The Curvature Formula

Calculus 3 · Axiom Academy

Quantifying the bendiness of a curve with — how fast the direction turns, per unit of distance travelled. 1. The Intuition Behind Curvature Curvature measures how quickly a curve changes direction . A straight line doesn't bend at all (zero curvature), while a tight circle bends constantly (high curvature). The cleanest picture: at every point, one circle hugs the curve better than any other — the osculating circle . Curvature is one divided by its radius. Gentle curves, slow direction changes. The best-fitting circle is huge, so is large and is small. Tight curves, rapid direction changes. The best-fitting circle is small, so is small and is large. Curvature is the reciprocal of the radius of curvature 2. The Unit Tangent Vector T(t) Given a curve , the unit tangent vector points in the direction the curve is travelling and has length 1. It is found by normalizing the velocity vector — dividing out the speed, so that only the direction survives. Points along the curve, but its length is the speed — it grows and shrinks as you drive faster or slower. A positive number, not a direction. Dividing by it is what strips the speed information away. Same direction as , always length exactly 1. Pure heading, no speed. Only a unit vector can turn without also stretching — so any change in is a change in direction . 3. Rate of Change of Direction: T (t)

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