Curves and Motion

Differential Geometry · Axiom Academy

When a particle moves through space, it traces out a curve. The physics of that motion tells us about the geometry of the curve. Watch as a particle moves along a circular path. Click anywhere inside the circle to start the motion. Adjust the speed slider to change how fast the particle moves. Notice the blue arrow showing the particle's velocity. Compare motion on different curves: a circle and a wavy path. Click on each curve to see how the particle moves. Drag the time slider to move the particle to any position along the spiral path. See how position, velocity, and acceleration all depend on time. When a particle moves through space, its path forms a curve. The curve is described by a position vector r(t) that depends on time. The velocity vector v(t) = r'(t) is always tangent to the curve. It tells us both the speed and direction of motion. The acceleration vector a(t) = r''(t) points toward where the curve is bending. Its magnitude relates to the curvature—how sharply the curve turns. Differential geometry studies curves using calculus. Physical concepts like velocity and acceleration become geometric tools for understanding shape, direction, and curvature.

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