Concepts in Curvature

Step 4: Find k and R

We're almost there! Now that we know $dT/ds$ , we can finally find the curvature of the curve. The curvature (shown as the greek letter Kappa) is defined as the magnitude of $dT/ds$ , which we already know from the last step.

\kappa = \left\lVert\frac{dT}{ds}\right\rVert = \frac {\left\lVert\frac{dT}{dt}\right\rVert} {\left\lVert\frac{ds}{dt}\right\rVert}

The last thing we need to do is find the radius of curvature ( $R$ ), which is defined as the reciprocal of the curvature. This is the radius of the circle that best approximates the curve at a given point.

R = \frac{1}{\kappa}

Congrats on making it this far! You now know how to find the curvature of a curve, or at least the concepts behind it. You can check out some pre-made examples of the curvature of different parametric curves in the examples section of this website, or enter your own in the playground section.

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$t = 0.000$ $\kappa = 1.500$ $R = 0.667$