Question

How is curvature formula derived?

Answer 1

Answer:

If the curve is a circle with radius R, i.e. x = R cost, y = R sin t, then k = 1/R, i.e., the (constant) reciprocal of the radius. In this case the curvature is positive because the tangent to the curve is rotating in a counterclockwise direction.

Answer 2

Answer:

The curvature of a circle is equal to the reciprocal of its radius. The binormal vector at t is defined as ⇀B(t)=⇀T(t)×⇀N(t), where ⇀T(t) is the unit tangent vector. The Frenet frame of reference is formed by the unit tangent vector, the principal unit normal vector, and the binormal vector.

If the curve is a circle with radius R, i.e. x = R cost, y = R sin t, then k = 1/R, i.e., the (constant) reciprocal of the radius. In this case the curvature is positive because the tangent to the curve is rotating in a counterclockwise direction.