Advanced Applications of Theory of Surfaces
Special topics on the geometric properties of surfaces are treated in this chapter. They are developed from the fundamental theory of surfaces which has been explained in the previous chapter. Knowledge of these topics is useful for treating free-form surfaces in advanced problems. First we discuss the umbilics and lines of curvature. On a free-form surface, there are points and regions which have the special characters inherent to its shape, and whose locations do not depend on the coordinate system adopted. The lines of curvature make orthogonal nets on a surface, and the pattern they form exhibits inherent features of the surface. The umbilics are singular points, or curves or regions seeing from the lines of curvature. On the free-form surface the umbilics appear more frequently than our expectation. There are other curves on the surface which depend not only on its inherent features, but also on its orientation with respect to its observers or its environments or its surface physical properties. These curves are useful for describing or evaluating the objects from engineering and aesthetic criteria.
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