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

Yamaguchi F. Curves and surfaces in computer aided geometric design - Tokyo, 1988. - 390 p.
Download (direct link): curvesandsurfacesincomputer1988.djvu
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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 /^ = 0 and /} = 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 ) ? + (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) 2 (EN + GL 2FM) + LN M2 = 0. (1.106)
This quadratic equation always has two real solutions and .
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
= Kn max k mjn
EG-F2 P2 Pi -{PUPJ
:- (107)
1^1
EN+GL-2FM
2 H = + =
EG-F2
Pu (e PwJ-2(P-Pw){e- P) + - PJ
|f| (1.108)
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)
= [ 2 1]
Aq>\ A 02 03 "w3
^10 A \2 \ 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 = 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
2 + 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
-VP 1 02 + A315 + A32
j^V(AP0,o) (<52^oo + <5^io+^2o)<52+(<52^oi +6An +A21)<5 + 82A 02 + <5 A 12+^22
Jr 2,) {63A00 + 2A10)d2 + (65A01 +2 An)3 + 6 SAq2 + 2 A12
/(^Xo) 6 A qq2 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(200 + 10 + 20) + 2(201++21)
-^V2(A2Po,o) 6(6SA00 + 2A10)S + 2(6SA0l+2An)
j^V2(A3Po,o) 36 A go <5 ~l~ 12A01
(1.115)
Os
^V3(AP0,o) 6(^00 + ^10+^20)
]?2,) 6(68A00 + 2Al0)
^,) ^
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 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 , , and D. Therefore, to generate a family of curves extending along the u-direction separated by a parametric distance (5 = l/2m,
1.3 Theory of Surfaces
55
= 1/2", mT^n) in the w-direction, it is sufficient to perform 1 curve-generating finite difference operation in the -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 , , and D. In other words, the process described above can be performed with the order of 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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