In
geometry, 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
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
,
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 can ...
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 vec ...
) 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
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 can ...
to the
tangent plane of the surface at P. The word "normal" is also used as an adjective: a
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
''normal'' to a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, the ''normal'' component of a
force, the normal vector, etc. The concept of normality generalizes to
orthogonality
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
(
right angles).
The concept has been generalized to
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
s 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
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 t ...
(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 o ...
between ''Q'' and its foot ''P''.
Normal to surfaces in 3D space
Calculating a surface normal
For a
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polyto ...
polygon (such as a
triangle), 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 is d ...
of two (non-parallel) edges of the polygon.
For a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
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
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 is d ...
If a (possibly non-flat) surface
in 3D space
is
parameterized by a system of
curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally i ...
with
and
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
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
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines conn ...
. In general, it is possible to define a normal almost everywhere for a surface that is
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there ex ...
.
Choice of normal
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
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term bound ...
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
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.
Most of t ...
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
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 space
For an
-dimensional
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperp ...
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
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
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
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
s 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
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 it ...
then the hypersurface is a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
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 space
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 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 ...
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
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
passing through
and generated by the normal vector space at
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
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.
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 m ...
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 t ...
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 dayligh ...
calculations (see
Lambert's cosine law
In optics, Lambert's cosine law says that the radiant intensity or luminous intensity observed from an ideal diffusely reflecting surface or ideal diffuse radiator is directly proportional to the cosine of the angle ''θ'' between the direction ...
), 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
Photometric stereo is a technique in computer vision for estimating the surface normals of objects by observing that object under different lighting conditions. It is based on the fact that the amount of light reflected by a surface is dependent ...
.
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 can ...
to the surface of an
optical medium at a given point.
In
reflection of light, the
angle of incidence and the
angle of reflection
Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves. The ...
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 vectorsfrom Microsoft's MSDN
* Clear pseudocode fo
calculating a surface normalfrom either a triangle or polygon.
{{Authority control
Surfaces
Vector calculus
3D computer graphics
Orthogonality