MATHEMATICS 317
Assignment #2
due at the beginning of class on Wednesday, January 20
th
1
(a)
For a curve defined by
,
0
)
,
(
=
y
x
f
show that the radius of curvature
R
at any
point of the curve is given by the expression
.
2
]
[
2
2
2
/
3
2
2
yy
x
xx
y
xy
y
x
y
x
f
f
f
f
f
f
f
f
f
R
-
-
+
=
(b)
Find the radius of curvature at the point
)
0
,
(
a
for the ellipse
.
1
2
2
2
2
=
+
b
y
a
x
2.
(a)
If
r
and
θ
are polar coordinates, show that the radius of curvature
R
at any point
of the curve defined by
)
(
θ
f
r
=
is given by the expression
.
2
]
[
2
2
2
/
3
2
2
f
f
f
f
f
f
R
′
′
-
′
+
′
+
=
(b)
Find the radius of curvature at any point lying on the cardioid
Unformatted text preview: ). cos 1 ( + = a r 3. (a) Prove that the torsion 2 2 2 2 3 3 2 2 ) ( ds d ds d ds d ds d ds d s r r r r r ⋅ × ⋅ = τ for a curve r ( s ) defined in terms of its intrinsic arc length parameter s . (b) Use the formula derived in (a) to find the torsion at any point of the circular helix defined by . sin cos ) ( k j i r bt t a t a t + + =...
View
Full Document
- Spring '10
- BLUMAN
- Trigraph, Conic section, Radius of curvature
