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).

- Normal to surfaces in 3D space
- Calculating a surface normal
- Choice of normal
- Transforming normals
- Hypersurfaces in n-dimensional space
- Varieties defined by implicit equations in n-dimensional space
- Example
- Uses
- Normal in geometric optics
- See also
- References
- External links

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 **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 * of a point *Q* to a curve or to a surface is the Euclidean distance between *Q* and its foot *P*.

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 3D 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.

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.

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.

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

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.

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.

- Surface normals are useful in defining surface integrals of vector fields.
- Surface normals are commonly used in 3D computer graphics for lighting calculations (see Lambert's cosine law), often adjusted by normal mapping.
- Render layers containing surface normal information may be used in Digital compositing to change the apparent lighting of rendered elements.
^{[ citation needed ]} - In computer vision, the shapes of 3D objects are estimated from surface normals using photometric stereo.
^{ [1] }

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.

- Dual space – In mathematics, vector space of linear forms
- Ellipsoid normal vector
- Normal bundle
- Pseudovector – Physical quantity that changes sign with improper rotation
- Vertex normal

In vector calculus, the **curl** is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field.

In mathematics, the **derivative** of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

In vector calculus, the **gradient** of a scalar-valued differentiable function *f* of several variables is the vector field whose value at a point is the vector whose components are the partial derivatives of at . That is, for , its gradient is defined at the point in *n-*dimensional space as the vector

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. Partial derivatives are used in vector calculus and differential geometry.

In vector calculus and physics, a **vector field** is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

An **ellipsoid** is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mathematics, a **quadric** or **quadric surface**, is a generalization of conic sections. It is a hypersurface in a (*D* + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in *D* + 1 variables; for example, *D* = 1 in the case of conic sections. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a *degenerate quadric* or a *reducible quadric*.

In mathematical optimization, the **method of Lagrange multipliers** is a strategy for finding the local maxima and minima of a function subject to equality constraints. It is named after the mathematician Joseph-Louis Lagrange. The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the **Lagrangian function**.

In mathematics, a **tangent vector** is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in **R**^{n}. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally, a tangent vector at the point is a linear derivation of the algebra defined by the set of germs at .

In mathematical physics, **Minkowski space** is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity.

In mathematics, the **covariant derivative** is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.

In differential geometry, the **second fundamental form** is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by . Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold.

In geometry, an **envelope** of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two "infinitesimally adjacent" curves, meaning the limit of intersections of nearby curves. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.

In mathematics, a **norm** is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a **function of a real variable** is a function whose domain is the real numbers , or a subset of that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval. The most widely considered such functions are the **real functions**, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.

In geometry and science, a **cross section** is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The boundary of a cross-section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space showing points on the surface of the mountains of equal elevation.

**Three-dimensional space** is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension.

A **parametric surface** is a surface in the Euclidean space which is defined by a parametric equation with two parameters . Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

In mathematics, and especially affine differential geometry, the **affine focal set** of a smooth submanifold *M* embedded in a smooth manifold *N* is the caustic generated by the affine normal lines. It can be realised as the bifurcation set of a certain family of functions. The bifurcation set is the set of parameter values of the family which yield functions with degenerate singularities. This is not the same as the bifurcation diagram in dynamical systems.

In physics, **deformation** is the continuum mechanics transformation of a body from a *reference* configuration to a *current* configuration. A configuration is a set containing the positions of all particles of the body.

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

- Weisstein, Eric W. "Normal Vector".
*MathWorld*. - An explanation of normal vectors from Microsoft's MSDN
- Clear pseudocode for calculating a surface normal from either a triangle or polygon.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.