# Curves and surfaces in computer aided geometric design - Yamaguchi F.

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+ cli0(w) (2W(1,0)+Clfl (w) (2W(1, l) 9

Q(u, 0) = C0j0(w) (2(0, 0) + Ñîä (è) (2(1,0) + C1>)?U(0,0) + C1>)?U(1,0) Q(u, 1) = C0j0 (u) (2(0,1) + Ñîä (è) (2(1, 1) + Cuo(u)Qu(0,l) + C1A(u)Qu(l,l)

These imply that:

P{u, w) = [C0 0(w) C0>1(m)]

Ãá(0,0) (2(0,1)'

Le(i,o) 0(1

[-1 C1>0(m) Clfl(M)]

',1)1 rc0)0

ëè êîä

- 0 QJu, 0) QJu, 1)" - -1 -

Qu( 0,w) O„w(0,0) Quw( 0,1) Ci,o(w)

Qu( l,w) Ow(l,0) Ow(l,l) _Ci,i (w)_

(3.96)

5) Next, let us assume that the following equations apply to the tangent vector functions in the directions across the boundary curves:

110

3. Hermite Interpolation

Qu( 0, w) = C0>0(w) Qu( 0,0) + C01(w) Qu( 0,1)

+ C1;0(w) 0U.(O,0) + C11(w) Quw(0,1) 0(1, w) = C0)0(w) 0.(1,0) + C0>i(w) 0.(1. i)

+ Clfo(w)0uiv(l,0) + C1)1(w)0uiv(l, 1) QJu, 0) = C0» 0.(0, 0) + Co» 0.(1, 0)

+ C1»0UIV(O,O) + C1»0UIV(1,O) Qw(u, 1) = C0 o(w) 0.(0,1) + C0» 0W(1, 1)

+ C1,o(«)Ou.(0, l) + Clfl(n)eBW(l, 1)

(3.97)

These imply that:

P(u,w) = [CU0(u) Ñ1Ä(ì)]

ÃÎ«.(0,0) 0u.(0,1)’

L0u.(l,0) 0u.(i,i)_ -[-1 Co» C0)1(w)]

Ï rci.0(w)l J Lclfl(w)J

- 0 Q(u, 0) 0(4,1)" - -1 -

0(0, w) 0(0,0) 0(0,1) Co,o(w)

_0(i,w) 0(1,0) 0(1,1) _C0,i(w)_

(3.98)

6) Denote the 1st, 2nd and 3rd surfaces in Eq. (3.83) by PA(u,w), PB(u,w) and Pc(u,w), respectively.

Assuming that Eqs. (3.95) and (3.97) apply, we obtain:

Pa(u, w) = PB(u, w) = Pc(u, w)

so that:

P(u, w) = Pc (u, w)

= [Co,o(w) C0ä (u) C10(u) C11(u)]

'0(0,0) 0(0,1) o.(0, o) 0.(0, i)“

0(1,0) o(i,i) o.(i,o) 0.(1, i) x e„(o,o) e„(o,i) q„(o,o) &J0.1)

_e»(i.o) e„d,i) a»(i,o) e„(u).

In Eq. (3.99), if we use the cubic Hermite interpolation functions H0 0(t), H0A(t), Hl 0(t) and Hltl(t) as blending functions, we obtain:

Ñî,î È

c0,iN

Cl;o(w)

Clt i(w)

. (3.99)

3.3 Surfaces

111

Ð(è,ìÎ = [ßî,î(è) Íîë(è) ß1>0(è) ß1Ä(è)]

"(2(0,0) 6(0,1) 6.(0,0) 6.(0,1)' 6(1,0) 6(1,1) 6.(1,0) 6w(l,l)

6.(0,0) 6.(0,1) 6..(0,0) 6..(0,1) .6.(1, o) 6.(1, i) e.w(i,o) e..(U)_

Using simpler notation, this becomes:

P(u,w) = [Hofo(u) Íîë(è) Huo(u) ß1Ä(è)]

x Bc [#o,o(w) ßîä (w) ß10(ø) ß1 x(w)]T = UMCBCMJWT

where Mc is the matrix defined in (3.10). We also have:

6(0,0) 6(0,1) 6(1,0) 6(1,1)

6.(0,0) 6.(1,0)

6.(0,1) 6.(1,1)

6.(0,0) 6.(0,1) 6..(0,0) 6..(0,1) 6.(1,0) 6.(1,1) 6..(1,0) 6..(U),

'tf0s0(w)-

ÍîëÌ

H,Aw)

Í1ËÌ

(3.100)

(3.101)

(3.102)

(3.103)

è = \èú W=[w3

è 1] w 1].

Since Bc is a matrix which expresses the surface patch boundary conditions, it is sometimes called the boundary condition matrix (Fig. 3.20).

Since, in the surface patch equation (3.100), H0 0, H0ä, Hl 0 and ß1Ä are expressed as cubic polynomials in the respective parameters, they are expressed

Fig. 3.20. Vectors that

define a Coons bi-cubic

surface patch

112

3. Hermite Interpolation

in the bi-cubic polynomial form of equation (1.109). Consequently, the surface described by Eqs. (3.100), (3.101) and (3.102) are called bi-cubic Coons patches.

The curved surface having the form (3.100) is a special case of a Coons surface; since it has a very simple form it is easy to use and is a standard type of Coons surface.

3.3.4 Twist Vectors and Surface Shapes

Cross partial derivative vectors, that is, twist vectors, do not occur in studies of curves but do come up in treatment of curved surfaces. In the Ferguson surface

patch and the 1964 Coons surface patch, the twist vectors are 0 at all 4 corners.

As we mentioned in discussing the description of the Ferguson surface patch, a tangent vector in the direction across a boundary curve can be found by simply interpolating between the tangent vectors in that direction at the 2 ends of the boundary curve using the functions H0 0 and H0 l. For example, Pu(0, w) is obtained as:

Pu( 0, w) = H0'0(w) Pu( 0,0) + ßîä (w) Pu( 0,1). (3.104)

In this interpolation, the rate of increase of Pu{0, w) in the w-direction is forcibly held to 0 at both ends (Fig. 3.21(a)). At both ends, the vectors showing the rate of tangent vector increase in the direction across the boundary curve are also specified; applying the cubic Hermite interpolation that was applied to the position vectors to the tangent vectors in the direction across the boundary curve gives a smoother interpolation. To do this, instead of equation (3.104), use:

Pu(0, w) = ß0>0(w) Pu(0,0) + H0 , (w) Pu(0,1) ¦+ ß1>0(w) Puw (0,0)

+ H1A(w)Puw{ 0,1). (3.105)

( a ) ( b )

Fig. 3.21. Effect of twist vectors on surface shape, (a) Case in which twist vectors are zero vectors; (b) case in which twist vectors are specified to be nonzero vectors

3.3 Surfaces

ÈÇ

If the tangent vector in the direction across the boundary curve is determined according to Eq. (3.105) in the course of deriving Ferguson’s surface formula, then surface patch formula (3.100), which is a special case of Coons’ 1967 surface, is obtained. If the twist vector is 0, that part will be flattened, but if the twist vector is nonzero then, from Fig. 3.21(b), it can be expected that the shape will become rounded. Since the effect of the twist vector on the surface shape is rather subtle, it is normally difficult to recognize that effect from a static image. It is easier to understand by moving the curved surface slowly on the display, or test-cutting the shape with an NC machine.

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