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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a normal is an
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
such as a line,
ray Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (gr ...
, or vector that is
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
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 In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
) or its length may represent the curvature of the object (a '' curvature vector''); its algebraic sign may indicate sides (interior or exterior). In three dimensions, a surface normal, or simply normal, to a surface at point P is a vector
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
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 In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
, the normal vector, etc. The concept of normality generalizes to orthogonality (
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...
s). The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
. The normal vector space or normal space of a manifold at point P is the set of vectors which are orthogonal to the tangent space at P. Normal vectors are of special interest in the case of smooth curves and smooth surfaces. The normal is often used in
3D computer graphics 3D computer graphics, or “3D graphics,” sometimes called CGI, 3D-CGI or three-dimensional computer graphics are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for th ...
(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 foot of a normal at a point of interest ''Q'' (analogous to the foot of a perpendicular) can be defined at the point ''P'' on the surface where the normal vector contains ''Q''. The ''
normal distance In geometry, the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both. The distance from a point to a line is the distance to the nearest point on that line. Th ...
'' of a point ''Q'' to a curve or to a surface is the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore ...
between ''Q'' and its foot ''P''.


Normal to surfaces in 3D space


Calculating a surface normal

For a convex
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
(such as a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
), a surface normal can be calculated as the vector
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
of two (non-parallel) edges of the polygon. For a plane given by the equation ax + by + cz + d = 0, the vector \mathbf n = (a, b, c) is a normal. For a plane whose equation is given in parametric form \mathbf(s,t) = \mathbf_0 + s \mathbf + t \mathbf, where \mathbf_0 is a point on the plane and \mathbf, \mathbf are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both \mathbf and \mathbf, which can be found as the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
\mathbf=\mathbf\times\mathbf. If a (possibly non-flat) surface S in 3D space \R^3 is parameterized by a system of curvilinear coordinates \mathbf(s, t) = (x(s, t), y(s, t), z(s, t)), with s and t real variables, then a normal to ''S'' is by definition a normal to a tangent plane, given by the cross product of the
partial derivative 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 (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s \mathbf=\frac \times \frac. 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 In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
\mathbf = \nabla F(x, y, z). since the gradient at any point is perpendicular to the level set S. For a surface S in \R^3 given as the graph of a function z = f(x, y), an upward-pointing normal can be found either from the parametrization \mathbf(x,y)=(x,y,f(x,y)), giving \mathbf = \frac \times \frac = \left(1,0,\tfrac\right) \times \left(0,1,\tfrac\right) = \left(-\tfrac, -\tfrac,1\right); or more simply from its implicit form F(x, y, z) = z-f(x,y) = 0, giving \mathbf = \nabla F(x, y, z) = \left(-\tfrac, -\tfrac, 1 \right). Since a surface does not have a tangent plane at a
singular point Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
, 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 normal

The normal to a (hyper)surface is usually scaled to have
unit length Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (a ...
, 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 In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its ...
.


Transforming normals

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 \mathbf, we can determine the matrix \mathbf that transforms a vector \mathbf perpendicular to the tangent plane \mathbf into a vector \mathbf^ perpendicular to the transformed tangent plane \mathbf, by the following logic: Write n′ as \mathbf. We must find \mathbf. \begin W\mathbb n \text M\mathbb t \quad \, &\text \quad 0 = (W \mathbb n) \cdot (M \mathbb t) \\ &\text \quad 0 = (W \mathbb)^\mathrm (M \mathbb) \\ &\text \quad 0 = \left(\mathbb^\mathrm W^\mathrm\right) (M \mathbb) \\ &\text \quad 0 = \mathbb^\mathrm \left(W^\mathrm M\right) \mathbb \\ \end Choosing \mathbf such that W^\mathrm M = I, or W = (M^)^\mathrm, will satisfy the above equation, giving a W \mathbb n perpendicular to M \mathbb t, or an \mathbf^ perpendicular to \mathbf^, 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 space

