Computer Aided Engineering Design

218 COMPUTER AIDED ENGINEERING DESIGN

7.2 Boundary Interpolation Surfaces

Ruled and lofted patches are some examples of boundary interpolation surfaces. Given two parametric
curves,r 1 (u) and r 2 (u),u in [0, 1], a linear blending of curves in parameter v provides a ruledpatch.
Discussed in section 6.6, the result is

r(u,v) = (1 –v)r 1 (u) + vr 2 (u) = r 1 (u) + v[r 2 (u)–r 1 (u)] (7.31)

Herer 2 (u)–r 1 (u) is the direction vector along the straight line rulings. If, in addition, the cross
boundary tangentst 1 (u) and t 2 (u) are also provided with respective curves r 1 (u) and r 2 (u), then the
Hermite blending of four conditions (positions and slopes) along v for every ucan be performed
as in Eq. (7.32) with Hermite functions in Eq. (7.6). The resultant patch is called a lofted surface.
Figure 7.12 differentiates between ruled and lofted patches for two boundary curves.

r(u,v) = φ 0 (v)r 1 (u) + φ 1 (v)r 2 (u) + φ 2 (v)t 1 (u) + φ 3 (v)t 2 (u) (7.32)

Figure 7.12 (a) A ruled patch and (b) a lofted patch

r 1 (u)

r 2 (u)

r 2 (u)

t 2 (u)

t 1 (u)

r 1 (u)

(a) (b)

Example 7.7: Let

r(u, 0) = {cos [π(1 –u)], sin [–πu], 0}

r(u, 1) = {(2u– 1), –2u (1 –u), 1}

be two given boundary curves of the ruled surface, with u∈ [0, 1]. The equation of the surface for
v∈ [0, 1] is given by

r(u,v) = {(1 –v) cos [π(1 –u)] + v(2u– 1), – (–v sin [πu]–2u)(1 –u)v,v}

The surface is shown in Figure 7.13.
It can be verified that the tangent vector and unit normal vector at (u = 0.5, v = 0.5) are

rru(0.5, 0.5) =

2

+ 1, 0, 0 ; (0.5, 0.5) = 0,

1

2

, 1

⎧π

⎨

⎩

⎫

⎬

⎭

⎧

⎨

⎩

⎫

⎬

⎭

v

rru× ⎛

⎝

⎞

⎠

⎧

⎨

⎩

⎫

⎬

⎭

= 0, ⇒

2

+ 1,^1

22

v + 1^

ππ

unit normal n = {0, 0.895, 0.448}