Computer Aided Engineering Design

DESIGN OF SURFACES 219

7.2.1 Coon’s patches

Coon’s patches can use either linear or Hermite blending in surface approximation using four boundary
curves. Given those curves as a 0 (v),a 1 (v),b 0 (u) and b 1 (u) that intersect at four corner points P 00 ,P 01 ,
P 10 and P 11 as shown in Figure 7.14(a), ruled surfaces can be obtained by combining any two pairs
of opposite curves

rbb

raa

101

201

( , ) = (1 – ) ( ) + ( )

( , ) = (1 – ) ( ) + ( )

uuu

uuuu

vvv

vv

(7.33)

A linear Coon’s patch r(u,v) is the sum of the two surfaces above, and a surface r 3 (u,v) is
subtracted as the correction surface so that the boundary conditions are met. The patch may be
expressed as

r(u,v) = r 1 (u,v) + r 2 (u,v)–r 3 (u,v) (7.34)

Note that

r(u, 0) = b 0 (u) = r 1 (u, 0) + r 2 (u, 0) –r 3 (u, 0)

=b 0 (u) + (1 –u)a 0 (0) + ua 1 (0) –r 3 (u, 0) = b 0 (u) + (1 –u)P 00 + uP 10 – r 3 (u, 0)

which implies

r 3 (u, 0) = (1 –u)P 00 + uP 10 (7.35a)

Similarly,

r(u, 1) = b 1 (u) = r 1 (u, 1) + r 2 (u, 1) –r 3 (u, 1)

=b 1 (u) + (1 –u)a 0 (1) + ua 1 (1) –r 3 (u, 1) = b 1 (u) + (1 –u)P 01 + uP 11 – r 3 (u, 1)

which gives

r 3 (u, 1) = (1 –u)P 01 + uP 11 (7.35b)

Figure 7.13 Ruled surface in Example 7.7

r(u, 1)

r(u, 0)

–1

–0.5

0

0.5

1

1

0.75

0.5

0.25

0

0

• 0.25


• 0.75 – 0.5

–1