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)