Surface normal

"Normal vector" redirects here. For a normalized vector, or vector of length one, see unit vector. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length

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. The intuitive idea of flatness is important in several fields In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent 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. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since 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. In Physics, a force is whatever can cause an object with Mass to Accelerate. The concept of normality generalizes to orthogonality. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i

1 Calculating a surface normal
2 n-dimensional surfaces
3 Uniqueness of the normal
4 Uses
5 Normal in geometric optics
6 References
7 External links

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. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

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).

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. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point Curvilinear coordinates are a Coordinate system for the Euclidean space based on some transformation that converts the standard Cartesian coordinate system to a coordinate In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant In Mathematics, an implicit function is a generalization for the concept of a function in which the Dependent variable has not been given "explicitly" In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar For example, a cone does not have a normal at its tip nor does it have a normal along the edge of its base. A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex However, the normal to the cone is defined almost everywhere. In Measure theory (a branch of Mathematical analysis) one says that a property holds almost everywhere if the set of elements for which the property does In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous. In Mathematics, more specifically in Real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions

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. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar

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 a different notion of boundary related to Manifolds see that article For an oriented surface, the surface normal is usually determined by the right-hand rule. A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back For the related yet different principle relating to electromagnetic coils see Right hand grip rule. If the normal is constructed as the cross product of tangent vectors (as described in the text above), it is a pseudovector. In Physics and Mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation but gains an

Uses

Surface normals are essential in defining surface integrals of vector fields. In Mathematics, a surface integral is a Definite integral taken over a Surface (which may be a curved set in Space) it can be thought In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space.
Surface normals are commonly used in 3D computer graphics for lighting calculations; see Lambert's cosine law. 3D computer graphics (in contrast to 2D computer graphics) are graphics that use a three-dimensional representation of geometric data that is stored in the computer Lighting includes both artificial Light sources such as lamps and natural illumination of interiors from Daylight. See also Lambertian reflectance Lambert's cosine law in Optics says that the Radiant intensity observed from a " Lambertian "
Render layers containing surface normal information may be used in Digital compositing to change the apparent lighting of rendered elements. What are Render Passes? When creating Computer-generated imagery or 3D computer graphics, final scenes appearing in movies and television productions are usually Digital compositing is the process of digitally assembling multiple images to make a final image typically for print motion pictures or screen display

Normal in geometric optics

Diagram of specular reflection

The normal is an imaginary line perpendicular to the surface^[1] of an optical medium. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent An optical medium is material through which Electromagnetic waves propagate 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. Reflection is the change in direction of a Wave front at an interface between two different media so that the wave front returns into the medium from which Angle of incidence is a measure of deviation of something from "straight on" for example in the approach of a ray to a surface or the angle In Optics, a ray is an idealized narrow Beam of light. Rays are used to model the propagation of Light through an optical system by dividing the real light The angle of reflection is the angle between the normal and the reflected ray. Reflection is the change in direction of a Wave front at an interface between two different media so that the wave front returns into the medium from which In Optics, a ray is an idealized narrow Beam of light. Rays are used to model the propagation of Light through an optical system by dividing the real light

References

^ The Law of Reflection (HTML). The Physics Classroom Tutorial. Retrieved on 2008-03-31. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common Events 307 - After divorcing his wife Minervina, Constantine marries Fausta, the daughter of the retired Roman Emperor

External links

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

Dictionary

-noun

(differential geometry) A unit vector at any given point P of a surface M which is perpendicular to the tangent plane T_P(M) of M at P. For a surface with parametrization <math> \vec x(u,v) </math>, it is given by

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