Cubic Hermite spline

In <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>, a cubic Hermite spline or cubic Hermite interpolator is a <a href="/facts/Spline_(mathematics)/yafkQtqp">spline</a> where each piece is a third-degree <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> specified in <a href="/facts/Hermite_interpolation/u36d6yH2">Hermite form</a>, that is, by its values and first <a href="/facts/Derivative_(mathematics)/7Ito70Dh">derivatives</a> at the end points of the corresponding <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> interval.
Cubic Hermite splines are typically used for <a href="/facts/Interpolation/8PyetX4f">interpolation</a> of numeric data specified at given argument values 
 
 
 
 
 x
 
 1
 
 
 ,
 
 x
 
 2
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 
 
 {\displaystyle x_{1},x_{2},\ldots ,x_{n}}
 
, to obtain a <a href="/facts/Continuous_function/sKbl02pB">continuous function</a>. The data should consist of the desired function value and derivative at each 
 
 
 
 
 x
 
 k
 
 
 
 
 {\displaystyle x_{k}}
 
. (If only the values are provided, the derivatives must be estimated from them.) The Hermite formula is applied to each interval 
 
 
 
 (
 
 x
 
 k
 
 
 ,
 
 x
 
 k
 +
 1
 
 
 )
 
 
 {\displaystyle (x_{k},x_{k+1})}
 
 separately. The resulting spline will be continuous and will have continuous first derivative.
Cubic polynomial splines can be specified in other ways, the <a href="/facts/Bezier_cubic/zJx3Uj60">Bezier cubic</a> being the most common. However, these two methods provide the same set of splines, and data can be easily converted between the Bézier and Hermite forms; so the names are often used as if they were synonymous. 
Cubic polynomial splines are extensively used in <a href="/facts/Computer_graphics/lapBE8wv">computer graphics</a> and <a href="/facts/Geometric_modeling/SGyQGku9">geometric modeling</a> to obtain <a href="/facts/Curve/xBQ0pTBW">curves</a> or motion <a href="/facts/Trajectory/STNN33FK">trajectories</a> that pass through specified points of the <a href="/facts/Plane_(geometry)/tAUtRFcn">plane</a> or three-dimensional <a href="/facts/Space_(geometry)/G6rt4bQ8">space</a>. In these applications, each coordinate of the plane or space is separately interpolated by a cubic spline function of a separate parameter t. 
Cubic polynomial splines are also used extensively in structural analysis applications, such as <a href="/facts/Euler%25E2%2580%2593Bernoulli_beam_theory/3DyQV9do">Euler–Bernoulli beam theory</a>. Cubic polynomial splines have also been applied to mortality analysis and mortality forecasting.
Cubic splines can be extended to functions of two or more parameters, in several ways. Bicubic splines (<a href="/facts/Bicubic_interpolation/Fov96ol8">Bicubic interpolation</a>) are often used to interpolate data on a regular rectangular grid, such as <a href="/facts/Pixel/UzwBvXBV">pixel</a> values in a <a href="/facts/Digital_image/3TNT47Mv">digital image</a> or <a href="/facts/Altitude/B5eNVurK">altitude</a> data on a terrain. <a href="/facts/B%25C3%25A9zier_patch/F1FlWnId">Bicubic surface patches</a>, defined by three bicubic splines, are an essential tool in computer graphics.
Cubic splines are often called csplines, especially in computer graphics. Hermite splines are named after <a href="/facts/Charles_Hermite/kj1bGlMr">Charles Hermite</a>.

Cubic Hermite spline open-in-new

Cubic Hermite spline