Cubic Hermite spline

In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form: i.e., by its values and first derivatives at the end points of the corresponding domain interval. R ′ ( 0 ) = 0 = a r {displaystyle R'(0)=0=ar}     (1) R ′ ( 1 ) = 0 = a ( 1 − r ) {displaystyle R'(1)=0=a(1-r)}     (2) In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form: i.e., by its values and first derivatives at the end points of the corresponding domain interval. Cubic Hermite splines are typically used for interpolation 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 smooth continuous function. 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 Bézier form 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 computer graphics and geometric modeling to obtain curves or motion trajectories that pass through specified points of the plane or three-dimensional space. 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 Euler–Bernoulli beam theory. Cubic splines can be extended to functions of two or more parameters, in several ways. Bicubic splines (Bicubic interpolation) are often used to interpolate data on a regular rectangular grid, such as pixel values in a digital image or altitude data on a terrain. Bicubic surface patches, 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 Charles Hermite. On the unit interval ( 0 , 1 ) {displaystyle (0,1)} , given a starting point p 0 {displaystyle {oldsymbol {p}}_{0}} at t = 0 {displaystyle t=0} and an ending point p 1 {displaystyle {oldsymbol {p}}_{1}} at t = 1 {displaystyle t=1} with starting tangent m 0 {displaystyle {oldsymbol {m}}_{0}} at t = 0 {displaystyle t=0} and ending tangent m 1 {displaystyle {oldsymbol {m}}_{1}} at t = 1 {displaystyle t=1} , the polynomial can be defined by where t ∈ . Interpolating x {displaystyle x} in an arbitrary interval ( x k , x k + 1 ) {displaystyle (x_{k},x_{k+1})} is done by mapping the latter to [ 0 , 1 ] {displaystyle } through an affine (degree 1) change of variable. The formula is