Osculating Circle and Osculating Sphere

Osculating Circle

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Osculating Sphere

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Center of Osculating Circle

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In the figure below, the locus of center of osculating circle is shown

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Properties on Osculating Circle

Center of Osculating Sphere

In the figure below, locus of center of osculating sphere is shown.

Properties on Osculating Sphere

Theorems

1. Show that principal normal to a curve is normal to the locus of center of curvature at points where curvature is stationary.
   or
   Show that principal normal to a curve is normal to the locus of center of osculating circle at points where curvature is stationary.
   or
   Show that principal normal to a curve is normal to the locus of center of circle of curvature at points where curvature is stationary.
   center of curvature =center of osculating circle=center of circle of curvature

2. Show that radius of osculating circle of a circular helix is equal to radius of osculating sphere.

3. Show that tangent to the locus of center of osculating sphere is parallel to the binormal of the curve at corresponding points.

4. Show that for a curve of constant curvature center of osculating sphere coincides with center of osculating circle.

5. Show that tangent to the locus of center of osculating sphere passes through the center of osculating circle.

6. If a curve lies on a sphere, show that \( \rho \) and \( \sigma \) are connected by the relation \( \rho \tau + \frac{d}{ds}(\sigma \rho')=0 \)

   or

   Show that necessary and sufficient condition for a curve to lie on a surface of sphere is \( \rho \tau + \frac{d}{ds}(\sigma \rho')=0 \)

Relation of locus of Osculating Circle and Osculating Sphere