There are two classical quadratic forms used in the analysis of smooth surfaces, the first and second fundamental forms. Knowledge of these forms provides an analysis and classification of surface shape. For a given surface or surface patch, , illustrated in Figure 2, from Equation 8, the x component is simply u, the y component is v and the z component is . The first fundamental form, I is defined

where

The subscripts denote partial differentiation

and define the tangent vectors to the surface at the point , as illustrated in Figure 2, see link above. They form the basis of the tangent plane which touches the surface at point . is the first fundamental form matrix, the metric or the metric tensor of the surface. The first fundamental form, , is a measure of the amount of movement of a surface at the coordinates in the parameter plane. It is invariant to translations and rotations of the surface in 3D space, and thus is an intrinsic parameter of the surface.

The second fundamental form is dependent on the position of the surface in 3D space. Thus it is an extrinsic property of the surface. This is given by

where the matrix elements are

The double subscripts denote second partial derivatives.

and the unit normal vector is

The second fundamental form matrix, , measures the change in normal vector and the change of surface position at a surface point at as a function of a small movement in the parameter space. The differential normal vector always lies in the tangent plane. The ratio is the normal curvature function, , which varies only as a function of the differential vector . If and are aligned for a particular direction of then that direction is a principal direction of the surface at that surface point. The extrema of the normal curvature function occur at that point and are called the principal curvatures, the maximum and the minimum. For example, consider the cylinder of Figure 1:

Figure 1: Viewing an ellipsoid: Principal curvatures of a cylinder

the maximum principal curvature occurs for a small movement of the normal vector ``around the cylinder'', i.e. perpendicular to the cylinder axis. The minimum principal curvature occurs for a small movement ``along the cylinder'', i.e parallel to the cylinder axis; in fact it is zero.