Computer Aided Engineering Design

Chapter 4

Design of Curves

Shapes are created by curves in that a surface, such as the rooftop of a car, the fuselage of an aircraft,
or a washbasin can be created by motion of curves in space in a specified manner. This may involve
sweep, revolution, deformation, contraction or expansion, and forming joints with other curves.
Theanalytical properties of curves were derived in Chapter 3, where it was assumed that the
equation of the curve is known. However, in design, an engineer first creates a shapeusing his
imagination, without knowing the equation of the curve at this stage. The computer should help the
designer in synthesizing the curve shape so that one can (a) replicate the imagined shape without
worrying about the equations of the curve (b) change or fine tune the shape to conform to technical,
manufacturing, aesthetic and other requirements.

The principles of curve design envisages the following:

1. The shapeof the curve should be controlled by placing only a few number of data points. The
   curve thus created should behave like an elastic string that a designer can manipulate to give
   a desired shape.


2. The curve should be syntheticallycomposed of polytnomial segments of lower degree to
   avoid undue oscillations and minimize computation time and complexity.


3. The curve model should have “affine” properties ensuring shape independence from the co-
   ordinate frame of reference. This makes it possible to treat the curve model as a real object
   in space that does not get distorted because of different frame of reference.


4. Since, in real design, complex shapes have to be created, it is more suitable to join together
   several segments of curves, fulfilling position, slope and/or curvature continuities at the
   joints. If we look at the profile of a car or an aircraft at any cross section, we can appreciate
   the smoothness with which various curve segments are joined together. Thus, curve models
   are developed which form simple building blocks for piecing them together to create a
   desired shape.


5. Parametric description is preferred over the implicit or explicit forms as it provides an
   articulate representation of curve segments in three dimensions. In additio, and trimming like
   operations can be handled with relative ease.
   Synthetic curves are suitable for designing generic forms that may not be represented by analytic
   curves. Mathematically, though both synthetic and analytic curves are polynomial representations,
   the former provides more control in that they may be derived from a given set of data points and/or