Computer Aided Engineering Design

178 COMPUTER AIDED ENGINEERING DESIGN

6.5 Curvature of a Surface: Gaussian and Mean Curvature

The curve C in Figure 6.12 lies on a surface passing through a point P. Let π denote the tangent plane
containing vectors ru and rv and n be the unit normal to the surface at P. The unit tangent vector t to
the curve at P also lies on π. Let κ denote the curvature and nc be the unit normal to curve at P. The
curvature vector k is in the direction of nc and can be decomposed into two components: (a) kn in the
direction of n and (b) kg in the plane π but perpendicular to t. Now

kn

t

= κκcng = = kk + with kn = n

d

ds n (6.25)

Herekn and kg are called the vectors of normal curvature and geodesic curvature, respectively. κn is
called the normal curvature of the surface at P. Since n and t are mutually perpendicular, n·t = 0.

Figure 6.11 Parabolic, elliptic and hyperbolic points on the torus.

b

1

2 π

a u

-^12 π or^32 π

Thus

d

ds

d

ds

n

tn

t

+ = 0⋅⋅ (6.26)

wheresis the arc length parameter of the curve C. From Eq. (6.25), since kg and n are perpendicular,
kg·n = 0 using which

d

ds

d

gnnds

t

nk k nkn nn t

n

= (⋅⋅⋅⋅⋅nn + ) = = κκ = = – (6.27)

Figure 6.12 Curvature of a surface

n

π

S

P

t

nc

κnn κnc

κgtg

C(u(t),v(t))

(tg⊥t)