Computer Aided Engineering Design

DESIGN OF SURFACES 207

n

rr

rr

(, ) =

(, ) (, )

| ( , ) ( , ) |

u

uu

uu

u

u

v

vv

vv

v

v

×

×

(7.16)

n

Q

P

vj

ui

Z

Y

X

O

rv(ui,vj)

ru(ui,vj)

Figure 7.3 Tangent plane on the surface patch.

The equation of the tangent plane at P = r(ui,vj) with Q(x,y,z) as any general point on the plane may
be obtained using Eqs. (6.6) and (6.7) as

r(ui,vj) = x(ui,vj)i + y(ui,vj)j + z(ui,vj)k = xiji + yijj + zijk

ru(ui,vj) = xu(ui,vj)i + yu(ui,vj)j + zu(ui,vj)kijk = xyziju + iju + iju

rv(ui,vj) = xv(ui,vj)i + yv(ui,vj)j + zv(ui,vj)kijk = xyzijvvv + ij + ij (7.17)

xxxx

yyyy

zzz

ij ij

u

ij

ij ij

u

ij

ij ij

u

ij

• 
  


• 
  


• 
  

= 0

v

v

z v

The Gaussian curvature K and the mean curvature H may also be obtained using Eq. (6.36) after
computing the expressions for G 11 , G 12 , G 22 and L,M,N depending on the derivatives of r(u,v),and
are detailed in Chapter 6.

G = (, ) (, ) G = (, ) (, ) G = (, ) (, )

= (, ) (, ) = (, ) (, ) = (, )

11 rr 12 rr 22 rr

rn rn rn

uu u

uu u

uu uu uu

Lu u Mu u Nu

vvvvvv

vvvvv

vvv

vvv

⋅⋅⋅

⋅⋅ ⋅((, )uv

(7.18)

From these the principal curvatures can be determined. Issues such as developability and point
classification (whether the given point on the surface is elliptic, hyperbolic, or parabolic) can also be
answered by examining whether the function LN–M^2 = 0 (parabolic point), LN–M^2 > 0 (elliptic
point) or LN–M^2 < 0 (hyperbolic point). The surface area of the patch can be determined using