Curvature of a parametric curve
Analytical expression
Let \( s: \mathbb R \rightarrow \mathbb R^3 \) be a vector valued function representing a parametric curve \( s(t) \) with \( t \in \mathbb R \) the curve's parameter; the curvature of \(s\) is given by \( \kappa: \mathbb R \rightarrow \mathbb R \) as follows:
\[ \kappa(t) = \frac{ \| s'(t) \times s''(t) \| }{ \| s'(t) \|^3 } \]
Where '×' is the cross product, \( s'(t) \) speed (velocity) at \( t \), \( s''(t) \) the acceleration, and \( \| \ \| \) the Euclidean norm of a vector.
2D version
The 2D cross product is not a vector like the 3D cross product but a real value \(\mathbb R\). In addition, the norm of the 2D cross product is the value of the cross product itself: \(\| s' \times s'' \| = s_x' s''_y - s_y's_x'' \). Finally the 2D cross product can be expressed as the determinant \(\left | \phantom{x} \right | \) of the 2D matrix below:
\[ \kappa(t) = \frac{ \left | \begin{matrix} s_x'(t) & s_x''(t) \\ s_y'(t) & s_y''(t) \\ \end{matrix} \right | }{ \| s'(t) \|^3 } \]Numerical computation
If you don't have the formula (i.e. analytical expression) of \(s(t)\), you can numerically compute it using finite differences:
with \(h\) "small" (ex \( h < 0.0001 \) )
Shader code
Glsl code to visualize the curvature:
shadertoy.com/view/Mlf3zl
// Under MIT License
// Copyright © 2015 Inigo Quilez
// Computes the curvature of a parametric curve f(x) as
// c(f) = | f' x f''| / |f'|^3
// More info here: https://en.wikipedia.org/wiki/Curvature
vec3 a, b, c, m, n;
// parametric curve value s(t):
vec3 mapD0(float t){
return 0.25 + a*cos(t+m)*(b+c*cos(t*7.0+n));
}
// curve derivative (velocity) s'(t)
vec3 mapD1(float t){
return -7.0*a*c*cos(t+m)*sin(7.0*t+n) - a*sin(t+m)*(b+c*cos(7.0*t+n));
}
// curve second derivative (acceleration) s''(t)
vec3 mapD2(float t){
return 14.0*a*c*sin(t+m)*sin(7.0*t+n) - a*cos(t+m)*(b+c*cos(7.0*t+n)) - 49.0*a*c*cos(t+m)*cos(7.0*t+n);
}
//----------------------------------------
float curvature( float t ){
vec3 r1 = mapD1(t); // first derivative
vec3 r2 = mapD2(t); // second derivative
return length(cross(r1,r2)) / pow(length(r1),3.0);
}
float curvature_reciprocal( float t ){
vec3 r1 = mapD1(t); // first derivative
vec3 r2 = mapD2(t); // second derivative
return pow(length(r1),3.0) / length(cross(r1,r2));
}
3d version: https://www.shadertoy.com/view/XlfXR4
Deriving the formula
A separate on article on how to interpret and derive the curvature formula where I do an in-depth explanation.
Related
- https://tutorial.math.lamar.edu/classes/calciii/curvature.aspx
- Khan Academy video series on curvature
- Khan Academy summary article on curvature
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