Show that for the curve x = a cos t (1+sin t), y = a sin t (1+cos t), the radius of curvature is a, at the point for which the value of the parameter 0 is –pie/4
Answers
Answer:
Curvature and Radius of Curvature
Consider a plane curve defined by the equation
y
=
f
(
x
)
.
Suppose that the tangent line is drawn to the curve at a point
M
(
x
,
y
)
.
The tangent forms an angle
α
with the horizontal axis (Figure
1
).
Definition of the curvature of a plane curve
Figure 1.
At the displacement
Δ
s
along the arc of the curve, the point
M
moves to the point
M
1
.
The position of the tangent line also changes: the angle of inclination of the tangent to the positive
x
−
axis
at the point
M
1
will be
α
+
Δ
α
.
Thus, as the point moves by the distance
Δ
s
,
the tangent rotates by the angle
Δ
α
.
(The angle
α
is supposed to be increasing when rotating counterclockwise.)
The absolute value of the ratio
Δ
α
Δ
s
is called the mean curvature of the arc
M
M
1
.
In the limit as
Δ
s
→
0
,
we obtain the curvature of the curve at the point
M
:
K
=
lim
Δ
s
→
0
∣
∣
∣
Δ
α
Δ
s
∣
∣
∣
.
From this definition it follows that the curvature at a point of a curve characterizes the speed of rotation of the tangent of the curve at this point.
For a plane curve given by the equation
y
=
f
(
x
)
,
the curvature at a point
M
(
x
,
y
)
is expressed in terms of the first and second derivatives of the function
f
(
x
)
by the formula
K
=
|
y
′
′
(
x
)
|
[
1
+
(
y
′
(
x
)
)
2
]
3
2
.
If a curve is defined in parametric form by the equations
x
=
x
(
t
)
,
y
=
y
(
t
)
,
then its curvature at any point
M
(
x
,
y
)
is given by
K
=
|
x
′
y
′
′
−
y
′
x
′
′
|
[
(
x
′
)
2
+
(
y
′
)
2
]
3
2
.
If a curve is given by the polar equation
r
=
r
(
θ
)
,
the curvature is calculated by the formula
K
=
∣
∣
r
2
+
2
(
r
′
)
2
−
r
r
′
′
∣
∣
[
r
2
+
(
r
′
)
2
]
3
2
.
The radius of curvature of a curve at a point
M
(
x
,
y
)
is called the inverse of the curvature
K
of the curve at this point:
R
=
1
K
.
Hence for plane curves given by the explicit equation
y
=
f
(
x
)
,
the radius of curvature at a point
M
(
x
,
y
)
is given by the following expression:
R
=
[
1
+
(
y
′
(
x
)
)
2
]
3
2
|
y
′
′
(
x
)
|
.
