Computer Aided Engineering Design

166 COMPUTER AIDED ENGINEERING DESIGN

Example 6.1. Some commonly known analytical surfaces can be represented in the parametric form.
(a) An x-y Plane:r(u,v) = ui + vj(in cartesian form, u,v) = u cos vi + usinvj (in polar form)
(u≠ 0, 0 ≤v≤ 2 π).
(b) A Sphere:A point P on the sphere is given by r(u,v) = acosu cos vi + a sin u cos vj+a sin vk,
wherea is the radius, v is the angle made by the radius vector r (u,v) with the x-y plane and uthe
angle made between the x-axis and the projection of the radius vector on the x-y plane as shown
in Figure 6.3. Angles u and v are called longitude and latitude, respectively. The circles of
latitude are v = constant (v = 0 is the equator), whereas u = constant, u∈ [0, 2π) are called
meridians.
(c) A Catenoid: Rotation of a catenary y = acosh (x/a) about its directrix (x-axis) results in a
catenoid.A point on the catenoid is then given by

r (u,v) = ui + acosh

u

a

⎛

⎝

⎞

⎠

cos vj + acosh

u

a

⎛

⎝

⎞

⎠

sin vk,

wherevis the angle of rotation in [0, 2π].
(d) The Pseudosphere: Tractrix is a planar curve having the property that the segment of its tangent
between the contact point P and some fixed straight line (called its asymptote) in the plane is

in parametric form using low order polynomials and then knitthese patches to form a composite
surface with continuous slopes and/or curvatures at their boundaries. Given that the parametric form
of a surface patch is known, this chapter deals with determining the differential properties of the
patch to facilitate composite fitting.

6.1 Parametric Representation of Surfaces

A surface patch (Figure 6.2) is a set of points whose position vectors are given by r = r(u,v) for
parametersu and v each varying in the interval [0, 1]. For constant u=uc,r(uc,v) is a parametric
curve in vwhile for v = vc,r(u,vc) is a curve that varies only with u. Thus, a parametric surface
r(u,v) may be regarded as a set of matted curves. Values of u and v determine the position of a point
on the surface and thus u and v may be regarded as the curvilinear or Gaussiancoordinates. For
scalar functions x(u,v),y(u,v) and z(u,v), a parametric surface may be represented in vector form
as

r(u,v) = x(u,v)i + y(u,v)j+z(u,v)k (6.1)

Figure 6.2 A parametric surface r (u,v)

ru

r(u 0 ,v 0 ) rv

vconstant

uconstant

v

u

P

n