Chapter 3
4. Design of Curves
The form of a real world object is often represented using points, curves, surfaces and solids.
Although, a form can be sketched manually, it may be useful to construct a mathematical or computer
model for a detailed description. In engineering design, we need to represent an object with precise
drawings, perform requisite analysis to possibly optimize the form and finally, manufacture it. Geometric
modeling of curves provides an invaluable tool for representation or visualization, analysis and
manufacture of any machine part by providing the basis for the representation of surfaces and solids,
and thus the real world objects (Figure 3.1). That curve design is fundamentally significant in
computer-aided design, it behoves to understand and layout the underlining intent: to seek a generic
mathematical representation of a curve in a manner that renders absolute shape control to the user.
In other words, the curve definition must be general to encompass any possible shape and also, the
user should be able to manipulate a curve’s shape locally without altering the curve overall.
There are, therefore, two issues to address in curve design: (a) representation and (b) shape
control. Analytic curves like conics (pair of straight lines, circles, ellipses, parabolae and hyperbolae
in two dimensions) and helix, helical spiral and many more in three dimensions are all well-defined
and well-studied. However, shapes and forms obtained using analytic curves are limited in engineering
applications and pose restrictions to curve design in real situations. Moreover, local shape control
with these curves is usually not possible. For instance, the coefficients a,b and c in the equation
ax + by + c = 0 of a straight line Ldetermine the slope and intercept of the line and changing their
(a)
(b)
Figure 3.1 A solid (a) represented as a network of curves (b)