Tangent Vectors

Differential Geometry · Axiom Academy

Understanding how curves move through space and the vectors that describe their direction 1. The Tangent Vector T = α'/|α'| For a smooth curve α(t) , the derivative α'(t) gives the velocity vector. The unit tangent vector is obtained by normalizing this velocity: This normalization ensures T has unit length at every point, capturing only the direction of motion, not the speed. 2. Unit Tangent Vector for Unit Speed Curves A curve is unit speed (or arc-length parametrized) when |α'(t)| = 1 for all t. In this special case: Unit speed parametrization simplifies many calculations because the tangent vector equals the velocity vector directly, and the parameter t represents actual distance traveled along the curve. 3. Geometric Interpretation of Tangent Direction The tangent vector T(t) points in the direction the curve is heading at parameter value t. As we move along the curve, T rotates to always point "forward" along the path. The tangent line at point α(t₀) is the line passing through that point in the direction of T(t₀): This line represents the best linear approximation to the curve near the point α(t₀). It's the limiting position of secant lines through α(t₀) and nearby points α(t₀ + h) as h → 0. 5. How the Tangent Changes Along the Curve As we move along a curve, the tangent vector T rotates. The rate of change of T measures how quickly the curve is bending:

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