Class template that performs the transformation between the polynomial coefficient matrix \( \mathbf{C}=\left[\mathbf{c}_0\ \mathbf{c}_1\ \ldots \ \mathbf{c}_P\right]\) and the Hermite coefficient matrix \( \mathbf{H}=\left[\mathbf{r}(0),\ \ldots \frac{d^\frac{P-1}{2}\mathbf{r}}{dt^\frac{P-1}{2}}(0),\ \mathbf{r}(1),\ \ldots \frac{d^\frac{P-1}{2}\mathbf{r}}{dt^\frac{P-1}{2}}(1)\right]\) of a spline \(\mathbf{r}(u)=\sum_{i=0}^P\mathbf{c}_it^i\) of odd degree \(P\), parametrized in \([0\ 1]\). More...
#include <Coeff2Hermite.h>
Static Public Member Functions | |
| template<short unsigned int dim> | |
| static Eigen::Matrix< double, dim, polyCoeff > | c2h (const Eigen::Matrix< double, dim, polyCoeff > &C) |
| template<short unsigned int dim> | |
| static Eigen::Matrix< double, dim, polyCoeff > | h2c (const Eigen::Matrix< double, dim, polyCoeff > &H) |
Static Public Attributes | |
| static const Eigen::Matrix< double, polyDegree+1, polyDegree+1 > | C2H |
| static const Eigen::Matrix< double, polyDegree+1, polyDegree+1 > | C2HT |
| static const Eigen::Matrix< double, polyDegree+1, polyDegree+1 > | H2C |
| static const Eigen::Matrix< double, polyDegree+1, polyDegree+1 > | H2CT |
Detailed Description
class model::Coeff2Hermite< polyDegree >
Class template that performs the transformation between the polynomial coefficient matrix \( \mathbf{C}=\left[\mathbf{c}_0\ \mathbf{c}_1\ \ldots \ \mathbf{c}_P\right]\) and the Hermite coefficient matrix \( \mathbf{H}=\left[\mathbf{r}(0),\ \ldots \frac{d^\frac{P-1}{2}\mathbf{r}}{dt^\frac{P-1}{2}}(0),\ \mathbf{r}(1),\ \ldots \frac{d^\frac{P-1}{2}\mathbf{r}}{dt^\frac{P-1}{2}}(1)\right]\) of a spline \(\mathbf{r}(u)=\sum_{i=0}^P\mathbf{c}_it^i\) of odd degree \(P\), parametrized in \([0\ 1]\).
Consider the spline: [ \mathbf{r}(u)=\sum_{i=0}^P\mathbf{c}_it^i ] where \(t\in[0,1]\) and P is odd. The matrix
\[ \left[\mathbf{c}_0\ \mathbf{c}_1\ \ldots \ \mathbf{c}_P\right] \]
is the coefficient matrix of the spline, while the matrix
\[ \mathbf{H}=\left[\mathbf{r}(0),\ \ldots \frac{d^\frac{P-1}{2}\mathbf{r}}{dt^\frac{P-1}{2}}(0),\ \mathbf{r}(1),\ \ldots \frac{d^\frac{P-1}{2}\mathbf{r}}{dt^\frac{P-1}{2}}(1)\right] \]
is the Hermite matrix of the spline. We now seek a relationship between the two.
