Normal Vector


The normal vector, often simply called the "normal," to a surface is a vector which is perpendicular to the surface at a given point. When normals are considered on closed surfaces, the inward-pointing normal (pointing towards the interior of the surface) and outward-pointing normal are usually distinguished.

The unit vector obtained by normalizing the normal vector (i.e., dividing a nonzero normal vector by its vector norm) is the unit normal vector, often known simply as the "unit normal." Care should be taken to not confuse the terms "vector norm" (length of vector), "normal vector" (perpendicular vector) and "normalized vector" (unit-length vector).

The normal vector is commonly denoted N or n, with a hat sometimes (but not always) added (i.e., N^^ and n^^) to explicitly indicate a unit normal vector.

The normal vector at a point (x_0,y_0) on a surface z=f(x,y) is given by

 N=[f_x(x_0,y_0); f_y(x_0,y_0); -1],

where f_x=partialf/partialx and f_y=partialf/partialy are partial derivatives.

A normal vector to a plane specified by


is given by

 N=del f=[a; b; c],

where del f denotes the gradient. The equation of a plane with normal vector n=(a,b,c) passing through the point (x_0,y_0,z_0) is given by

 [a; b; c]·[x-x_0; y-y_0; z-z_0]=a(x-x_0)+b(y-y_0)+c(z-z_0)=0.

For a plane curve, the unit normal vector can be defined by


where T^^ is the unit tangent vector and phi is the polar angle. Given a unit tangent vector


with sqrt(u_1^2+u_2^2)=1, the normal is


For a plane curve given parametrically, the normal vector relative to the point (f(t),g(t)) is given by


To actually place the vector normal to the curve, it must be displaced by (f(t),g(t)).

For a space curve, the unit normal is given by


where T^^ is the tangent vector, s is the arc length, and kappa is the curvature. It is also given by


where B^^ is the binormal vector (Gray 1997, p. 192).

For a surface with parametrization x(u,v), the normal vector is given by


Given a three-dimensional surface defined implicitly by F(x,y,z)=0,

 n^^=(del F)/(sqrt(F_x^2+F_y^2+F_z^2)).

If the surface is defined parametrically in the form


define the vectors

 a=[x_phi; y_phi; z_phi]
 b=[x_psi; y_psi; z_psi].

Then the unit normal vector is


Let g be the discriminant of the metric tensor. Then


See also

Binormal Vector, Contact Angle, Curvature, Frenet Formulas, Multivariable Calculus, Norm, Tangent Vector, Torsion Explore this topic in the MathWorld classroom

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Gray, A. "Tangent and Normal Lines to Plane Curves." §5.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 108-111, 1997.

Referenced on Wolfram|Alpha

Normal Vector

Cite this as:

Weisstein, Eric W. "Normal Vector." From MathWorld--A Wolfram Web Resource.

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