Class template that computes tangent, curvature and torsion of a parametric curve \(\mathbf{r}(u)\) in arbitrary dimension. More...
#include <ParametricCurve.h>
Public Member Functions | |
| double | get_j (const double &u) const |
| double | get_jSquare (const double &u) const |
| VectorDim | get_rl (const double &u) const |
| double | get_ju (const double &u) const |
| VectorDim | get_rll (const double &u) const |
| double | get_kappa (const double &u) const |
| the scalar curvature | |
| template<short unsigned int qOrder, template< short unsigned int, size_t > class QuadratureRule> | |
| double | arcLength () const |
| template<short unsigned int qOrder, template< short unsigned int, size_t > class QuadratureRule> | |
| VectorDim | rm () const |
| VectorDim | rm_integrand (const double &u) const |
Detailed Description
class model::ParametricCurve< Derived, dim >
Class template that computes tangent, curvature and torsion of a parametric curve \(\mathbf{r}(u)\) in arbitrary dimension.
Let \(u\in[0,1]\) and \(\mathbf{r}(u): [0,1]\rightarrow \mathbb{R}^d\) a parametric curve. The first derivative \(\mathbf{r}_{,u}\) with respect to the parameter \(u\) represents the (non unitary) parametric tangent to the curve. The arc length \(l\) along the curve can be calculated as: of the parameter \(u\) at each time \(t\):
\[ l(u)=\int_0^u\sqrt{\mathbf{r}_{,v}\cdot\mathbf{r}_{,v}}dv \]
The length of the curve is therefore:
\[ L=\int_0^1\sqrt{\mathbf{r}_{,u}\cdot\mathbf{r}_{,u}}du \]
The kernel of the integral above represents the jacobian of the transformation between \( u \) and \( l \):
\[ \begin{eqnarray} J&=\frac{d l}{d u}=\sqrt{\mathbf{r}_{,u}\cdot\mathbf{r}_{,u}} \\ J_{,u}&=\frac{\mathbf{r}_{,u}\cdot\mathbf{r}_{,uu}}{J}\\ J_{,uu}&=\frac{\mathbf{r}_{,uu}\cdot\mathbf{r}_{,uu}+\mathbf{r}_{,u}\cdot\mathbf{r}_{uuu}-J^2_{,u}}{J} \end{eqnarray} \]
Derivatives with respect to the arc length are obtained by chain rule:
\[ \begin{eqnarray} \mathbf{r}_{,l}&=\frac{\mathbf{r}_{,u}}{J}\\ \mathbf{r}_{,ll}&=\frac{\mathbf{r}_{,uu}}{J^2}-J_{,u}\frac{\mathbf{r}_{,u}}{J^3}\\ \mathbf{r}_{,lll}&=\frac{\mathbf{r}_{,uuu}}{J^3}-\frac{\mathbf{r}_{,u}J_{,uu}}{J^4}-3\frac{J_{,u}}{J^2}\left(\frac{\mathbf{r}_{,uu}}{J^2}-J_{,u}\frac{\mathbf{r}_{,u}}{J^3}\right) \end{eqnarray} \]
In matrix form
\[ \left[ \begin{array}{c} \mathbf{r}\\ \mathbf{r}_{,u}\\ \mathbf{r}_{,uu}\\ \end{array}\right]= \left[ \begin{array}{ccc} 1&0&0\\ 0&J&0\\ 0&J_{,u}&J^2 \end{array}\right] \left[ \begin{array}{c} \mathbf{r}\\ \mathbf{r}_{,l}\\ \mathbf{r}_{,ll}\\ \end{array}\right] \]
Notice that tangent and curvature are orthogonal:
\[\mathbf{r}_{,l}\cdot\mathbf{r}_{,ll}=\frac{1}{J^3}\left(\mathbf{r}_{u}\cdot\mathbf{r}_{,uu}-J_{,u}\frac{\mathbf{r}_{,u}\cdot\mathbf{r}_{,u}}{J}\right)= \frac{1}{J^3}\left(\mathbf{r}_{u}\cdot\mathbf{r}_{,uu}-\frac{\mathbf{r}_{,u}\cdot\mathbf{r}_{,uu}}{J}\frac{J^2}{J}\right)=0 \]
The Frenet-Serret frame (TNB) frame is defined as:
\[\begin{eqnarray} \hat{\mathbf{t}}&=\mathbf{r}_{,l}&\text{tangent unit vector}\\ \hat{\mathbf{n}}&=\frac{\mathbf{r}_{,ll}}{\left|\mathbf{r}_{,ll}\right|}=\frac{\mathbf{r}_{,ll}}{\kappa}&\text{principal unit normal}\\ \hat{\mathbf{b}}&=\hat{\mathbf{t}}\times\hat{\mathbf{n}}=\frac{\mathbf{r}_{,u}\times\mathbf{r}_{,uu}}{J^3\kappa}&\text{binormal unit vector}\\ \end{eqnarray} \]
The curvature
\[ \kappa=\left|\mathbf{r}_{,ll}\right| = \frac{1}{J^2}\sqrt{\left|\mathbf{r}_{,uu}\right|^2-J_{,u}^2} \]
\[ \begin{eqnarray} \kappa_{,u}&= -2\frac{J_{,u}}{J^3}\sqrt{\left|\mathbf{r}_{,uu}\right|^2-J_{,u}^2}+\frac{1}{J^2}\frac{\mathbf{r}_{,uu}\cdot\mathbf{r}_{uuu}-J_{,u}J_{,uu}}{\sqrt{\left|\mathbf{r}_{,uu}\right|^2-J_{,u}^2}}\\ &=-2\frac{J_{,u}}{J}\kappa+\frac{\mathbf{r}_{,uu}\cdot\mathbf{r}_{uuu}-J_{,u}J_{,uu}}{J^4\kappa} \end{eqnarray} \]
Torsion measures the deviation from a planar curve:
\[ \tau=\frac{\mathbf{r}_{,u}\cdot\left(\mathbf{r}_{uu}\times\mathbf{r}_{uuu}\right)}{J^6\kappa^2} \]
Definition at line 108 of file ParametricCurve.h.
Member Function Documentation
◆ arcLength()
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inline |
Definition at line 182 of file ParametricCurve.h.
◆ get_j()
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inline |
The jacobian of the transformation r(u)->r(l):
\[J=\frac{d l}{d u}=\sqrt{\mathbf{r}_{,u}\cdot\mathbf{r}_{,u}} \]
Definition at line 124 of file ParametricCurve.h.
◆ get_jSquare()
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inline |
Definition at line 134 of file ParametricCurve.h.
◆ get_ju()
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inline |
The first derivative of the jacobian:
\[J_{,u}=\frac{\mathbf{r}_{,u}\cdot\mathbf{r}_{,uu}}{J} \]
Definition at line 144 of file ParametricCurve.h.
◆ get_kappa()
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inline |
the scalar curvature
The scalar curvature at u:
\[\kappa=\left|\mathbf{r}_{,ll}\right| \]
Definition at line 170 of file ParametricCurve.h.
◆ get_rl()
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inline |
Definition at line 139 of file ParametricCurve.h.
◆ get_rll()
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inline |
The curvature vector at u:
\[\mathbf{r}_{,ll}=\frac{\mathbf{r}_{,uu}}{J^2}-J_{,u}\frac{\mathbf{r}_{,u}}{J^3} \]
Definition at line 155 of file ParametricCurve.h.
◆ rm()
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inline |
Definition at line 190 of file ParametricCurve.h.
◆ rm_integrand()
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inline |
Definition at line 196 of file ParametricCurve.h.
The documentation for this class was generated from the following file:
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