Interpolation curves - Lagrange

The Lagrange polynomials form a basis for interpolation curves, from chapter 5.3 in Blending techniques in Curve and Surface constructions

A Lagrange interpolation curve is expressed by the following formula:: \( C(t) = \displaystyle\sum_{i=0}^{d}L_{d,i}(t)\ c_i\), where \(d\) is the polynomial degree, and where \(c_i\) are \(d+1\) interpolation points and \(L_{d,i}(t)\) are \(d+1\) scalar Lagrange polynomials/basis functions. In the figure above, a set of Lagrange polynomials/basis functions is plotted. In the plots, the parameter domain is \([0,\ d]\), and the set/vector with the parameter values in the interpolation points \( \{t_0, t_1, ...,t_d\} \) we have, in the figure above, chosen to be \( \{ 0, 1, 2, ..., d\} \). In the figure below, a curve is plotted using a set of \(d+1\) interpolation points \(c_i,\ i=0,1,...,d\ \) which all are marked as gray pentagons.

In the menu at the top right, you can determine the degree \(d\).
In the figure above, you can press the left mouse button when the cursor is over one of the vertical black lines that represent a parameter value in an interpolation point. The line moves horizontally as long as you hold the button down. When you release the button, the line is locked and both figures are updated.
In the figure below, you can move the interpolation points by pressing the left mouse button when the cursor is over one of the gray pentagons that represent an interpolation point. The pentagon moves as long as you hold the button down. When you release the button, the pentagon is locked and the curve is updated.

The plot is inspired by chapter 5.3 in the book.

A Lagrange interpolation curve has the following formula: \( C(t) = \displaystyle\sum_{i=0}^{d} L_{d,i}(t)\ c_i\), where \(d\) is the polynomial degree and where \(t \in[t_0, t_d]\), see chapter 5.3 in the book. The Lagrange polynomials are the basis functions with the formula: \(L_{d,i}(t) = \displaystyle\prod_{\substack{j=0\\ j\neq i}}^{d} \frac{(t-t_j)}{(t_i-t_j)} \) The following property is what provides interpolation: \(L_{d,i}(t_i) = 1\) and \(L_{d,i}(t_j) = 0, \ j \neq i \). The construction of the Lagrange polynomials is the simplest and most logical to achieve precisely these properties..

The derivative of a Lagrange polynomial is: \( L'_{d,i}(t) = \displaystyle\sum_{\substack{j=0\\ j \neq i}}^{d} \left(\frac{1}{t_i-t_j} \displaystyle\prod_{\substack{k=0\\ k\neq (i,j)}}^{d} \frac{(t-t_j)}{(t_i-t_k)}\right) = L_{d,i}(t) \displaystyle\sum_{\substack{j=0\\ j \neq i}}^{d}\frac{1}{t-t_i} \).

For degree 3 we will get: \( L_{3,0}(t) = \frac{(t-t_1)(t-t_2)(t-t_3)}{(t_0-t_1)(t_0-t_2)(t_0-t_3)} \), \( L_{3,1}(t) = \frac{(t-t_0)(t-t_2)(t-t_3)}{(t_1-t_0)(t_1-t_2)(t_1-t_3)} \), \( L_{3,2}(t) = \frac{(t-t_0)(t-t_1)(t-t_3)}{(t_2-t_0)(t_2-t_1)(t_2-t_3)} \) and \( L_{3,3}(t) = \frac{(t-t_0)(t-t_1)(t-t_2)}{(t_3-t_0)(t_3-t_1)(t_3-t_2)} \). In matrix form, the Lagrange polynomials are:

\( \begin{bmatrix} \frac{-t_1t_2t_3}{(t_0-t_1)(t_0-t_2)(t_0-t_3)} & \frac{t_1t_2+t_1t_3+t_2t_3}{(t_0-t_1)(t_0-t_2)(t_0-t_3)} & \frac{-(t_1+t_2+t_3)}{(t_0-t_1)(t_0-t_2)(t_0-t_3)} & \frac{1}{(t_0-t_1)(t_0-t_2)(t_0-t_3)}\\ \frac{-t_0t_2t_3}{(t_1-t_0)(t_1-t_2)(t_1-t_3)} & \frac{t_0t_2+t_0t_3+t_2t_3}{(t_1-t_0)(t_1-t_2)(t_1-t_3)} & \frac{-(t_0+t_2+t_3)}{(t_1-t_0)(t_1-t_2)(t_1-t_3)} & \frac{1}{(t_1-t_0)(t_1-t_2)(t_1-t_3)}\\ \frac{-t_0t_1t_3}{(t_2-t_0)(t_2-t_1)(t_2-t_3)} & \frac{t_0t_1+t_0t_3+t_1t_3}{(t_2-t_0)(t_2-t_1)(t_2-t_3)} & \frac{-(t_0+t_1+t_3)}{(t_2-t_0)(t_2-t_1)(t_2-t_3)} & \frac{1}{(t_2-t_0)(t_2-t_1)(t_2-t_3)}\\ \frac{-t_0t_1t_2}{(t_3-t_0)(t_3-t_1)(t_3-t_2)} & \frac{t_0t_1+t_0t_2+t_1t_2}{(t_3-t_0)(t_3-t_1)(t_3-t_2)} & \frac{-(t_0+t_1+t_2)}{(t_3-t_0)(t_3-t_1)(t_3-t_2)} & \frac{1}{(t_3-t_0)(t_3-t_1)(t_3-t_2)} \end{bmatrix} \begin{bmatrix} 1 \\ t\\ t^2 \\ t^3 \end{bmatrix} \). This matrix is then the basis shift matrix from a monomial 3rd degree basis to the Lagrange polynomials of degree 3.

Conversely, the basis shift matrix from the Lagrange polynomials of degree 3 to a monomial basis is: \( \begin{bmatrix} -1 & -1 & -1 & -1 \\ t_{0} & t_{1} & t_{2} & t_{3} \\ t_{0}^{2} & t_{1}^{2} & t_{2}^{2} & t_{3}^{2} \\ t_{0}^{3} & t_{1}^{3} & t_{2}^{3} & t_{3}^{3} \end{bmatrix}\).