Graphics Reference
In-Depth Information
Section 9.3
9.3.1.
Find the principal normal, the signed curvature, and the center of curvature for the
following curves p(t):
(a)
p(t) = (t,t
2
)
(parabola)
(a)
(cycloid)
p(t) = (t - sin t,1 - cos t)
(b)
(catenary)
p(t) = (t,cosh t)
9.3.2.
Show that a noncircular ellipse has four vertices.
Section 9.4
9.4.1.
Prove that the length of a curve, its curvature, and its torsion are invariant under a rigid
motion.
9.4.2.
Find the curvature, torsion, Frenet frame, and the equations for the osculating, normal,
and rectifying plane for the
twisted cubic
p(t) = (t,t
2
,t
3
) at the origin.
9.4.3.
Consider the curve p(t) = (t - sin t,1 - cos t,t). Show that its curvature and torsion is
defined by the following formulas:
12
/
t
Ê
Ë
ˆ
¯
4
14
+
sin
1
2
()
=
()
=-
k
t
,
t
t
.
32
/
t
t
Ê
Ë
ˆ
¯
4
14
+
sin
2
14
+
sin
2
2
Define a curve in
R
3
to be a
generalized helix
if it admits a regular parameterization F(t)
with the property that the tangent vector F¢(t) makes a constant angle q with a fixed
unit vector
u
, where 0 <q<p/2. Prove that a curve is a generalized helix if and only if
the ratio t
/
k is a nonzero constant (see [Wein00] or [Lips69]).
9.4.3.
9.4.4.
Let g(t) be a curve. Show that if g¢(t) and g≤(t) are linearly dependent for all t, then g(t)
is a straight line.
Section 9.6
9.6.1.
Show that the equation of the involute of the circle p(t) = (a cos t,a sin t), a > 0, is
(
(
)
(
)
)
qt
()
=
a
cos
t t
+
sin
t a
,
sin
t t
-
cos
t
.
9.6.2.
Show that the equation of the involute of the catenary p(t) = (t, cosh t), is
(
)
qt
()
=-
t
sinh
t
cosh ,
t
2
cosh
t
.
Section 9.7
9.7.1.
Let p(t) be a closed convex plane curve. Let
d
()
=
()
+
()
pt pt dnt
,
d
>
0
