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

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gives, for an equation corresponding to Eq. (2.16):

70 2. Lagrange Interpolation

Then, the divided difference expression for the parametric Lagrange polynomial corresponding to Eq. (2.21) is:

+ («-f0)(f-t,)... —1)6[«0> h, ¦¦¦, *„]• (2-27)

Problem 2.1. Using the divided difference expression for a parametric Lagrange polynomial, find the formula for the case n = 3, tt = i/n and confirm that the coefficient polynomial formula agrees with formula (2.12).

Solution: First find the divided differences.

e[to^i3=-%zr2-=-^1~g° =4Q,-Qo)

0 1 3-°

62-61

^2 ti 2 1

63 62 II IP 1 tp

Q It 2, Ó = v-f1-== ¦3 (Ñç ¦- Qi)

3_2 i——

ç

nrt t t 1 — — 6[^o?^i] 3 (Q2 — Qi) — 3 (6i — Q0)

Hlt0, tl, t2j — - - — —

= ~(Qo~2Qi + Q2)

ëã* * + _ 6[^2> ^ç] Q[ti, ?2] _ 3 (63 Q2) 3(^2 61)

UUl, 12, r3J — - - - -

3 1 1 ——

3

= -^{Ql-^Q2 + Qz)

Q[_t0, ti, t2, ?3]

2 v

Q[ti, t2, ?3] — Q[_t0, ti, f2]

j(e,-2Q2 + e3)-j(Qo~2Q1 + Q2)

'' T^o

: 2"^ — ^o + 3 Qi — 3 62 + 63)-

References (Chap. 2) 71

Setting n = 3 in formula (2.27), and substituting 6o[*o>*i>*2] and

¹o, *i, *2, ^ç] gives:

^(O=eo+(f-fo)e[fo, *i]+(*-*o)(*-*i)e[*o, h, ó

+ (t-*o) (t-*i) (t-t2)Q[t0, tu t2, t3]

=eo+3«(e1-e0)+|t(t-{)(eo-2e1+e2)

+Y((^—y) (/—y) 60 + Ç (2i — 3 62 + 63)

-òÈÕ'-óÊ^ò'Í)1'-"»'

= L0 (t) ??0 + Lx (t) + L2 (r) ??2 + L3 (r) Q3.

This confirms that the coefficient polynomial that has been obtained agrees with the Lagrange coefficient polynomial.

References (Chap. 2)

8) See for example Akira Sakurai, Introduction to Spline Functions, Tokyo Denki Daigaku Press, p. 14—15.

9) Forrest, A. R.: “Mathematical Principles for Curve and Surface Representation”, in Curved Surfaces in Engineering. I. J. Brown (ed.), IPC Science and Technology Press Ltd., Guildford, Surrey, England, 1972, p. 5.

3. Hermite Interpolation

3.1 Hermite Interpolation

Hermite interpolation is a generalized form of Lagrange interpolation. Whereas Lagrange interpolation interpolates only between values of a function f0, fu ..., /„ at different abscissas x0, xu ..., xn, Hermite interpolation also interpolates between higher order derivatives (Fig. 3.1). The following discussion deals with Hermite interpolation of function values and slopes.

Fig. 3.1. Hermite interpolation

When function values /0, /l5 ...,/„ and slopes /q, //, ..., /„' are given at

different abscissas x0, xl, ..., xn, the Hermite polynomial fH(x) used to

interpolate between these data is given by the following formula7):

/»w=i iwf (3.i)

i = 0 r = 0

where:

Hll{x) = [l-2L'l{xi) (õ-õ^Ùõ) (3.2)

H*J(x) = (x — xi)L2i(x) (3.3)

Ö(õ) is given by Eq. (2.4). From the fact that L;(x) is an nth-degree formula, we see that the polynomial in (3.1) is a (2 n + l)-degree polynomial which satisfies

2 (n + 1) conditions.

3.2 Curves 73

The most common form of Hermite interpolation used in CAD interpolates not between (n +1) points as in formula (3.1), but rather involves interpolation up to the /cth-order derivative between 2 points x0 and xx:

/íÌ= i I H'.Mff'K (3.4)

i = 0 r = 0

This formula gives a (2/c +l)-degree polynomial. Hr i(x) can be expressed in terms of Kronecker’s delta as:

(3.5)

which means:

d* è , ë f1 (i=J and r = s) ^

-7—Hr,i(xj) = )n r,. , , 3-6)

ax (, Î (1Ô7 or r + s)

In this chapter, we will discuss Ferguson curves and surfaces and Coons

surfaces based on Hermite interpolation.

3.2 Curves

3.2.1 Derivation of a Ferguson Curve Segment

As we found in Sect. 1.2.2, a parametric cubic curve is the lowest degree polynomial that can describe a space curve. A parametric cubic curve is given as follows:

P(t) = [t3 t2 t 1] [À Â Ñ D]T

= [t3 t2 t 1]M (3.7)

In this formula, M is a 4 x 3 matrix. Differentiating by t gives:

P(t) = [_3t2 It 1 0]M. (3.8)

The curve is supposed to have a position Q0 and tangent vector Q0 at t = 0, and a position Q{ and tangent vector Ql at t = 1 (Fig. 3.2). Substituting these conditions into formulas (3.7) and (3.8) and expressing the resulting relations in matrix form, we obtain:

~Qo~ "0 0 0 1~

Qi 1 1 1 1

Qo 0 0 1 0

Qi 3 2 1 0

74

3. Hermite Interpolation

Fig. 3.2. A Ferguson curve segment

This equation can be inverted to find M as:

M =

'0 0 0 1~ -1 -Qo

1 1 1 1 Qi

0 0 1 0 Qo

3 2 1 0 Qi

2-2 1 1

-3 3 -2 -1

0 0 10

10 0 0

Substituting this M into formula (3.7) gives the parametric cubic curve that satisfies the specified boundary conditions:

(3.9)

'2-2 1 Ã Qo Qo

— 3 3 -2 -1 Qi Qi

P{t) = [t31211] 0 0 10 Qo = [t3 t2 t 1 ]MC Qo

! 0 0 0_ Qi _6i.

where:

M =

'2-2 1 1' -3 3 -2 -1

0 0 10 10 0 0

(Ç.Þ)

The curve P(t) can also be written in the following form.

P(t) = lH0'0(t) Huo(t) Hlfl(t)]

= ßî,î(0 Qo + ßîä (0 Qi +1.0 (0 Qo + ä (0 Qx

where:

H0'0(t) = 2t3-3t2 + l=(t-l)2 (2t+l) H0A{t)= -2t3 + 3t2 =t2(3-2t) Hi,o{t) = t3 — 2t2 + t =(t-l)2t Hltl(t) = t3-t2 =(t-l)t2

(3.11)

(3.12)

(3.13)

3.2 Curves

75

Since they will be referred to frequently, let us write out Mc 1 and the first and second derivatives of H0 0(t), ..., ß1Ä(?):

M'1 =

0 0 0 1' 1111

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