# Normal

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to a curve (or surface) at a point of it

A straight line passing through the point and perpendicular to the tangent (or tangent plane) of the curve (or surface) at this point. A smooth plane curve has at every point a unique normal situated in the plane of the curve. If a curve in a plane is given in rectangular coordinates by an equation , then the equation of the normal to the curve at has the form A curve in space has infinitely many normals at every point of it. These fill a certain plane (the normal plane). The normal lying in the osculating plane is called the principal normal; the one perpendicular to the osculating plane is called the binormal.

The normal at to a surface given by an equation is defined by If the equation of the surface has the form , then the parametric representation of the normal is The notion of a normal obviously extends to -dimensional submanifolds of Euclidean -space , giving an -dimensional affine subspace as the normal -plane to the manifold at the corresponding point. For submanifolds of (pseudo-) Riemannian manifolds, the normal planes are considered as subspaces of the tangent space of the ambient space, where orthogonality is defined by means of the (ambient) (pseudo-) Riemannian metric. See also Normal bundle; Normal plane; Normal space (to a surface).