# Normal vector

In the geometry is a **normal vector** , and **normal vector** , a vector which is orthogonal (i. E. Perpendicular, vertical) on a straight line , curve , plane (curved) surface , or a higher-dimensional generalization is of such an object. A straight line with this vector as a directional vector is called a **normal. **A **normal ****unit ****vector** or a **unit** normal is a normal vector of length 1.

This article first deals with the case of straight lines in the plane and of planes in three-dimensional space ( linear algebra and analytic geometry ), then the case of curves in the plane and surfaces in space ( differential geometry ).

## Linear algebra and analytic geometry

In this section, the variables for vectors are indicated by vector arrows, as is common in school mathematics.

### Normal and normal vector of a straight line

A normal vector of a straight line in the plane is a vector different from the zero vector that is perpendicular to this straight line, i.e. the direction vector of a straight line that is perpendicular to , i.e. an *orthogonal* or *normal* to .

Has the direction vector , the two vectors and normal vectors. If one traverses the straight line in the direction of , then points to the left and to the right.

Is the straight line in the slope form by the equation

given, the vector is a direction vector of the straight lines and and are normal vectors. For so any normal has the slope . If , that is, horizontal, then every normal is vertical, so it has an equation of the form .

Is the straight line in general form

given, then is a normal vector.

A normal unit vector can be calculated from a normal vector by dividing by its length ( norm , amount). The vector is therefore normalized.

The second normal unit vector is obtained by multiplying the above normal unit vector by . Each normal vector can be formed by multiplying a normal unit vector by a real number not equal to zero.

### Normal and normal vector of a plane

A *normal vector of* a plane in three-dimensional space is a vector different from the zero vector that is perpendicular to this plane, i.e. the directional vector of a straight line that is perpendicular to , i.e. an *orthogonal* or *normal* to .

Is the plane in the coordinate form by the equation

given, then is a normal vector.

If and is given by two spanning vectors (point-direction form or parametric form ), the condition that the normal vector is perpendicular to and leads to a linear system of equations for the components of :

Each of the different solutions provides a normal vector.

Another way to determine normal vectors is to use the cross product :

is a vector that is perpendicular to and , and in this order they form a legal system .

Has the equation

- ,

so is an upward and a downward normal vector.

As in the case of the straight line in the plane, a normal vector is obtained from a normal vector by dividing it by its length, a second by multiplying by and all other normal vectors by multiplying by real numbers.

A plane is clearly defined by a normal vector and a point lying on the plane, see normal form and Hessian normal form .

## Normal vectors of curves and surfaces

### Flat curves

In analysis and differential geometry , the normal vector to a plane curve (at a certain point) is a vector that is orthogonal (perpendicular) to the tangential vector at this point. The straight line in the direction of the normal vector through this point is called the **normal** , it is orthogonal to the tangent .

If the curve is given as a graph of a differentiable function , the tangent at the point has the slope , so the slope of the normal is

The normal in the point is then given by the equation

so through

given.

Is the plane curve in parameter form given , then a tangent vector at the point and a normal vector pointing to the right. As is customary in differential geometry, the point here denotes the derivative according to the curve parameter.

In the case of space curves , the normal vectors at a point (as in the case of straight lines in space) form a two-dimensional sub - vector space . In elementary differential geometry, one chooses a unit vector that points in the direction in which the curve is curved. This is called the *principal normal (unit) vector* , see Frenet's formulas .

### Areas in three-dimensional space

Correspondingly, the normal vector of a curved surface at one point is the normal vector of the tangential plane at this point.

Is the area through the parametric representation

given, the two vectors are

- and

Clamping vectors of the tangential plane at the point . (Here it is assumed that the area at is regular , i.e. that and are linearly independent .) A normal vector in the point is a vector that is perpendicular to and , e.g. B. the *main normal vector* given by the cross product and then normalized

Here the vertical lines indicate the Euclidean norm of the vector.

If the area is given implicitly by an equation,

- ,

where is a differentiable function, then is the gradient

a normal vector of the surface in the point (provided that it does not vanish there).

If the area is given as a graph of a differentiable function , then

an upward normal vector at the point . This is obtained by using that the mapping is a parameterization or that the area is given by the equation

is pictured.

## Generalizations

The concept of the normal vector can be generalized to

- affine subspaces (generalized levels) in Euclidean spaces of higher dimensions (mathematics) (especially on hyperplanes ),
- Surfaces, hypersurfaces and submanifolds in Euclidean spaces of higher dimensions,
- Surfaces, hypersurfaces and submanifolds of Riemannian manifolds ,
- Non-smooth objects, such as convex bodies and rectifiable sets.

## Applications

In the field of computer graphics , normal vectors are used, among other things, to determine whether a surface is facing the user or not in order to exclude the latter from the image calculation ( back-face culling ). They are also required to calculate incidence of light and reflections.