Curvature Calculations

Differential Geometry · Axiom Academy

EXAMPLE Curvature Calculations Master the curvature formula through step-by-step calculations for fundamental curves Excellent work! You've completed these curvature calculations. Here's what we learned: Circle Curvature: For a circle of radius R, the curvature is constant at κ = 1/R. Larger circles have smaller curvature (are "less curved"). Helix Curvature: The curvature κ = a/(a² + b²) depends on both the radius a and the vertical spacing b. As the helix becomes more vertical (larger b), curvature decreases. General Formula: The formula κ = |r' × r''| / |r'|³ works for any parametrized curve. The cross product captures how the curve is turning in space. Practical Insight: Parabolas and ellipses have variable curvature that depends on position, unlike circles which have constant curvature. Understanding curvature is fundamental to differential geometry. Practice these calculations with different curves to build intuition!

This is the written version of the interactive lesson above. See the full Differential Geometry course.