MATLAB Code Implementation for Curvature Calculation

Reference MATLAB code for curvature calculation:

function curvature = calculate_curvature(x, y, z)

% Calculate first and second derivatives using numerical differentiation
% The gradient function computes central differences for interior points
% and one-sided differences for boundary points
dx = gradient(x);
dy = gradient(y);
dz = gradient(z);
ddx = gradient(dx);
ddy = gradient(dy);
ddz = gradient(dz);

% Implement curvature formula for 3D curves
% The numerator computes the squared magnitude of the cross product between
% first and second derivatives (|r' × r''|^2)
numerator = (dy .* ddz - dz .* ddy).^2 + (dz .* ddx - dx .* ddz).^2 + (dx .* ddy - dy .* ddx).^2;
% The denominator contains the cubed magnitude of the first derivative (|r'|^3)
denominator = (dx.^2 + dy.^2 + dz.^2).^1.5;
% Final curvature calculation using the standard formula: k = |r' × r''| / |r'|^3
curvature = numerator ./ denominator;

% Return curvature values for all points along the curve
return

This code efficiently computes curvature values for discrete curve points. Curvature measures how rapidly a curve deviates from a straight line at a given point, which is crucial for analyzing curve geometry, shape characteristics, and motion trajectories. The implementation uses vectorized operations for optimal performance. For detailed mathematical background and applications, refer to relevant literature in differential geometry and physics.