In geometry, a normal is an object (e.g. 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 line perpendicular to the tangent line to the curve at the point.

A normal vector of length one is called a unit normal vector. A curvature vector is a normal vector whose length is the curvature of the object. Multiplying a normal vector by −1 results in the opposite vector, which may be used for indicating sides (e.g., interior or exterior).

In three-dimensional space, a surface normal, or simply normal, to a surface at point P 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 ${\displaystyle P}$ is the set of vectors which are orthogonal to the tangent space at ${\displaystyle P.}$ Normal vectors are of special interest in the case of smooth curves and smooth surfaces.

The normal is often used in 3D computer graphics (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.

Normal to surfaces in 3D space

Calculating a surface normal

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 ${\displaystyle ax+by+cz+d=0,}$ the vector ${\displaystyle \mathbf {n} =(a,b,c)}$ is a normal.

For a plane whose equation is given in parametric form

where ${\displaystyle \mathbf {r} _{0}}$ is a point on the plane and ${\displaystyle \mathbf {p} ,\mathbf {q} }$ are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both ${\displaystyle \mathbf {p} }$ and ${\displaystyle \mathbf {q} ,}$ which can be found as the cross product ${\displaystyle \mathbf {n} =\mathbf {p} \times \mathbf {q} .}$

If a (possibly non-flat) surface ${\displaystyle S}$ in 3D space ${\displaystyle \mathbb {R} ^{3}}$ is parameterized by a system of curvilinear coordinates ${\displaystyle \mathbf {r} (s,t)=(x(s,t),y(s,t),z(s,t)),}$ with ${\displaystyle s}$ and ${\displaystyle t}$ real variables, then a normal to S is by definition a normal to a tangent plane, given by the cross product of the partial derivatives

If a surface ${\displaystyle S}$ is given implicitly as the set of points ${\displaystyle (x,y,z)}$ satisfying ${\displaystyle F(x,y,z)=0,}$ then a normal at a point ${\displaystyle (x,y,z)}$ on the surface is given by the gradient

since the gradient at any point is perpendicular to the level set ${\displaystyle S.}$

For a surface ${\displaystyle S}$ in ${\displaystyle \mathbb {R} ^{3}}$ given as the graph of a function ${\displaystyle z=f(x,y),}$ an upward-pointing normal can be found either from the parametrization ${\displaystyle \mathbf {r} (x,y)=(x,y,f(x,y)),}$ giving

or more simply from its implicit form ${\displaystyle F(x,y,z)=z-f(x,y)=0,}$ giving ${\displaystyle \mathbf {n} =\nabla F(x,y,z)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right).}$ 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.

Orientation

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 two normal orientations, 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.

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 ${\displaystyle \mathbf {M} ,}$ we can determine the matrix ${\displaystyle \mathbf {W} }$ that transforms a vector ${\displaystyle \mathbf {n} }$ perpendicular to the tangent plane ${\displaystyle }$Zdroj:https://en.wikipedia.org?pojem=Normal_(geometry)
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