Curvature Intuition Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

The osculating circle (circle of curvature) at a point on a curve has radius

\rho

. If the curvature at point

P

\kappa = 0.4

^{-1}

, find

\rho

. If the curvature doubles, what happens to

\rho

1
Step 1: $\rho = \dfrac{1}{\kappa} = \dfrac{1}{0.4} = 2.5$ cm.
2
Step 2: If $\kappa$ doubles to $0.8$ cm $^{-1}$ , then $\rho = \dfrac{1}{0.8} = 1.25$ cm. The radius halves.

Answer

\rho = 2.5

cm; doubling curvature halves

\rho

1.25

cm.

The osculating circle at a point is the best-fitting circle to the curve at that point. Its radius

\rho = 1/\kappa

is inversely proportional to curvature. Doubling curvature halves the radius of curvature.

About Curvature Intuition

A measure of how quickly a curve bends or deviates from being a straight line at a given point.

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