| |
|
|
Polar Coordinate System
|
Polar
and Cartesian coordinates relations |
|
Conversion from polar
to rectangular coordinates |
|
Conversion from
rectangular to polar coordinates |
|
Polar coordinates of a point |
|
Equation
of a circle in polar form |
|
General
equation of a circle in polar coordinates |
|
Polar
equation of a circle with a center on the polar axis running
through the pole |
|
Polar
equation of a circle with a center at the pole |
|
|
|
|
|
|
|
Polar coordinate system
|
| 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. |
 |
|
| Polar
and Cartesian coordinates relations, |
 |
| Note,
since the inverse tangent function (arctan or tan-1)
returns values in the range -p/2
< q <
p/2, then |
|
for points lying in the 2nd or 3rd quadrant |
 |
|
|
and for points lying in the 4th quadrant |
 |
|
|
| Example:
Convert Cartesian coordinates
(-1,
-Ö3)
to polar coordinates. |
| Solution: |
 |
|
and since the point lies in the 3rd quadrant,
then |
|
|
|
 |
|
 |
|
 |
|
|
|
| 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, |
|
r2
+ r02
-
2rr0
cos(q
-
j)
=
R2 |
|
|
|
 |
|
|
| 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 |
 |
then, |
|
|
|
|
 |
|
|
| Polar
equation of a circle with a center at the pole |
| Since,
r2
=
x2
+ y2
and x2
+ y2
=
R2
then |
r
=
R |
|
|
| is
polar
equation of a circle with radius R
and a center at the pole (origin). |
|
| Example:
Convert the polar
equation of a circle r
=
-4
cosq
into Cartesian coordinates. |
| Solution:
As, r
=
-4
cosq |
|
then r2
=
-4r
cosq, |
| and
by using polar to Cartesian conversion formulas, r2
=
x2
+ y2
and x
=
r cosq |
| obtained
is
x2
+ y2
=
-4x |
|
x2
+ 4x
+ y2
= 0
|
|
or
(x
+ 2)2 + y2
= 4
|
|
the
equation of a circle with radius R
= 2 and
the
|
|
center
(-2,
0).
|
|
 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Functions
contents B
|
|
 |
|
| Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |
