Surface normal

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"Normal vector" redirects here. For a normalized vector, or vector of length one, see unit vector.
A polygon and two of its normal vectors.
A normal to a surface at a point is the same as a normal to the tangent plane to that surface at that point.

A surface normal, or simply normal, to a flat surface is a vector which is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality.

In the two-dimensional case, a normal line perpendicularly intersects the tangent line to a curve at a given point.

Contents

Calculating a surface normal

For a polygon (such as a triangle), a surface normal can be calculated as the vector cross product of two (non-parallel) edges of the polygon.

For a plane given by the equation ax + by + cz = d, the vector (a,b,c) is a normal. For a plane given by the equation r = a + αb + βc, where a is a vector to get onto the plane and b and c are non-parallel vectors lying on the plane, the normal to the plane defined is given by b × c (the cross product of the vectors lying on the plane).

If a (possibly non-flat) surface S is parametrized by a system of curvilinear coordinates x(s, t), with s and t real variables, then a normal is given by the cross product of the partial derivatives

{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}.

If a surface S is given implicitly, as the set of points (x,y,z) satisfying F(x,y,z) = 0, then, a normal at a point (x,y,z) on the surface is given by the gradient

\nabla F(x, y, z).

If a surface does not have a tangent plane at a point, it does not have a normal at that point either. For example, a cone does not have a normal at its tip nor does it have a normal along the edge of its base. However, the normal to the cone is defined almost everywhere. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous.

n-dimensional surfaces

The definition of a normal to a two-dimensional surface in three-dimensional space can be extended to n − 1-dimensional "surfaces" in n-dimensional space. Such a hypersurface may be defined implicitly as the set of points (x_1, x_2, \ldots, x_n) satisfying the equation F(x_1, x_2, \ldots x_n) = 0. If F is continuously differentiable, then the surface obtained is a differentiable manifold, and its surface normal is given by the gradient of F,

\nabla F(x_1, x_2, \ldots, x_n) = \left( \tfrac{\partial F}{\partial x_1}, \tfrac{\partial F}{\partial x_2}, \ldots, \tfrac{\partial F}{\partial x_n} \right) .

Uniqueness of the normal

A vector field of normals to a surface.

A normal to a surface does not have a unique direction; the vector pointing in the opposite direction of a surface normal is also a surface normal. For a surface which is the topological boundary of a set in three dimensions, one can distinguish between the inward-pointing normal and outer-pointing normal, which can help define the normal in a unique way. For an oriented surface, the surface normal is usually determined by the right-hand rule. If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector.

Uses

Normal in geometric optics

Diagram of specular reflection

The normal is an imaginary line perpendicular to the surface1 of an optical medium. The word normal is used here in the mathematical sense, meaning perpendicular. In reflection of light, the angle of incidence is the angle between the normal and the incident ray. The angle of reflection is the angle between the normal and the reflected ray.

References

  1. ^ "The Law of Reflection" (HTML). The Physics Classroom Tutorial. Retrieved on 2008-03-31.

External links

Wikipedia content modification information:

  • This page was last modified on 19 October 2008, at 02:58.

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