In differential geometry, the **osculating plane** is a concept related to curves in three-dimensional space. Specifically, the osculating plane at a given point on a curve is the plane that best approximates the curve near that point. The osculating plane can be defined using the following components: 1. **Tangent Vector**: At any point on a smooth curve, the tangent vector represents the direction in which the curve is moving.
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The osculating plane of a regular space curve at a point of nonzero curvature is the affine plane through the curve point spanned by its unit tangent vector and principal normal vector . Its normal is the binormal vector , so the plane is described byEquivalently, any regular parameter gives the span of and when these vectors are linearly independent. The second-order Taylor expansion then lies in this plane, which explains the name: it captures the curve's instantaneous bending.
A planar circle has its own containing plane as its osculating plane everywhere. For the helix at , the point is and a normal is , giving . At zero curvature the ordinary Frenet frame does not define a unique osculating plane.