TABLE OF CONTENTS Global properties are those that depend on the over-all characteristics of a geometric object. Here we will consider surface area, volume, and certain characteristic tests for surfaces.
The bicubic patch and other parametric forms are well adapted to computing geometric properties, since we do not have to compute explicit points. To compute surface area, we use an elementary property of vectors shown in Figure 12.15.
where n (u,w) is a vector function defining the patch normals and dA is the scalar element of area.
Therefore,
Finally, using Gaussian quadrature, we obtain
g i and h j are the weight values associated with a specific n-point formula.
For the volume of a closed region, we have from vector calculus
where p is the position vector, n is the normal vector, and dV is the scalar element of volume (Figure 12.16).
The factor of 1/3 arises from the solid geometry formula for a pyramid (that is, the volume is 1/3 the product of the base area and the perpendicular height). The scalar product of the vectors satisfies the requirement of perpendicularity between the base and height. So we obtain
and we may again apply quadrature as we did to obtain Equation 12.80 from Equation 12.79.
Here are two simple tests we can apply to surfaces to determine if they posses special characteristics. A surface is...
