Math 454, Spring ’08, homework assignment #3, due Feb 28
Problem 1
Let
C
be a regular parametrized simple space curve of class
C
3
. Prove
that for any point
P
on
C
, there exists a plane
π
passing through
P
and such that
all nearby points on
C
lie on the same side of
π
. (
Hint:
it is one of the “coordinate
planes” determined by the orthonormal frame
{
t
,
n
,
b
}
. Use the local canonical form.)
Problem 2
Use vector analysis and the Frenet–Serret formulas for a regular space
curve
γ
(
s
) to compute the following quantities in terms of the vectors
t
(
s
)
,
n
(
s
)
,
b
(
s
)
and the curvature and torsion,
κ
(
s
)
, τ
(
s
):
1.
b
(
s
)
×
t
(
s
);
2.
n
(
s
);
3.
b
(
s
);
4.
γ
(
s
).
Prime denotes differentiation with respect to
s
and you may assume that all the nec
essary derivatives exist.
Problem 3
Let
γ
(
t
) be a circular helix as in Pressley, p. 26. Find the equations of the
osculating plane, rectifying plane, and the normal plane at the point
P
= (
x
0
, y
0
, z
0
)
of the helix. Your answer must be expressed in terms of the Cartesian coordinates of
P
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 Spring '08
 PROTSAK
 Math, Geometry, Matrices, Orthogonal matrix

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