Theory of Curves

Curves are not only the fundamental geometric elements, but also they give us various information on shapes. In CAD/CAM, free-form curves have to be mathematically defined or extracted so that their properties can be controlled and evaluated by the computer. Accordingly, knowledge of the fundamentals of the differential geometry of curves is required to apply geometry to practical problems and it also gives the concepts and methods needed to understand the theory of surfaces which is explained in the next two chapters [1]. The curves are usually expressed in parametric forms, and arc length of the curve is used for the parameter in theoretical treatments because of its simplicity of expression, but for practical uses the parameter is changed from arc length 5 to a more manageable variable parameter t which monotonically increases with arc length. When treating interference problems of curves or surfaces, the curves expressed in implicit forms are sometimes convenient in numerical calculations or theoretical analysis. Therefore, we introduce a method of converting from a parametric form to an implicit one in the last section.