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

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e is called the unit normal vector.-

1.3.2 The First Fundamental Matrix of a Surface

As was stated in the preceding section, a curve on the curved surface P(u, w) can be given by u = u(t), w = w(t), (aSLt^b). The following notation is used for this.

[R-P(u0, w0), Pu, Pw] = 0.

(1.82)

Pu x Pw Pu x Pw

(1.83)

\/(PuxPJ2 1/JJ+jf+jJ'

(1.84)

From Eq. (1.81), the tangent vector to this curve is given by:

= éÀ.

(1.85)

Here A is defined by:

(1.86)

The square of the magnitude s of the tangent vector is given by: S2 = P{t)2 = P(t) ' P{t)T

= uFuT.

(1.87)

1.3 Theory of Surfaces

Here F is the matrix defined by:

FJ H Ð.-ÐË

Lpu pw pi J'

(1.8

This matrix F is called the first fundamental matrix of the curved surface.

Using the first fundamental matrix, the unit tangent vector t of a curve on a curved surface is:

P{t)

UA

\Ht)\ (uFuTy

(1.89)

Therefore, the cosine of the angle 9 between the two curves u1=u1(t) and u2 = u2 (t) on the curved surface is, using (1.89):

cosv = t, -t2=-

éõË ¦ ATit[

u.Ful

i«\Fu[y {u2Fu\Y («!*«[)' {AiFulY

The length of a curve on the curved surface is given by:

I = J sdt = j (uFuT)^ dt.

(1.90)

(1.91)

Next, let us consider a small area of a curved surface surrounded by the parametric curves u = u0, u = u0 + Au, w = w0, w = w0 + zlw. Referring to Fig. 1.26, we see that the area AS of this small segment of surface is:

AS = \PU xPJAuAw.

P(u0,W0 + du))

48

1. Basic Theory of Curves and Surfaces

We also have:

| Pu x Pw\2 = Pi Pi-~(PU ¦ PJ2 *> = |F|.

(1.92)

Therefore, the curved surface area corresponding to the region R on the uw plane is:

1.3.3 Determining Conditions for a Tangent Vector to a Curve on ÷ Surface

As shown in ihe preceding section, since the unit tangent vector of a curve on a curved surface is given by Eq. (1.89), the condition that determines the unit tangent vector t is:

are satisfied, the unit tangent vectors are determined for all curves u = u(t). If the second condition is satisfied but the first is not, then, as can be seen from Eq. (1.81) and (1.83), the tangent plane and normal line are determined, but the unit tangent vector t is not determined due to twisting of the curve u = u(t). If the first condition is satisfied but the second is not, there can be a cusp or the w-curve and the w-curve can be parallel, with the result that the tangent plane is undetermined7’.

S = N\F\2dudw.

(1.93)

R

uFuT> 0.

(1.94)

In the case P2 ô 0, changing the form of the equation slightly gives :

\2 ID xp 12

w) + u pi vv2 >0 (Ð2ô0). (1.95)

= Ð2é2 + 2PU ¦ Pwiiw + P2w2

This equation implies that if the conditions

« Ô 0 and Pu x Pw Ô 0

(1.96)

*> Using the vector identity (A xB) {CxD) = {AC) (BD)-{AD) (BC)

1.3 Theory of Surfaces

49

1.3.4 Curvature of a Surface

From Eq. (1.9) we have:

P = sP' = St.

Differentiating by t we obtain the equation:

P = st + si = st + s2t'

= st + s2K/i. (1-97)

Now let us find P for the curve u(t) = [u(t)w(t)'] on the curved surface P(u, w). From Eq. (1.81) we have:

P = Pu1A + PwW.

Therefore:

dPu n dPw P = H + Puii + —-^-w + Pww dt dt

= Ðèèé2 +Puwuw + Puti + Pwww2 + Pwuwii + Pww = Puu u2 + Puwuw + Pwu liw + Pwww2 + Puti + Pww. (1-98)

Taking the inner product of the unit normal vector e to the curved surface and P from Eq. (1.97), since e and t are perpendicular with each other:

e ¦ P=s2k/i ¦ e. (1.99)

Also taking the inner product of the unit normal vector e and P from Eq. (1.98), since e is perpendicular to both Pu and Pw :

e ¦ P = e ¦ Puuu2 + e ¦ Puwiiw + e ¦ Pwuiiw + e ¦ Pwww2

= iiGiiT

(1.100)

where:

(1.101)

From Eq. (1.99) and (1.100) we obtain: s2Ktt-e = uGuT.

(1.102)

50

1. Basic Theory of Curves and Surfaces

Fig. 1.27. The intersection curve Ñ of the plane containing the tangent vector P and the unit normal vector e with a curved surface

The matrix G is called the second fundamental matrix of the curved surface. Normally, in the curved surface representation encountered in CAD, it often happens that PUW = PWU. In such a case, G is a symmetrical matrix.

Let a space curve u = u(t) be given on the curved surface P(u, w) and let Ñ be the curve formed by the intersection of the curved surface with the plane which includes the tangent vector P = uA at a point P on the curve and the unit normal vector e (refer to Fig. 1.27). The curvature of the curve Ñ is called the normal curvature relative to the direction éÀ at point P. The normal curvature is the projected length of the curvature vector of the curve è to e. Letting êï be the normal curvature, from Eq. (1.102) we have:

The sign of êï is plus if the curve Ñ is concave in the direction of e, negative if it is convex. As shown in Fig. 1.27, êï<0 at point P.

In general, if the direction of the intersection curve at point P changes, the curvature êï will also change. The direction in which êï takes an extreme value is called the principal direction of the normal curvature. In Eq. (1.103), if we take:

? = m, *7 = w, L = e • Puu, M = e ¦ Puw = e ¦ Pwu,

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