DIFFERENTIAL GEOMETRY OF SURFACES 177duMMLN
L
dduMMLN
L
= d
- –
and =
- –
(^22) –– –
vv
here P is called a hyperbolic point on the surface. All regular points of a pseudosphere and all points
of a catenoidare hyperbolic (Figure 6.4). All points of the hyperbolic paraboloid and hyperboloid of
one sheet, or a non-developable ruled surface (discussed later) are also hyperbolic points.
Case 4. L = M = N = 0: The tangent plane is not only tangent to the surface at P, but also has a contact
of higher order with the surface. In this case, point P on the surface is called a flat point, because
points in the small neighborhood of P are also the points of tangency to the same tangent plane. A
monkey saddle z = x(x^2 – 3y^2 ) has a flat point at (0, 0, 0).
Example 6.2. Show that the lines u = ±^12 π are parabolic lines of the torus x = (b+acosu) cos v,
y = (b + acosu) sin v,z = asinu with b > a. These lines divide the torus into two domains.
Show that the exterior domain –π/2 < u < π/2 consists of elliptic points and the interior domain
π/2 < u < 3π/ 2 consists of hyperbolic points.
The expression for M^2 – LN is computed using Eqs. (6.19) and (6.22) to obtain
x (u,v)= (b + acosu) cos v, y (u,v)= (b + acosu) sin v, z (u,v)= asinu
xu=–asinu cos v, yu=–asinu sin v, zu=acosu
xv=–(b + acosu) sin v, yv= (b + acosu) cos v, zv= 0
xuu=–acosu cos v, yuu=–a cos usinv, zuu=–asinu
xuv=a sin u sin v, yuv=–acosv sin u, zuv= 0
xvv=–(b + acosu) cos v, yvv=–(b+acosu) sin v, zvv= 0
D
xyz
xyz
xyz
D
xyz
xyz
xyz
D
xyz
xyz
xyz
uu uu uu
uuu
uuu
11 = , 12 = uuu , 22 = uuu^
vvv
vvv
vvv
vvvvvv
vvv
Substitution and simplification of the determinants yields
D 11 = a^2 (b + acosu), D 12 = 0, D 22 = a(b + a cos u)^2 cos u
MLN
aubau
ABC
2
33
22 2
- = –
cos ( + cos )
whereA,B and C are defined in Eq. (6.21b). For u = ±^12 π, cos (u) = 0 and hence M^2 – LN = 0. This
corresponds to Case 2 above for all values of the parameter vand hence u = ±^12 π are parabolic points
on the surface of the torus.
For–
2
< <
2ππuu, cos ( ) > 0, since aand (b + acosu) are both greater than 0, M (^2) – LN < 0 for
all values of v. This shows (Case 1) that the exterior part of the torus has all elliptic points. For
π π
2
< <
3
2
u , cos u < 0, with aand (b + acosu) > 0. Thus, M^2 – LN > 0 corresponds to Case 3 above.
Hence, the surface patch corresponding to π
π
2
< <
3
2
u has all hyperbolic points.