# surface normal

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. 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. 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.

## 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.

For a plane given by the equation , the vector 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

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

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.

## Uniqueness of the normal

A vector field of normals to a surface.
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 an oriented surface, 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.

## n-dimensional surfaces

The definition of a normal to a two-dimensional surface in three-dimensional space can be extended to -dimensional "surfaces" in -dimensional space. Such a hypersurface may be defined implicitly as the set of points satisfying the equation . If is continuously differentiable, then the surface obtained is a differentiable manifold, and its surface normal is given by the gradient of ,

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1 (the unit length). A unit vector is often written with a superscribed caret or “hat”, like this (pronounced "i-hat").
The intuitive idea of flatness is important in several fields.

## Flatness in mathematics

The flatness of a surface is the degree to which it approximates a mathematical plane.
spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B.
perpendicular (or orthogonal) to each other if they form congruent adjacent angles. The term may be used as a noun or adjective. Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B.
Point can refer to:
• Point and counterpoint, meaning or purpose, especially in a discussion or dispute
• Point of order, a matter raised during a debate concerning the rules of debating themselves

In mathematics, the tangent space of a manifold is a concept which 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.
line can be described as an ideal zero-width, infinitely long, perfectly straight curve (the term curve in mathematics includes "straight curves") containing an infinite number of points. In Euclidean geometry, exactly one line can be found that passes through any two points.
Plane may refer to:
• Short for aircraft or airplane, referred to by aircraft engineers as a fixed-wing aircraft
• Plane (mathematics), theoretical surface which has infinite width and length, zero thickness, and zero curvature

In physics, force is an action or agency that causes a body of mass m to accelerate. It may be experienced as a lift, a push, or a pull. The acceleration of the body is proportional to the vector sum of all forces acting on it (known as net force or resultant force).
In mathematics, orthogonal, as a simple adjective, not part of a longer phrase, is a generalization of perpendicular. It means at right angles, from the Greek ὀρθός orthos
POLYGONE is an Electronic Warfare Tactics Range located on the border between France and Germany. It is one of only two in Europe, the other being RAF Spadeadam.

The range, also referred to as the Multi-national Aircrew Electronic Warfare Tactics Facility (MAEWTF), is
A triangle is one of the basic shapes of geometry: a polygon with three corners or and three sides or edges which are straight line segments.

In Euclidean geometry any three non-collinear points determine a triangle and a unique plane, i.e.
cross product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the two input vectors. By contrast, the dot product produces a scalar result.
plane is a two-dimensional manifold or surface that is perfectly flat. Informally it can be thought of as an infinitely vast and infinitesimally thin sheet oriented in some space.
coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring.
Curvilinear coordinates are a coordinate system for the Euclidean space based on some transformation that converts the standard Cartesian coordinate system to a coordinate system with the same number of coordinates in which the coordinate lines are curved.
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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).
In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable.
gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
cone is a three-dimensional geometric shape consisting of all line segments joining a single point (the apex or vertex) to every point of a two-dimensional figure (the base).
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 not hold is a null set, i.e. is a set with measure zero.
In mathematics, more specifically in real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions which is stronger than regular continuity.
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 formally, it is the set of points in the closure of S, not belonging to the interior of S.
R3 is non-orientable, if a figure such as the figure can be moved around the surface and back to where it started so that it looks like , its mirror image. (This figure was chosen because it cannot be continuously moved to its mirror-image within a plane).
right hand grip rule.
In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in 3-D.