Normal (geometry)

In geometry, a normal is an object like a line, ray, or vector that is perpendicular to another object. For instance, the normal line to a plane curve at a certain point is a straight line that meets the curve at a right angle to its tangent line.

A normal vector is a vector that stands straight up to a surface or curve at a specific point. When this vector has a length of one, it is called a unit normal vector or normal direction. In three-dimensional space, a surface normal is a vector that is perpendicular to the tangent plane of a surface at a given point. These ideas help us understand directions and angles in many areas, such as 3D computer graphics, where normals help decide how surfaces look when lit by a light source.

Normal to space curves

Main article: Frenet–Serret formulas

Further information: Curvature vector

The normal direction to a space curve is shown by a special formula. It uses the tangent vector, which shows the direction of the curve at any point, and the radius of curvature, which tells us how much the curve bends. These help us find the normal direction, which is always at a right angle to the curve at that point.

Normal to planes and polygons

For a convex polygon like a triangle, we can find a special direction that points straight out from its surface. This direction is called a surface normal and can be found using a math operation called the cross product on two sides of the polygon.

For a flat surface, or plane, described by a special math rule, there is a special set of numbers that points straight out from the surface. This set of numbers is also called a normal. If the plane is described using different math rules with directions along the surface, the normal can again be found using the cross product of those directions.

Normal to general surfaces in 3D space

In geometry, a normal is a line or vector that stands straight up from a surface at a point. It is always at a right angle to the surface.

When we have a surface in 3D space, we can find a normal by looking at how the surface changes in two directions. By studying these changes, we can figure out which way is "up" from the surface.

For simpler shapes like flat graphs, we can also find normals using a special math tool called a gradient. This helps us understand the steepest direction from the surface.

A surface doesn’t always have a clear normal at every point — for example, the tip of a cone doesn’t have a single, obvious direction that counts as "up." But for most smooth surfaces, we can find a normal almost everywhere.

Orientation

Normals are often made to have the same length, but they can point in two opposite directions. For surfaces that close around a space, like a ball, we can choose normals that point either inward or outward.

When dealing with changes to a surface, like twisting or stretching, we need to adjust the normals carefully. The right way to do this is to use a special math trick with the transformation matrix to keep the normals standing straight up from the new shape.

Hypersurfaces in n-dimensional space

A normal is a line, ray, or vector that stands straight up from a surface, making a 90-degree angle with it. In higher dimensions, we can think of a hyperplane — a flat space with one less dimension than the space it lives in — having normals that are also straight up from it.

For a hyperplane in n-dimensional space, a normal vector is any vector that is orthogonal, or at a right angle, to all the vectors that lie on the hyperplane. This means the normal vector points straight out from the surface, showing its direction.

Varieties defined by implicit equations in n-dimensional space

A differential variety is a special shape made by finding where certain equations are true in space with n points. We can think of it as the places where several math rules all match at once.

At certain points on these shapes, we can find a normal vector space. This is a way to describe directions that are straight up and down compared to the shape at that point. It helps us understand how the shape bends and turns.

Uses

Surface normals help us understand how light shines on objects. In 3D computer graphics, they are used to make objects look realistic by calculating how light affects their surfaces. This is important for creating shadows and bright spots in images.

In computer vision, scientists use surface normals to guess the shape of objects in pictures. They also help in digital compositing, where different parts of an image are combined to change how lighting looks.

Normal in geometric optics

Main article: Specular reflection

In the study of light, a normal ray is a straight line that points straight out from a surface and is at a right angle to that surface at a specific spot. When light hits a surface and bounces off, the way we measure the angles involves this normal ray. The angle between the normal and the incoming light ray is called the angle of incidence, and the angle between the normal and the light ray that bounces away is called the angle of reflection.