## Normal Vector

The normal to a Plane specified by

 (1)

is given by
 (2)

The normal vector at a point on a surface is
 (3)

In the Plane, the unit normal vector is defined by

 (4)

where is the unit Tangent Vector and is the polar angle. Given a unit Tangent Vector
 (5)

with , the normal is
 (6)

For a function given parametrically by , the normal vector relative to the point is therefore given by
 (7) (8)

To actually place the vector normal to the curve, it must be displaced by .

In 3-D Space, the unit normal is

 (9)

where is the Curvature. Given a 3-D surface ,
 (10)

If the surface is defined parametrically in the form
 (11) (12) (13)

define the Vectors
 (14)

 (15)

Then the unit normal vector is
 (16)

Let be the discriminant of the Metric Tensor. Then
 (17)

Gray, A. Tangent and Normal Lines to Plane Curves.'' §5.5 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 85-90, 1993.