For an (n-1)-dimensional hyperplane in n-dimensional space \R^n given by its parametric representation \mathbf\left(t_1, \ldots, t_\right) = \mathbf_0 + t_1 \mathbf_1 + \cdots + t_\mathbf_, where \mathbf_0 is a point on the hyperplane and \mathbf_i for i = 1, \ldots, n - 1 are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector \mathbf n in the
null space In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kern ...
of the matrix P = \begin\mathbf_1 & \cdots &\mathbf_\end, meaning P\mathbf n = \mathbf 0. 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 a_1x_1+\cdots+a_nx_n = c, then the vector \mathbb = \left(a_1, \ldots, a_n\right) is a normal. The definition of a normal to a surface in three-dimensional space can be extended to (n - 1)-dimensional hypersurfaces in \R^n. A hypersurface may be locally defined implicitly as the set of points (x_1, x_2, \ldots, x_n) satisfying an equation F(x_1, x_2, \ldots, x_n) = 0, where F is a given scalar function. If F is
continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
then the hypersurface is a differentiable manifold in the
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural a ...
of the points where the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
is not zero. At these points a normal vector is given by the gradient: \mathbb n = \nabla F\left(x_1, x_2, \ldots, x_n\right) = \left( \tfrac, \tfrac, \ldots, \tfrac \right)\,. The normal line is the one-dimensional subspace with basis \.


Varieties defined by implicit equations in ''n''-dimensional space

A differential variety defined by implicit equations in the n-dimensional space \R^n is the set of the common zeros of a finite set of differentiable functions in n variables f_1\left(x_1, \ldots, x_n\right), \ldots, f_k\left(x_1, \ldots, x_n\right). The
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
of the variety is the k \times n matrix whose i-th row is the gradient of f_i. By the implicit function theorem, the variety is a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
in the neighborhood of a point where the Jacobian matrix has rank k. At such a point P, the normal vector space is the vector space generated by the values at P of the gradient vectors of the f_i. In other words, a variety is defined as the intersection of k 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 P of the variety is the affine subspace passing through P and generated by the normal vector space at P. These definitions may be extended to the points where the variety is not a manifold.


Example

Let ''V'' be the variety defined in the 3-dimensional space by the equations x\,y = 0, \quad z = 0. This variety is the union of the x-axis and the y-axis. At a point (a, 0, 0), where a \neq 0, the rows of the Jacobian matrix are (0, 0, 1) and (0, a, 0). Thus the normal affine space is the plane of equation x = a. Similarly, if b \neq 0, the '' normal plane'' at (0, b, 0) is the plane of equation y = b. At the point (0, 0, 0) the rows of the Jacobian matrix are (0, 0, 1) and (0, 0, 0). Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the z-axis.


Uses

* Surface normals are useful in defining
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one ...
s of vector fields. * Surface normals are commonly used in
3D computer graphics 3D computer graphics, or “3D graphics,” sometimes called CGI, 3D-CGI or three-dimensional computer graphics are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for th ...
for
lighting Lighting or illumination is the deliberate use of light to achieve practical or aesthetic effects. Lighting includes the use of both artificial light sources like lamps and light fixtures, as well as natural illumination by capturing dayl ...
calculations (see Lambert's cosine law), often adjusted by normal mapping. * Render layers containing surface normal information may be used in
digital compositing Digital compositing is the process of digitally assembling multiple images to make a final image, typically for print, motion pictures or screen display. It is the digital analogue of optical film compositing. Mathematics The basic operation us ...
to change the apparent lighting of rendered elements. * In
computer vision Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the human ...
, the shapes of 3D objects are estimated from surface normals using photometric stereo.


Normal in geometric optics

The is the outward-pointing ray
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
to the surface of an
optical medium An optical medium is material through which light and other electromagnetic waves propagate. It is a form of transmission medium. The permittivity and permeability of the medium define how electromagnetic waves propagate in it. Properties The ...
at a given point. 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 also

* * * * *


References


External links

* * A
explanation of normal vectors
from Microsoft's MSDN * Clear pseudocode fo
calculating a surface normal
from either a triangle or polygon. {{Authority control Surfaces Vector calculus 3D computer graphics Orthogonality