TABLE OF CONTENTS Local properties of a surface include the normal, tangent plane, and principal curvatures at a point. These turn out to be the most useful for properties for geometric modeling and 3D modeling. First, we will explore some special differential geometry of surfaces.
Just as two local properties, curvature and torsion, determine a unique curve, in a similar way local properties called the first and second fundamental forms determine a unique surface. We denote these as Form I and Form II. They are presented here without derivation or proof; see Lane (1940) or O Neill (1966), among many others, for a more complete discussion. If p (u, w) represents a parametric surface, then its first fundamental form is
Form I:
where
E , F , and G are known as the coefficients of the first fundamental form. In the metric theory of surfaces, the first fundamental form arises when calculating the arc length of a curve on a surface.
If n (u, w) is the unit normal to a surface at a point p (u,w), then the second fundamental form is
Form II:
where
and where p u= ? p (u,w)/ ?u, etc.
L, M, and N are known as the coefficients of the second fundamental form.
Note that n u and n w are perpendicular to n. Then, because p u and p w are perpendicular to n for all u,...
