Graphics Reference
In-Depth Information
Section 9.14
9.14.1.
Show that if
S
is a surface of constant mean curvature h π 0, then the parallel surface
S
1/2h
has constant Gauss curvature 4 h
2
. (This is a theorem of Bonnet.)
Section 9.15
9.15.1.
(a)
Prove Lemma 9.15.3.
(b)
Consider a curve
()
=
()
+
()(
.
m
uquruu
a
Show that the condition m¢ • a¢ = 0 defines a unique such curve if a¢ π 0 and that
this is the line of striction.
9.15.2.
Find the line of striction for the Moebius strip
q
q
q
Ê
Ë
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
ˆ
¯
()
=+
F q
,
t
2
t
cos
cos ,
q
2
+
t
sin
sin
q
,
t
sin
.
2
2
2
9.15.3.
Show that every singular point of a noncylindrical ruled surface lies on its line of
striction.
9.15.4.
Show that the surface obtained by revolving a segment about a line is developable.
9.15.5.
Let g(s) be a space curve parameterized by arc-length. Let (T(s),N(s),B(s)) be its moving
trihedron. Find conditions under which the following surfaces are developable:
(a)
F(s,t) =g(s) + tN(s)
(b)
F(s,t) =g(s) + tB(s)
Show that the hyperbolic paraboloid z = x
2
- y
2
is a doubly ruled surface.
9.15.4.
Section 9.16
Let g(s) be a curve in
R
3
parameterized by arc-length. Assume that k(s) > 0. Let
(
U
1
,
U
2
,
U
3
) be a frame field that extends the Serret frame (T,N,B) in a neighborhood of
the curve. Let
X
=
U
1
. Show that the Equations (9.94) in Theorem 9.16.6 reduce to the
Serret-Frenet equations.
9.16.1.
