Surface normal

Encyclopedia

A

is a vector that is perpendicular

to that surface. A normal to a non-flat surface at a point

to that surface at

normal to a plane, the normal component of a force

, the

.

In the two-dimensional case, a

The

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

.

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 given by the equation

i.e.,

.

For a hyperplane

in

where

of

That is, any vector orthogonal to all in-plane vectors is by definition a surface normal.

If a (possibly non-flat) surface

by a system of curvilinear coordinates

variables, then a normal is given by the cross product of the partial derivative

s

If a surface

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, and (the surface) is a level set of .

For a surface

of the independent variables (e.g., ), its normal can be found in at least two equivalent ways.

The first one is obtaining its implicit form , from which the normal follows readily as the gradient

.

(Notice that the implicit form could be defined alternatively as;

these two forms correspond to the interpretation of the surface being oriented

upwards or downwards, respectively, as a consequence of the difference in the sign of the partial derivative .)

The second way of obtaining the normal follows directly from the gradient of the explicit form,;

by inspection,

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.

## Hypersurfaces in

The definition of a normal to a surface in three-dimensional space can be extended to -dimensional hypersurface

s in a -dimensional space. A

defined implicitly as the set of points satisfying an equation , where is a given scalar function

. If is continuously differentiable, then the hypersurface obtained is a differentiable manifold

, and its hypersurface normal can be obtained from the gradient

of , in the case it is not null, by the following formula

of a set in three dimensions, one can distinguish between the

, 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

.

resulting surface from the original normals. All points

to

So use the inverse transpose of the linear transformation (the upper 3x3 matrix) when transforming surface normals.

to the surface 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.

That Normal force will then Be perpendicular to the surface

**surface normal**, or simply**normal**, to a flat surfaceFlatness

Flatness may refer to:*Flatness *Flatness *Flatness *Flatness *Flatness , a geometrical tolerance required in certain manufacturing situations*Flatness...

is a vector that is perpendicular

Perpendicular

In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to that surface. A normal to a non-flat surface at a point

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

*P*on the surface is a vector perpendicular to the tangent planeTangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....

to that surface at

*P*. The word "normal" is also used as an adjective: a lineLine (geometry)

The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

normal to a plane, the normal component of a force

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

, the

**normal vector**, etc. The concept of**normality**generalizes to orthogonalityOrthogonality

Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...

.

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 graphicsComputer graphics

Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....

to determine a surface's orientation toward a light source for flat shading, or the orientation of each of the corners (vertices

Vertex (geometry)

In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...

) to mimic a curved surface with Phong shading

Phong shading

Phong shading refers to an interpolation technique for surface shading in 3D computer graphics. It is also called Phong interpolation or normal-vector interpolation shading. Specifically, it interpolates surface normals across rasterized polygons and computes pixel colors based on the interpolated...

.

## Calculating a surface normal

For a convexConvex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

polygon

Polygon

In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

(such as a triangle

Triangle

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 segments. A triangle with vertices A, B, and C is denoted ....

), a surface normal can be calculated as the vector cross product

Cross product

In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

of two (non-parallel) edges of the polygon.

For a plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

given by the equation , the vector is a normal.

For a plane given by the equation

- ,

i.e.,

**a**is a point on the plane and**b**and**c**are (non-parallel) vectors lying on the plane, the normal to the plane is a vector normal to both**b**and**c**which can be found as the cross productCross product

In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

.

For a hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

in

*n*+1 dimensions, given by the equation,where

**a**_{0}is a point on the hyperplane and**a**_{i}for*i*= 1, ... ,*n*are non-parallel vectors lying on the hyperplane, a normal to the hyperplane is any vector in the null spaceNull space

In linear algebra, the kernel or null space of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of n-dimensional Euclidean space...

of

*A*where*A*is given by- .

That is, any vector orthogonal to all in-plane vectors is by definition a surface normal.

If a (possibly non-flat) surface

*S*is parameterizedCoordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

by a system of curvilinear coordinates

Curvilinear coordinates

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 invertible at each point. This means that one can convert a point given...

**x**(*s*,*t*), with*s*and*t*realReal number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

variables, then a normal is given by the cross product of the partial derivative

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...

s

If a surface

*S*is given implicitlyImplicit function

The implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval...

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

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

since the gradient at any point is perpendicular to the level set, and (the surface) is a level set of .

