📂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.

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• $T,N,B$ represents the Tangent, Normal, Binormal respectively.
• $\text{span}$ represents the space generated by vectors.

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.