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Tangent Plane and Normal Plane 📂Geometry

Tangent Plane and Normal Plane

Definition 1

A curve $\alpha$ is given.

  1. The plane $\text{span} \left\{ T, N \right\}$ perpendicular to $B$ is called the Osculating Plane.
  2. The plane $\text{span} \left\{ N, B \right\}$ perpendicular to $T$ is called the Normal Plane.
  3. The plane $\text{span} \left\{ B, T \right\}$ perpendicular to $N$ is called the Rectifying Plane.

Explanation

These planes should be considered as planes that move together while $\alpha (s)$ of $s$ is proceeding. Especially, since the normal plane is perpendicular to the tangent, one can imagine that the curve $\alpha$ always ‘pierces’ straight through the normal plane.


  1. Millman. (1977). Elements of Differential Geometry: p31. ↩︎