Computer Aided Engineering Design

DIFFERENTIAL GEOMETRY OF CURVES 67

values implies only reorienting the line. Another example is of a second-degree polynomial S≡ ax^2

• 2 hxy + by^2 + 2 gx + 2 fy + c = 0 which is representative of all conic sections in the x-y plane.
  Coefficients a,h,b,g,f and c in some combination represent a class of conic sections. Arbitrary
  values of these coefficients would yield only a few shapes mentioned above. More precisely, the
  shape of Sdepends on two invariants, D and I 2 , where

D

ahg

hb f

gfc

= (3.1)

andI 2 = h^2 – ab. One reason D and I 2 are called invariants is that their values remain unaltered if a
translationx = x′ + p,y = y′ + q and/or rotation (x = x′ cos θ – y′ sin θ,y = x′ sin θ + y′ cos θ) is applied
toS. (a) For D= 0, if I 2 = 0, S represents a set of parallel lines. For positive I 2 ,S is a pair of
intersecting lines while for negative I 2 ,S is a point. (b) For D≠ 0, if I 2 = 0, S represents a parabola,
ifI 2 >0,S is a hyperbola and if I 2 <0,S is an ellipse or a circle. Apart from the limited shapes analytic
curves have to offer, direct or active control on their shape is not available to a user. However,
segments of analytic curves like an arc, an elliptic or parabolic segment, if so desired, are often used
in the wireframe modeling of solids (see Chapter 8).
Shapes of the reaction turbine blades, car windshields, aircraft fuselage, potteries, temple minarets,
kitchenware, cathode ray tubes, air-conditioning ducts, seats for cars, scooters or bicycles, instrument
panels for aircrafts provide many examples of some household and industrial products where a free-
form surface is desired. This surface may be composed of a network of curves, and a designer
requires an active control to arrive at a desired shape of a curve. It is interesting to observe how a
potter creating a clay pot on a rotating wheel merely adjusts and manipulates his finger pressure at
a few points to obtain a desired shape.
The way active control on curve’s shape can be sought is by choosing a set of data points and
requiring to interpolateorbest fit a curve through it. Curve interpolation and curve fitting methods
have been two of the oldest methods available in curve design. For given n data points (xi,yi),i =
0, ..., n – 1, interpolation requires to pass the curve through all the points by choosing a polynomial
g(x) of degree n – 1 and determining the unknown coefficients. Alternatively, in curve fitting, one
may choose a polynomial of a smaller degree m (< n – 1) such that the curve depicts the best possible
trend or distribution of data points. In both methods, a user gets a distinct advantage in that the shape
of the curve is governed by the placement of data points, that is, the user may actively control the
position of data points to affect the change in shape of the interpolated or best fit curve. Curve
interpolation and fitting lay the groundwork for curve design and thus are discussed in detail below.

3.1 Curve Interpolation

Given a set of nordered data points (xi,yi),i = 0, ..., n−1, let y = p(x) be a polynomial of degree
n−1 in x with unknowns a 0 ,a 1 , ..., an− 1. That p(x) traverses through data points above implies

ypx aaxax ax 000102 = ( ) = + + 02 + 3 03 +... + axn–1 0 n–1

ypx a axax ax 1 = ( 1 0112) = + + 12 + 313 +... + axn–1 1 n–1

ypx aaxax ax 220122 = ( ) = + + 22 + 3 23 +... + axn–1 2 n–1

...
ypx aaxaxnn–1 = ( –1) = + 0 1 n–1 + 2 n^2 –1 +... + axn–1 nn–1–1 (3.2)