-10-27). Elementary Topics in Differential Geometry. New York Heidelberg: Springer Science & Business Media. ISBN 0-387-90357-7. <a href="0-387-90357-7" target="_blank">0-387-90357-7</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">http://faculty.engineering.ucdavis.edu/farouki/wp-content/uploads/sites/41/2013/02/Introduction-to-PH-curves.pdf Archived 2015-06-05 at the Wayback Machine, p. 16 "taxonomy of offset curves" <a href="http://faculty.engineering.ucdavis.edu/farouki/wp-content/uploads/sites/41/2013/02/Introduction-to-PH-curves.pdf" target="_blank">http://faculty.engineering.ucdavis.edu/farouki/wp-content/uploads/sites/41/2013/02/Introduction-to-PH-curves.pdf</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Brechner, Eric (1990). Envelopes and tool paths for three-axis end milling (PhD). Rensselaer Polytechnic Institute. <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">Tiller, Wayne; Hanson, Eric (1984). "Offsets of Two-Dimensional Profiles". IEEE Computer Graphics and Applications. 4 (9): 36–46. doi:10.1109/mcg.1984.275995. S2CID 9046817. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">Elber, Gershon; Lee, In-Kwon; Kim, Myung-Soo (May–Jun 1997). "Comparing offset curve approximation methods". IEEE Computer Graphics and Applications. 17 (3): 62–71. doi:10.1109/38.586019. <a href="https://ieeexplore.ieee.org/document/586019" target="_blank">https://ieeexplore.ieee.org/document/586019</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Levien, Raph (September 9, 2022). "Parallel curves of cubic Béziers". Retrieved September 9, 2022. <a href="https://raphlinus.github.io/curves/2022/09/09/parallel-beziers.html" target="_blank">https://raphlinus.github.io/curves/2022/09/09/parallel-beziers.html</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
<li id="fn:28">Kimmel, R.; Bruckstein, A.M. (1993). "Shape offsets via level sets" (PDF). Computer-Aided Design. 25 (3). Elsevier BV: 154–162. doi:10.1016/0010-4485(93)90040-u. ISSN 0010-4485. S2CID 8434463. <a href="https://www.cs.technion.ac.il/~ron/PAPERS/offsets_cad1993.pdf" target="_blank">https://www.cs.technion.ac.il/~ron/PAPERS/offsets_cad1993.pdf</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></li>
<li id="fn:29">Faux, I. D.; Pratt, Michael J. (1979). Computational Geometry for Design and Manufacture. Halsted Press. ISBN 978-0-47026-473-7. OCLC 4859052. <a href="978-0-47026-473-7" target="_blank">978-0-47026-473-7</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></li>
<li id="fn:30">Brechner, Eric L. (1992). "5. General Offset Curves and Surfaces". In Barnhill, Robert E. (ed.). Geometry Processing for Design and Manufacturing. SIAM. pp. 101–. ISBN 978-0-89871-280-3. <a href="978-0-89871-280-3" target="_blank">978-0-89871-280-3</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></li>
<li id="fn:31">Pottmann, Helmut; Wallner, Johannes (2001). Computational Line Geometry. Springer Science & Business Media. pp. 303–304. ISBN 978-3-540-42058-3. <a href="978-3-540-42058-3" target="_blank">978-3-540-42058-3</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></li>
<li id="fn:32">Chirikjian, Gregory S. (2009). Stochastic Models, Information Theory, and Lie Groups, Volume 1: Classical Results and Geometric Methods. Springer Science & Business Media. pp. 171–175. ISBN 978-0-8176-4803-9. <a href="978-0-8176-4803-9" target="_blank">978-0-8176-4803-9</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></li>
<li id="fn:33">Sarfraz, Muhammad, ed. (2003). Advances in geometric modeling. Wiley. p. 72. ISBN 978-0-470-85937-7. <a href="978-0-470-85937-7" target="_blank">978-0-470-85937-7</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></li>
<li id="fn:34">Brechner, Eric L. (1992). "5. General Offset Curves and Surfaces". In Barnhill, Robert E. (ed.). Geometry Processing for Design and Manufacturing. SIAM. pp. 101–. ISBN 978-0-89871-280-3. <a href="978-0-89871-280-3" target="_blank">978-0-89871-280-3</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></li>
<li id="fn:35">Brechner, Eric L. (1992). "5. General Offset Curves and Surfaces". In Barnhill, Robert E. (ed.). Geometry Processing for Design and Manufacturing. SIAM. pp. 101–. ISBN 978-0-89871-280-3. <a href="978-0-89871-280-3" target="_blank">978-0-89871-280-3</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></li>
<li id="fn:36">Kinch, Richard J. (1995). "MetaFog: Converting METAFONT Shapes to Contours" (PDF). TUGboat. 16 (3): 233–243. <a href="https://www.tug.org/TUGboat/tb16-3/tb48kinc.pdf" target="_blank">https://www.tug.org/TUGboat/tb16-3/tb48kinc.pdf</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></li>
<li id="fn:37">http://design.tutsplus.com/tutorials/illustrator-cs5-variable-width-stroke-tool-perfect-for-making-tribal-designs--vector-4346 application of the generalized version in Adobe Illustrator CS5 (also video) <a href="http://design.tutsplus.com/tutorials/illustrator-cs5-variable-width-stroke-tool-perfect-for-making-tribal-designs--vector-4346" target="_blank">http://design.tutsplus.com/tutorials/illustrator-cs5-variable-width-stroke-tool-perfect-for-making-tribal-designs--vector-4346</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></li>
<li id="fn:38">http://www.cc.gatech.edu/~jarek/papers/localVolume.pdf, p. 3 <a href="http://www.cc.gatech.edu/~jarek/papers/localVolume.pdf" target="_blank">http://www.cc.gatech.edu/~jarek/papers/localVolume.pdf</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></li>
<li id="fn:39">Brechner, Eric L. (1992). "5. General Offset Curves and Surfaces". In Barnhill, Robert E. (ed.). Geometry Processing for Design and Manufacturing. SIAM. pp. 101–. ISBN 978-0-89871-280-3. <a href="978-0-89871-280-3" target="_blank">978-0-89871-280-3</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></li>
<li id="fn:40">Brechner, Eric (1990). Envelopes and tool paths for three-axis end milling (PhD). Rensselaer Polytechnic Institute. <a href="#fnref:40" class="footnote-back-ref">↩</a></li>
<li id="fn:41">Brechner, Eric L. (1992). "5. General Offset Curves and Surfaces". In Barnhill, Robert E. (ed.). Geometry Processing for Design and Manufacturing. SIAM. pp. 101–. ISBN 978-0-89871-280-3. <a href="978-0-89871-280-3" target="_blank">978-0-89871-280-3</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></li>
</ol>

Parallel curve open-in-new

Parallel curve