Curvature of a plane curve is defined in an elementary integral calculus course as the magnitude of the rate of change of the angle a tangent makes with the horizontal. Some texts for this course go on to define the osculating circle as the tangent circle whose radius equals the curvature. The center of such a circle is called the center of curvature.
The orbit of the center of curvature is a curve called the evolute of the original curve. During a webinar on evolutes, I was asked about the osculating sphere, a term with which I was completely unfamiliar. So, this present webinar is a distillation of my search for the connections between the osculating plane, circle, and sphere for a space curve.
After a quick reminder of the osculating circle and center of curvature for a plane curve, we show that these constructs and the curvature itself can be found by searching for the circle that makes second-order contact with the curve. The osculating sphere, however, is that sphere making third-order contact with the space curve. We give a general derivation for the osculating sphere, and then an example.
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