Computer Aided Engineering Design

196 COMPUTER AIDED ENGINEERING DESIGN

Intersection of a Cylinder and a Sphere: Viviani Curve (1692)
Let the equations of the sphere and the cylinder be given by

x^2 + y^2 + z^2 = 4a^2

(x–a)^2 + y^2 = a^2

The parametric equation of the cylinder can be written as

x = a(1 + cos u),y= a sin u,z

The parametric equation of the curve common to the cylinder and the sphere can be written as

x = a(1 + cos u),y = a sin u, z = 2a sin (u/2)

This curve is known as the Viviani curve as shown in Figure 6.25.

With r(u) = {a(1 + cos u),a sin u, 2a sin (u/2)}

⇒ r ̇(u) = {–a sin u, a cos u, a cos (u/2)}

̇ ̇r(u) = {–a cos u,–a sin u,–(a/2) sin (u/2)}

̇ ̇ ̇r(u) = {a sin u,–a cos u,–(a/4) cos (u/2)}

⇒ κτ =

| |

| |

=

(13 + 3 cos )

(3 + cos )

, =

( )

| |

=

6 cos ( /2)

(^3) (13 + 3 cos )
1
2
3
2
2
r ̇ ̇ ̇r
̇r
̇r ̇ ̇r ̇ ̇ ̇r
r ̇ ̇ ̇r
× ×⋅
×
u
au
u
au
2
1.5
1
0.5
0
2
1.5^1 0.5
0 0.5 1 1.5 2
Viviani curve
Cylinder
Sphere
Figure 6.25 (a) Viviani curve and (b) intersection between two cylinders
(a) (b)