Computer Aided Engineering Design

DIFFERENTIAL GEOMETRY OF CURVES 73

3.3 Representing Curves

Curves may be expressed mathematically using one of the three forms, viz. explicit, implicit or
parametric. In two dimensions, explicit equations are of the form

y = f(x)

wherein the slope at a point (x,y) is given as ∂
∂

∂

∂

y

x

f
x
or. Consider, for instance, the equation of
a straight line in the two-point form

yy

yy

xx

• = xx
  ( – )
  ( – )
  1 21 ( – )
  21
  1

So long as x 1 and x 2 are not equal, the representation works well in that there is a unique value of y
for every x. However, as x 2 approaches x 1 , the slope approaches infinity. Thus, for x 1 = x 2 , even though
the line is vertical, that the y value can be non-unique is not apparent, that is, explicit representations
by themselves cannot accommodate vertical lines or tangents.
Implicit equations are of the type

g(x,y) = 0

for instance the equation of a straight line ax + by + c = 0 or the circle x^2 + y^2 −r^2 = 0. To determine
the intersection of the line and circle above, the implicit forms need to be first converted into the
respective explicit versions. Two possibilities would exist for the roots; either they both are complex
or both are real. In case the roots are real and equal, the line would be tangent to the circle. For
unequal and real roots, the line will intersect the circle at two points. In general, additional processing
is required to determine the intersection points for any two curves. Moreover, a concern with both
explicit and implicit form of representations is that they cannot, by themselves, represent a curve
segment which is what the designers are usually interested in. For instance, it would be very difficult

given set. The entire set of data points may then be piecewise interpolated and the resultant would be
a composite curve with cubic segments juxtaposed sequentially (Figure 3.4). There, however, would
be continuity related issues at a data point common to the two adjacent cubic segments. Though both
segments would pass through the data point (position continuity), the slope and/or curvature would
be discontinuous and the composite curve may not be smooth overall. It is here that an insight into
the differential properties of curves would be of help.

Figure 3.4 A schematic showing a composite curve with two cubic segments

interpolating the data points

Data points

Two cubic segments

Junction point