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 that 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.

The normal is often used in computer graphics to determine a surface's orientation toward a light source for flat shading, or the orientation of each of the corners (vertices) to mimic a curved surface with Phong shading.

[edit] Calculating a surface normal

For a convex 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 $a x + b y + c z = d$ , 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).

For a hyperplane in n+1 dimensions, given by the equation r = a₀ + α₁a₁ + α₂a₂ + ... + α_na_n, where a₀ is a vector to get onto the hyperplane and a_i for i = 1, ... , n are non-parallel vectors lying on the hyperplane, the (unscaled) normal to the hyperplane can be approximated by $(A A T + b b T) - 1 b$ where A = [a₁, a₂, ... , a_n] and b is an arbitrary vector in the space not in the linear span of a_i.

If a (possibly non-flat) surface S is parameterized 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.

[edit] Hypersurfaces in n-dimensional space

The definition of a normal to a surface in three-dimensional space can be extended to $n - 1$ -dimensional hypersurfaces in a $n$ -dimensional space. A hypersurface may be locally defined implicitly as the set of points $\scriptstyle(x_1, x_2, \ldots, x_n)$ satisfying an equation $\scriptstyle F(x_1, x_2, \ldots x_n) = 0$ , where $F$ is a given scalar function . If $F$ is continuously differentiable, then the hypersurface obtained is a differentiable manifold, and its hypersurface normal can be obtained from the gradient of $F$ , in the case it is not null, by the following formula

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

[edit] 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.

[edit] Uses

Surface normals are essential in defining surface integrals of vector fields.
Surface normals are commonly used in 3D computer graphics for lighting calculations; see Lambert's cosine law.
Render layers containing surface normal information may be used in Digital compositing to change the apparent lighting of rendered elements.

[edit] Normal in geometric optics

Diagram of specular reflection

The normal is an imaginary line perpendicular to the surface^[1] 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.

[edit] See also

Geometric algebra, which describes a normal directly as a bivector rather than as a normal pseudovector.

[edit] References

^ "The Law of Reflection" (HTML). The Physics Classroom Tutorial. http://www.glenbrook.k12.il.us/gbssci/phys/Class/refln/u13l1c.html. Retrieved 2008-03-31.

[edit] External links

An explanation of normal vectors from Microsoft's MSDN
Some instructional videos and a free tool for performing normals-based relighting.

[0] "The Law of Reflection" (HTML). The Physics Classroom Tutorial. http://www.glenbrook.k12.il.us/gbssci/phys/Class/refln/u13l1c.html. Retrieved 2008-03-31.

[1]

Surface normal

From Wikipedia, the free encyclopedia

Contents

[edit] Calculating a surface normal

[edit] Hypersurfaces in n-dimensional space

[edit] Uniqueness of the normal

[edit] Uses

[edit] Normal in geometric optics

[edit] See also

[edit] References

[edit] External links

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