For a surface

*S*given explicitly as a functionFunction (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

of the independent variables (e.g., ), its normal can be found in at least two equivalent ways.

The first one is obtaining its implicit form , from which the normal follows readily as the gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

.

(Notice that the implicit form could be defined alternatively as;

these two forms correspond to the interpretation of the surface being oriented

Orientability

In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...

upwards or downwards, respectively, as a consequence of the difference in the sign of the partial derivative .)

The second way of obtaining the normal follows directly from the gradient of the explicit form,;

by inspection,

- , where is the upward unit vector.

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

Cone (geometry)

A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

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

Almost everywhere

In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...

. In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous.

## Hypersurfaces in *n*-dimensional space

The definition of a normal to a surface in three-dimensional space can be extended to -dimensional hypersurfaceHypersurface

In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...

s in a -dimensional space. A

*hypersurface*may be locallyLocal property

In mathematics, a phenomenon is sometimes said to occur locally if, roughly speaking, it occurs on sufficiently small or arbitrarily small neighborhoods of points.-Properties of a single space:...

defined implicitly as the set of points satisfying an equation , where is a given scalar function

Scalar field

In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...

. If is continuously differentiable, then the hypersurface obtained is a differentiable manifold

Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

, and its hypersurface normal can be obtained from the gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

of , in the case it is not null, by the following formula

## Uniqueness of the normal

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 boundaryBoundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the 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 surfaceOrientability

In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...

, the surface normal is usually determined by the right-hand rule

Right-hand rule

In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century....

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

Pseudovector

In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image...

.

## Transforming normals

When applying a transform to a surface it is sometimes convenient to derive normals for theresulting surface from the original normals. All points

*P*on tangent plane are transformedto

*P′*. We want to find**n′**perpendicular to*P*. Let**t**be a vector on the tangent plane and*M*be the upper 3x3 matrix (translation part of transformation does not apply to normal or tangent vectors)._{l}So use the inverse transpose of the linear transformation (the upper 3x3 matrix) when transforming surface normals.

## Uses

- Surface normals are essential in defining surface integralSurface integralIn mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...

s of vector fieldVector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

s. - Surface normals are commonly used in 3D computer graphics3D computer graphics3D computer graphics are graphics that use a three-dimensional representation of geometric data that is stored in the computer for the purposes of performing calculations and rendering 2D images...

for lightingLightingLighting or illumination is the deliberate application of light to achieve some practical or aesthetic effect. Lighting includes the use of both artificial light sources such as lamps and light fixtures, as well as natural illumination by capturing daylight...

calculations; see Lambert's cosine lawLambert's cosine lawIn optics, Lambert's cosine law says that the radiant intensity observed from a Lambertian surface or a Lambertian radiator is directly proportional to the cosine of the angle θ between the observer's line of sight and the surface normal. A Lambertian surface is also known as an ideal diffusely...

. - Surface normals are often adjusted in 3D computer graphics3D computer graphics3D computer graphics are graphics that use a three-dimensional representation of geometric data that is stored in the computer for the purposes of performing calculations and rendering 2D images...

by normal mappingNormal mappingIn 3D computer graphics, normal mapping, or "Dot3 bump mapping", is a technique used for faking the lighting of bumps and dents. It is used to add details without using more polygons. A common use of this technique is to greatly enhance the appearance and details of a low polygon model by...

. - Render layersRender layers-What are Render Passes?:When creating computer-generated imagery or 3D computer graphics, final scenes appearing in movies and television productions are usually produced by Rendering more than one "layer" or "pass," which are multiple images designed to be put together through digital...

containing surface normal information may be used in Digital compositingDigital compositingDigital compositing is the process of digitally assembling multiple images to make a final image, typically for print, motion pictures or screen display...

to change the apparent lighting of rendered elements.

## Normal in geometric optics

The**normal**is an imaginary line perpendicularPerpendicular

In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

to the surface of an optical medium. The word normal is used here in the mathematical sense, meaning perpendicular. In reflection of light, the angle of incidence

Angle of incidence

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 at which the wing or horizontal tail of an airplane is installed on the fuselage, measured relative to the axis of the fuselage.-Optics:In geometric...

is the angle between the normal and the incident ray. The angle of reflection is the angle between the normal and the reflected ray.

That Normal force will then Be perpendicular to the surface

## External links

- An explanation of normal vectors from Microsoft's MSDN
- Clear pseudocode for calculating a surface normal from either a triangle or polygon.