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What is Curvature? (3D)

This animation, created using MATLAB, illustrates the curvature using a color spectrum. While it requires a vector function and its first and second derivative, the curvature of a parametric curve in 3D is a scalar-valued function shown in black on the right. It can be interpreted as a relative measure of how sharply the curve bends at some particular point. For the twisted cubic, the sharpest bend occurs at the origin, which is also the peak of the curvature graph. For the ellipse, the sharpest bend occurs at the ends that are stretched by a factor of 2 (in the x direction). For the "chandelier", the curvature is highest near the z axis when the curve is rapidly turning as indicated by the red/orange colors.