Normal (geometry)

  (Redirected from Surface normal)

In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one (a unit vector) or its length may represent the curvature of the object (a curvature vector); its algebraic sign may indicate sides (interior or exterior).

A polygon and its two normal vectors
A normal to a surface at a point is the same as a normal to the tangent plane to the surface at the same point.

In three dimensions, a surface normal, or simply normal, to a surface at point is a vector perpendicular to the tangent plane of the 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 (right angles).

The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point is the set of vectors which are orthogonal to the tangent space at Normal vectors are of special interest in the case of smooth curves and smooth surfaces.

The normal is often used in 3D computer graphics (notice the singular, as only one normal will be defined) to determine a surface's orientation toward a light source for flat shading, or the orientation of each of the surface's corners (vertices) to mimic a curved surface with Phong shading.

The normal distance of a point Q to a curve or to a surface is the Euclidean distance between Q and its perpendicular projection on the object (at the point P on the object where the normal contains Q). The normal distance is a type of perpendicular distance generalizing the distance from a point to a line and the distance from a point to a plane. It can be used for curve fitting and for defining offset surfaces.

Normal to surfaces in 3D spaceEdit

 
A curved surface showing the unit normal vectors (blue arrows) to the surface

Calculating a surface normalEdit

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   the vector   is a normal.

For a plane whose equation is given in parametric form

 
where   is a point on the plane and   are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both   and   which can be found as the cross product  

If a (possibly non-flat) surface   in 3-space   is parameterized by a system of curvilinear coordinates   with   and   real variables, then a normal to S is by definition a normal to a tangent plane, given by the cross product of the partial derivatives

 

If a surface   is given implicitly as the set of points   satisfying   then a normal at a point   on the surface is given by the gradient

 
since the gradient at any point is perpendicular to the level set  

For a surface   in   given as the graph of a function   an upward-pointing normal can be found either from the parametrization   giving

 
or more simply from its implicit form   giving   Since a surface does not have a tangent plane at a singular point, it has no well-defined normal at that point: for example, the vertex of a cone. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous.

Choice of normalEdit

 
A vector field of normals to a surface

The normal to a (hyper)surface is usually scaled to have unit length, but it does not have a unique direction, since its opposite is also a unit 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. For an oriented surface, the normal is usually determined by the right-hand rule or its analog in higher dimensions.

If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector.

Transforming normalsEdit

When applying a transform to a surface it is often useful to derive normals for the resulting surface from the original normals.

Specifically, given a 3×3 transformation matrix   we can determine the matrix   that transforms a vector   perpendicular to the tangent plane   into a vector   perpendicular to the transformed tangent plane   by the following logic:

Write n′ as   We must find  

 

Choosing   such that   or   will satisfy the above equation, giving a   perpendicular to   or an   perpendicular to   as required.

Therefore, one should use the inverse transpose of the linear transformation when transforming surface normals. The inverse transpose is equal to the original matrix if the matrix is orthonormal, that is, purely rotational with no scaling or shearing.

Hypersurfaces in n-dimensional spaceEdit

For an  -dimensional hyperplane in  -dimensional space   given by its parametric representation

 
where   is a point on the hyperplane and   for   are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector   in the null space of the matrix   meaning   That is, any vector orthogonal to all in-plane vectors is by definition a surface normal. Alternatively, if the hyperplane is defined as the solution set of a single linear equation   then the vector   is a normal.

The definition of a normal to a surface in three-dimensional space can be extended to  -dimensional hypersurfaces in   A hypersurface may be locally defined implicitly as the set of points   satisfying an equation   where   is a given scalar function. If   is continuously differentiable then the hypersurface is a differentiable manifold in the neighbourhood of the points where the gradient is not zero. At these points a normal vector is given by the gradient:

 

The normal line is the one-dimensional subspace with basis  

Varieties defined by implicit equations in n-dimensional spaceEdit

A differential variety defined by implicit equations in the  -dimensional space   is the set of the common zeros of a finite set of differentiable functions in   variables

 
The Jacobian matrix of the variety is the   matrix whose  -th row is the gradient of   By the implicit function theorem, the variety is a manifold in the neighborhood of a point where the Jacobian matrix has rank   At such a point   the normal vector space is the vector space generated by the values at   of the gradient vectors of the  

In other words, a variety is defined as the intersection of   hypersurfaces, and the normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point.

The normal (affine) space at a point   of the variety is the affine subspace passing through   and generated by the normal vector space at  

These definitions may be extended verbatim to the points where the variety is not a manifold.

ExampleEdit

Let V be the variety defined in the 3-dimensional space by the equations

 
This variety is the union of the  -axis and the  -axis.

At a point   where   the rows of the Jacobian matrix are   and   Thus the normal affine space is the plane of equation   Similarly, if   the normal plane at   is the plane of equation  

At the point   the rows of the Jacobian matrix are   and   Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the  -axis.

UsesEdit

Normal in geometric opticsEdit

 
Diagram of specular reflection

The normal ray is the outward-pointing ray perpendicular to the surface of an optical medium at a given point.[2] In reflection of light, the angle of incidence and the angle of reflection are respectively the angle between the normal and the incident ray (on the plane of incidence) and the angle between the normal and the reflected ray.

See alsoEdit

ReferencesEdit

  1. ^ Ying Wu. "Radiometry, BRDF and Photometric Stereo" (PDF). Northwestern University.
  2. ^ "The Law of Reflection". The Physics Classroom Tutorial. Archived from the original on April 27, 2009. Retrieved 2008-03-31.

External linksEdit