https://github.com/morganherlocker/cubic-spline
http://blog.ivank.net/interpolation-with-cubic-splines.html
https://github.com/jawj/cubicsplines
http://blog.mackerron.com/2011/01/01/javascript-cubic-splines/
https://github.com/osuushi/Smooth.js
https://gist.github.com/skhokhlov/66aa6254f930f98b2c917abaf63eecb9
https://en.wikipedia.org/wiki/Spline_(mathematics)
For the elastic rulers being the model for the spline interpolation, one has that to the left of the left-most "knot" and to the right of the right-most "knot" the ruler can move freely and will therefore take the form of a straight line with q′′ = 0. As q′′ should be a continuous function of x, natural splines in addition to the n − 1 linear equations (15) should have
There exist other end conditions, clamped spline, which specifies the slope at the ends of the spline, and the popular "not-a-knot spline", which requires that the third derivative is also continuous at the x1 and xN−1 points. For the "not-a-knot" spline, the additional equations will read:
https://en.wikipedia.org/wiki/Spline_interpolation
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, that is, by its values and first derivatives at the end points of the corresponding domain interval.[1]
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}} x_{1},x_{2},\ldots ,x_{n}, to obtain a continuous function. The data should consist of the desired function value and derivative at each x k {\displaystyle x_{k}} 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})} (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 Bezier cubic 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.
A data set, ( x k , p k ) {\displaystyle (x_{k},{\boldsymbol {p}}{k})} (x_k,\boldsymbol{p}k) for k = 1 , … , n {\displaystyle k=1,\ldots ,n} k=1,\ldots,n, can be interpolated by applying the above procedure on each interval, where the tangents are chosen in a sensible manner, meaning that the tangents for intervals sharing endpoints are equal. The interpolated curve then consists of piecewise cubic Hermite splines and is globally continuously differentiable in ( x 1 , x n ) {\displaystyle (x{1},x{n})} {\displaystyle (x_{1},x_{n})}.
The choice of tangents is not unique, and there are several options available.
Finite difference
Cardinal spline
Catmull–Rom spline
In computer graphics, Catmull–Rom splines are frequently used to get smooth interpolated motion between key frames. For example, most camera path animations generated from discrete key-frames are handled using Catmull–Rom splines. They are popular mainly for being relatively easy to compute, guaranteeing that each key frame position will be hit exactly, and also guaranteeing that the tangents of the generated curve are continuous over multiple segments.
Kochanek–Bartels spline
Monotone cubic interpolation