Curvature of Curves

Differential Geometry · Axiom Academy

Understanding how curves bend in space through curvature and the osculating circle Differential Geometry - Unit 1: Curves in Space Let r (t) be a parameterized curve and T be its unit tangent vector. The curvature is defined as: Here, s represents the arc length parameter. This definition captures the geometric idea that curvature measures how fast the curve is turning away from its tangent line. While the definition = |d T /ds| is conceptually clear, it's not always practical for computation. For a curve r (t) with arbitrary parameter t, we have a more useful formula: Where r is the first derivative and r is the second derivative with respect to the parameter t. The cross product r × r gives us a vector perpendicular to the osculating plane. For a plane curve y = f(x), this simplifies to a familiar form: At each point on a curve, there exists a unique circle that best approximates the curve's shape at that point - this is called the osculating circle (from the Latin "to kiss"). The osculating circle has three important properties: It passes through the point on the curve It has the same tangent as the curve at that point It has the same curvature as the curve at that point 4. Example: Curvature of a Circle Let's verify our formula works by computing the curvature of a circle of radius R, parameterized as: 5. Example: Curvature of a Helix Consider a circular helix with radius a and pitch 2 b:

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