\[ \frac{d^n\mathbf{r}}{dt^n}=\sum_{i=0}^P \underbrace{i(i-1)\ldots(i-n+1)}_{n\ terms} \mathbf{c}_it^{i-n} \]
Since only terms having \(i-n+1>0\) survive in the sum, that is \(i\ge n\) then we obtain:
\[ \frac{d^n\mathbf{r}}{dt^n}=\sum_{i=n}^P \frac{i!}{(i-n)!} \mathbf{c}_it^{i-n} \]
Letting \(t=0\) and \(t=1\), one finds that the Hermite coefficients of order n (n=0...(P-1)/2) are:
\[ \frac{d^n\mathbf{r}}{dt^n}(0)=\sum_{i=n}^P \frac{i!}{(i-n)!} \mathbf{c}_i0^{i-n}=n!\mathbf{c}_n \]
\[ \frac{d^n\mathbf{r}}{dt^n}(1)=\sum_{i=n}^P \frac{i!}{(i-n)!} \mathbf{c}_i1^{i-n}=\sum_{i=n}^P \frac{i!}{(i-n)!} \mathbf{c}_i \]
In matrix form this reads:
\[\underbrace{\left[\begin{array}{l} {d^0\mathbf{r}}/{dt^0}(0)\\ {d^1\mathbf{r}}/{dt^1}(0)\\ {d^2\mathbf{r}}/{dt^2}(0)\\ \vdots\\ {d^0\mathbf{r}}/{dt^0}(1)\\ {d^1\mathbf{r}}/{dt^1}(1)\\ {d^2\mathbf{r}}/{dt^2}(1)\\ \vdots\\ \end{array}\right]}_{\mathbf{H}^T}= \left[\begin{array}{cccc} 0!& 0 & 0 & \ldots\\ 0 & 1! & 0 & \ldots\\ 0 & 0 & 2! & \ldots\\ \vdots & \vdots & \vdots & \ddots\\ \frac{0!}{(0-0)!}& \frac{1!}{(1-0)!} & \frac{2!}{(2-0)!} & \ldots\\ 0& \frac{1!}{(1-1)!} & \frac{2!}{(2-1)!} & \ldots\\ 0& 0 & \frac{2!}{(2-2)!} & \ldots\\ \vdots & \vdots & \vdots & \ddots\\ \end{array}\right] \underbrace{\left[\begin{array}{l} \mathbf{c}_0\\ \mathbf{c}_1\\ \mathbf{c}_2\\ \vdots\\ \end{array}\right]}_{\mathbf{C}^T} \]
For a linear spline (P=1, n=0) we find:
\[ \mathbf{H}=\left[\mathbf{c}_0,\ \mathbf{c}_0+\mathbf{c}_1\right] \]
\[ \mathbf{C}=\left[\mathbf{h}_0,\ \mathbf{h}_1-\mathbf{h}_0\right] \]
For a cubic spline (P=3, n=0,1) we find:
\[ \mathbf{H}=\left[\mathbf{c}_0,\ \mathbf{c}_1,\ \mathbf{c}_0+\mathbf{c}_1+\mathbf{c}_2+\mathbf{c}_3,\ \mathbf{c}_1+2\mathbf{c}_2+3\mathbf{c}_3\right] \]
\[ \mathbf{C}=\left[\mathbf{h}_0,\ \mathbf{h}_1,\ -3\mathbf{h}_0-2\mathbf{h}_1+3\mathbf{h}_2-\mathbf{h}_3,\ 2\mathbf{h}_0+\mathbf{h}_1-2\mathbf{h}_2+\mathbf{h}_3\right] \]
Definition at line 166 of file Coeff2Hermite.h.
Member Function Documentation
◆ c2h()
|
inlinestatic |
The Hermite matrix corresponding to the polynomial coefficients C (coefficients in columns)
Definition at line 193 of file Coeff2Hermite.h.
◆ h2c()
|
inlinestatic |
The Hermite matrix corresponding to the polynomial coefficients C (coefficients in columns)
Definition at line 203 of file Coeff2Hermite.h.
Member Data Documentation
◆ C2H
|
static |
Definition at line 184 of file Coeff2Hermite.h.
◆ C2HT
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static |
Definition at line 185 of file Coeff2Hermite.h.
◆ H2C
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static |
Definition at line 187 of file Coeff2Hermite.h.
◆ H2CT
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static |
Definition at line 188 of file Coeff2Hermite.h.
The documentation for this class was generated from the following file:
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