Computer Aided Engineering Design

188 COMPUTER AIDED ENGINEERING DESIGN

(a) (b)

Example 6.5.Find the equation of surfaces parallel to the sphere

x(u,v) = a cos u cos v, y(u,v) = acosu sin v, z(u,v) = asinu

and the catenoid

r(u,v) = acosu cosh (v/a)i + a sin u cosh (v/a)j + vk

A surface S* parallel to the surface S is obtained from

r*(u,v,t) = r(u,v) + tn(u,v)

wheret is the separation of the parallel surface and n(u,v) is the unit normal to S at the corresponding

point with n

rr

rr

(, ) =

| |

u u

u

v v

v

×

×

⋅

For the sphere,

ru=–asinu cos vi–asinu sin vj+acosuk

rv=–acosu sin vi + a cos u cos vj + 0k

ru×rv=–a^2 cos^2 u cos vi–a^2 cos^2 u sin vj–a^2 cos u sin uk

|ru×rv | = a^2 cos u

r*(u,v,t) = r(u,v) + tn(u,v) = cos u cos v(a–t)i + cos u sin v(a–t)j + sin u(a–t)k

Two parallel spheres are shown with t = 0, the original surface S is between Figure 6.18(a). In the
example,a = 1, t = –^12 , and t =^12. Using the above method, we can determine the equation of the
surface parallel to the catenoid separated by a distance t as

r*(u,v,t) = [acosu cosh (v/a) + t cos u sech (v/a)]i

+ [asinu cosh (v/a) + t sin u sech (v/a)]j + [v–ttanh (v/a)]k

Parallel surfaces for the catenoid have been generated for a= 1, 0 < u < 3π/2, –1.5 < v 1.5 and three
values of t at –^12 , 0 and^12 in Figure 6.18 (b).

Figure 6.18 Parallel surfaces (a) spheres and (b) catenoids

6.8 Surfaces of Revolution

A large number of common objects such as cans and bottles, funnels, wine glasses, pitchers, football,
legs of furniture, torus, ellipsoid, paraboloid and sphere are all surfaces of revolution.