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148
CHAPTER 2. CURVES
and then using the identities
r
2
=
x
2
+
y
2
and
y
=
r
sin
θ
, we can write
x
2
+
y
2
=
y
which, after completing the square, can be rewritten as
x
2
+
p
y
−
1
2
P
2
=
1
4
.
We recognize this as the equation of a circle centered at (0
,
1
2
) with radius
1
2
.
The polar equation
r
= sin 2
θ
may appear to be an innocent variation on the preceding, but it turns out
to be quite diFerent. Again the curve begins at the origin when
θ
= 0 and
r
increases with
θ
, but this time it reaches its maximum
r
= 1 when
θ
=
π
4
,
which is to say along the diagonal (
−→
p
(
π
4
)
= (
1
√
2
,
1
√
2
)), and then decreases,
hitting
r
= 0 and hence the origin when
θ
=
π
2
. Then
r
turns negative,
which means that as
θ
goes from
π
2
to
π
, the point
−→
p
(
θ
) lies in the fourth
quadrant (
x >
0,
y <
0); for
π < θ <
3
π
2
,
r
is again positive, and the point
makes a “loop” in the third quadrant, and ±nally for
3
π
2
< θ <
2
π
, it
traverses a loop in the second quadrant. After that, it traces out the same

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