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Polar Coordinate System
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Polar
and Cartesian coordinates relations |
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Conversion from polar
to rectangular coordinates |
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Conversion from
rectangular to polar coordinates |
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Polar coordinates of a point |
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Equation
of a circle in polar form |
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General
equation of a circle in polar coordinates |
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Polar
equation of a circle with a center on the polar axis running
through the pole |
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Polar
equation of a circle with a center at the pole |
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Polar coordinate system
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| The
polar coordinate system is a two-dimensional coordinate system
in which each point P
on a plane is determined by the length of its position vector r
and the angle q
between it and the positive direction of the x-axis,
where 0 <
r
< + oo
and 0
<
q
< 2p. |
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| Polar
and Cartesian coordinates relations, |
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| Note,
since the inverse tangent function (arctan or tan-1)
returns values in the range -p/2
< q <
p/2, then |
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for points lying in the 2nd or 3rd quadrant |
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and for points lying in the 4th quadrant |
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| Example:
Convert Cartesian coordinates
(-1,
-Ö3)
to polar coordinates. |
| Solution: |
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and since the point lies in the 3rd quadrant,
then |
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| Equation
of a circle in polar form |
| General
equation of a circle in polar coordinates |
| The
general equation of a circle with a center
at |
| (r0,
j)
and radius R. |
| Using
the law of cosine, |
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r2
+ r02
-
2rr0
cos(q
-
j)
=
R2 |
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| Polar
equation of a circle with a center on the polar axis running
through the pole |
| Polar
equation of a circle with radius R
and a center on the polar axis running through the pole O
(origin). |
| Since |
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then, |
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| Polar
equation of a circle with a center at the pole |
| Since,
r2
=
x2
+ y2
and x2
+ y2
=
R2
then |
r
=
R |
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| is
polar
equation of a circle with radius R
and a center at the pole (origin). |
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| Example:
Convert the polar
equation of a circle r
=
-4
cosq
into Cartesian coordinates. |
| Solution:
As, r
=
-4
cosq |
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then r2
=
-4r
cosq, |
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by using polar to Cartesian conversion formulas, r2
=
x2
+ y2
and x
=
r cosq |
| obtained
is
x2
+ y2
=
-4x |
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x2
+ 4x
+ y2
= 0
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or
(x
+ 2)2 + y2
= 4
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the
equation of a circle with radius R
= 2 and
the
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center
(-2,
0).
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Functions
contents A
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
