Lecture Notes for 120 - UCLA Department of Mathematics
Lecture Notes for 120 - UCLA Department of Mathematics
Lecture Notes for 120 - UCLA Department of Mathematics
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5.5. RULED SURFACES 129<br />
and<br />
<strong>for</strong> a constant v 0 .<br />
The surface is then given by<br />
2<br />
c = 4<br />
q (s, v) = 4<br />
which shows explicitly that it is a helicoid.<br />
3<br />
0<br />
0 5<br />
hv + v 0<br />
2<br />
s cos v<br />
3<br />
s sin v 5<br />
Exercises.<br />
(1) Let q (s) be a unit speed asymptotic line <strong>of</strong> a ruled surface q (u, v) =<br />
↵ (v)+uX (v). Notethatu-curves are asymptotic lines.<br />
(a) Show that<br />
⇤<br />
=0<br />
det ⇥ d↵<br />
¨q, X,<br />
dv + u dX<br />
dv<br />
(b) Assume <strong>for</strong> the remainder <strong>of</strong> the exercise that K < 0. Show that<br />
there is a unique asymptotic line through through every point that<br />
is not tangent to X.<br />
(c) Show that this asymptotic line can locally be reparametrized as<br />
where<br />
h<br />
du det<br />
dv =<br />
↵ (v)+u (v) X (v)<br />
d↵<br />
dX<br />
X,<br />
dv<br />
+ u (v)<br />
dv ,<br />
d 2 ↵<br />
dv 2<br />
2det ⇥ d↵<br />
dv , X, dX<br />
dv<br />
+ u (v) d2 X<br />
dv<br />
⇤<br />
2<br />
(2) Show that a generalized cylinder q (u, v) =↵ (v)+uX where X is a fixed<br />
unit vector admits a parametrization q (s, t) =c (t) +sX (t), where c is<br />
parametrized by arclength and lies a plane orthogonal to X.<br />
(3) Consider a parameterized surface q (u, v). ShowthattheGausscurvature<br />
vanishes if and only if @N<br />
@u , @N<br />
@v<br />
are linearly dependent everywhere.<br />
(4) Consider<br />
q (u, v) = u + v, u 2 +2uv, u 3 +3u 2 v<br />
(a) Determine when it defines a surface.<br />
(b) Show that the Gauss curvature vanishes.<br />
(c) What type <strong>of</strong> ruled surface is it?<br />
(5) Consider the Monge patch<br />
nX<br />
z = (ax + by) k + cx + dy + f<br />
k=2<br />
(a) Show that the Gauss curvature vanishes.<br />
(b) Depending on the values <strong>of</strong> a and b determine the type <strong>of</strong> ruled<br />
surface.<br />
(6) Consider the equation<br />
xy =(z c) 2<br />
(a) Determine when it defines a surface.<br />
i<br />
⇤