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

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N = e • Pww, E = P'2, F = PU PW, G = P2

To find the extreme value of k„, take the partial derivatives with respect to ? and rj and set äêï/ä^ = 0 and äêï/äã} = 0 to obtain:

s2k„ = uGut

iiGu1 iiGu1 K’=~F~~=JFiF'

(1.103)

then:

L^2 + 2M^rj + Nrj2 — Kn (E^2+ 2F^r]+ Gr]2) = 0.

(1.104)

(L — êïÅ) ? + (M — KnF) r] = 0 1 (M-KnF)? + (N-KnG)tj = Q J

(1.105)

1.3 Theory of Surfaces 51

Eliminating ? and rj from these equations gives:

(EG — F2) ê2 — (EN + GL — 2FM) êï + LN — M2 = 0. (1.106)

This quadratic equation always has two real solutions êïòàõ and êïòò. êïòàõ

and Knmin are the maximum and minimum values of the normal curvature,

called the principal curvatures.

From the relation between the roots and the coefficients we have:

LN -M2 {e ¦ PJ (e ¦ Pww) - (e ¦ PUJ2

Ê = Kn max k„ mjn —

EG-F2 P2 Pi -{PUPJ

:-Öã (1Ë07)

1^1

EN+GL-2FM

2 H = êïòò + êïòàõ =

EG-F2

Pu (e ¦PwJ-2(P„-Pw){e- P„) + ÐÖå- PJ

|f| (1.108)

Ê is called the total curvature or the Gauss curvature, while H is called the mean curvature. It is easy to show that the two directions in which the principal curvatures are obtained are perpendicular to each other.

1.3.5 Calculation of a Point on a Surface

Many of the curved surface patch equations which are normally used are of the bi-cubic surface type:

P(u,w)= X X A3_U3_jUl\ i=0 j=0

(1.109)

= (A00w3 + A01w2 + A02w + A03)u3 + (A10w3 + A11w2 + A12w + A13)u2 + (A2oW3 + A21w2 + A22w + A23)u + A30w3 + A31w2 + A32w + A33

(1.110)

= [Ó è2 è 1]

Î î Aq>\ A 02 À 03 "w3

^10 A \2 À\ç w2

^20 A 21 A22 A 23 w

_^30 A$i A32 ^33_ 1

(1.111)

For a curved surface of this type, just as in the case of a curve, a point on a curved surface can be calculated rapidly by finite difference calculation.

52 1. Basic Theory of Curves and Surfaces

Taking w to be fixed in Eq. (1.110), we have:

P(u, w) = Au3 +Bu2 + Cu + D

where :

A=A00w3 + A01w2 + A02w + A03 B = A10w3 + A11w2 + A12w + A13 Ñ = A20w3 + A21w2 + A22w + A23 D = A30w3 + A31w2 + A32w + A33

which is the equation of a cubic curve. Then, using the finite difference matrix in Eq. (1.48), a point on a curve on the curved surface for which w is fixed can be found.

Letting 8 be the finite difference increment in the w-direction and denoting a forward finite difference by A, the finite difference matrix for u = 0 is, from Eq. (1.48):

‘ Po.» " D A30w3 + A31w2 + A32w + A33

Àä2 + B8 + C (d2A0o + 3A10 + A20)w3 + (S2A01 + 5An+A21)w2 + (S2Aq2 + SA l2 + A22) w + S2A03 + SA13 + A23

6A5+2B (6<5/400 + 2/410)w3 +(6 SA01 +2 A n)w2 + (6 dAQ2-^-2A]2)w + 6 3Aq3-\- 2 A13

6 A 6A00w3 + 6A01w2 + 6A02w + 6A03

(1.112)

Therefore, the finite difference matrix for w = 0 is:

Po,o ^33

1AP°'° d2A03 + SAl3+A23

~*2Po,o = 6dA03 + 2A13

6A 03

Next, in Eq. (1.112), let us take the finite difference in the w-direction at w = 0. Letting <5 also be the finite difference increment in this direction and denoting the forward finite difference in this direction by V, then the first order finite differences are:

1.3 Theory of Surfaces

53

Ã -VP 1 Àç0ä2 + A315 + A32

j^V(AP0,o) (<52^oo + <5^io+^2o)<52+(<52^oi +6An +A21)<5 + 82A 02 + <5 A 12+^22

Jr ÏÄ2Ðî,î) {63A00 + 2A10)d2 + (65A01 +2 An)3 + 6 SAq2 + 2 A12

/(^Xo) 6 A qq§2 6 Aq^ d 6 Aq2

(1.114)

The second and third order finite differences are:

^ 172 p S2 °’° 6 A308 + 2 A31

^V2(^Po,o) 6(ä2À00 + äÀ10 + À20)ä + 2(ä2À01+äÀï+À21)

-^V2(A2Po,o) 6(6SA00 + 2A10)S + 2(6SA0l+2An)

j^V2(A3Po,o) 36 A go <5 ~l~ 12A01

(1.115)

Os î

^V3(AP0,o) 6(^00 + ^10+^20)

]ã?×Ä2Ðî,î) 6(68A00 + 2Al0)

^Ó×ÀÚÐî,î) Çá^îî

Let us name the 4x3 matrices on the right-hand sides of Eqs. (1.113),

(1.114), (1.115) and (1.116), respectively as finite difference matrix A, finite difference matrix B, finite difference matrix Ñ and finite difference matrix D (Fig. 1.28).

Calculations on the curve P(u, 0) in the u-direction at w = 0 can be carried out by finite differences using matrix A, that is, Eq. (1.113). Next, to find the finite difference matrix to be used to calculate the curve P(u, <5), as shown in Fig. (1.28) perform finite difference calculations for each element corresponding to matrices À, Â, Ñ and D. Therefore, to generate a family of curves extending along the u-direction separated by a parametric distance ó (5 = l/2m,

1.3 Theory of Surfaces

55

ó = 1/2", mT^n) in the w-direction, it is sufficient to perform 1 curve-generating finite difference operation in the ì-direction for every 2m~" finite difference operations between matrices (Fig. 1.29).

Next, to find a family of curves extending in the w-direction, as shown in Fig. 1.30 matrix transpositions are performed using the vectors of finite difference matrices À, Â, Ñ and D. In other words, the process described above can be performed with the order of è and w reversed; this corresponds to the transformations A-Jx in the finite difference matrices.

1.3.6 Subdivision of Surface Patches

Let us now divide up a curved surface patch, using a bi-cubic surface described by Eq. (1.111) (Fig. 1.31).

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