Normal plane (geometry)

In geometry, a normal plane is a special flat surface that includes the normal vector of another surface at a certain point. This means the normal plane touches the surface exactly at that point and stands straight up from it.

The normal plane can also describe a flat surface that is perpendicular to the tangent vector of a curved line in space. This special plane also contains the normal vector, and its properties are important in studying the shape and behavior of space curves, as explained by the Frenet–Serret formulas. Understanding normal planes helps us describe how curves and surfaces relate to each other in three-dimensional space.

Normal section

The normal section of a surface at a particular point is the curve made when that surface meets a normal plane.

The shape of this normal section is known as the normal curvature. For certain shapes like bowls or tubes, the biggest and smallest of these shapes are called the principal curvatures. When a surface looks like a saddle, the biggest shapes on each side are the principal curvatures. Multiplying the principal curvatures gives the Gaussian curvature of the surface, which is negative for saddle shapes. The average of the principal curvatures is the mean curvature. If this average is zero, the surface is called a minimal surface.