Polar Coordinates
This is a coordinate system in a plane, or two dimensions.
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Coordinates
Start with a point O in the plane, the pole. Through the pole O
choose a ray (half a line) bounded by O. This is the polar
axis. To any point P corresponds a pair of real numbers called its
polar coordinates, r and theta, determined as follows. Connect
P to O. Measure the distance r from pole O to point P, and measure the
angle theta between the axis and OP in a counterclockwise direction in
radians (1 radian = 180/Pi degrees). Here is a diagram to illustrate
this:
* P(r,theta)
/
/
/
r/
/<.
/ `.
/ theta\
/ .
O o+'++> r
0 1 2 3
In this example, we have plotted the point (3,Pi/3).
Conversely, given a coordinate pair (r,theta), construct from pole O
the ray making angle theta with the polar axis, and measure a distance
of r along that ray. The endpoint is P.
Observe that (r,theta) and (r, theta+Pi) are two different pairs of
coordinates that represent the same point. Also observe that
(0, theta) for any value of theta are infinitely many different pairs
of coordinates that represent the same point, the pole O.
To transform from rectangular coordinates to polar ones and vice versa,
use the following formulas:
x = r cos(theta),
y = r sin(theta),
r = ±sqrt(x^{2}+y^{2}),
theta = arctan(y/x).
The sign of r is determined by which of the values of the arctangent
function is chosen:
Sign of x 
Sign of y 
Quadrant of theta 
Sign of r 
+  + 
I  + 
+  + 
III  – 
–  + 
II  + 
–  + 
IV  – 
–  – 
I  – 
–  – 
III  + 
+  – 
II  – 
+  – 
IV  + 
The quadrant of theta can always be chosen to make r positive, if it
is so desired.
Note: Many equations which are simple in Cartesian coordinates
are very complicated in polar coordinates, and vice versa. To solve
any particular problem, one system may be much more tractable than
the other. The ability to convert from one to the other quite
readily allows us to try both, then use only the easier one.
Let the coordinates of P_{1} be
(r_{1},theta_{1}), those of P_{2} be
(r_{2},theta_{2}), and those of P_{3} be
(r_{3},theta_{3}).
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Points
The distance from P_{1} to P_{2} is
d = sqrt(r_{1}^{2}+r_{2}^{2}2r_{1}r_{2}cos[theta_{2}theta_{1}]).
The coordinates of the point dividing the line segment
P_{1}P_{2} in the ratio a/b are:
(sqrt[b^{2}r_{1}^{2}+a^{2}r_{2}^{2}+2abr_{1}r_{2}cos(theta_{2}theta_{1})]/[a+b],
arctan([br_{1}sin(theta_{1})+ar_{2}sin(theta_{2})]/[br_{1}cos(theta_{1})+ar_{2}cos(theta_{2})]).
As a special case, when a = b, the midpoint of the line segment has
coordinates
(sqrt[r_{1}^{2}+r_{2}^{2}+2r_{1}r_{2}cos(theta_{2}theta_{1})]/2,
arctan([r_{1}sin(theta_{1})+r_{2}sin(theta_{2})]/[r_{1}cos(theta_{1})+r_{2}cos(theta_{2})]).
P_{1}, P_{2}, and P_{3} are collinear if and
only if the determinant

r_{1}cos(theta_{1}) 
r_{1}sin(theta_{1}) 
1 

= 0, 
r_{2}cos(theta_{2}) 
r_{2}sin(theta_{2}) 
1 
r_{3}cos(theta_{3}) 
r_{3}sin(theta_{3}) 
1 
or else
r_{1}r_{2}sin(theta_{2}theta_{1}) + r_{2}r_{3}sin(theta_{3}theta_{2}) + r_{3}r_{1}sin(theta_{1}theta_{3}) = 0.
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Directions
A direction is determined an angle alpha that a line in that direction
makes with the polar axis. The angle alpha is called the
inclination of the direction, and m = tan[alpha] is called the
slope of the direction.
In this context, we can think of the possibility of infinite slope,
which occurs if the inclination is Pi/2, so the direction is vertical.
We interpret 1/m to be 0 in that case.
Two directions are parallel if and only if any of the following
relationships hold:
alpha_{1} = alpha_{2},
m_{1} = m_{2}.
The angle between two directions is given by
alpha_{2}  alpha_{1} = arctan([m_{2}m_{1}]/[1+m_{1}m_{2}]).
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Translations
If we wish to move the pole of our polar coordinate system from (0,0)
to (r_{0},theta_{0}), and keep the new polar axis
parallel to the old one, call the coordinates with respect to the new
system (r',theta'). Then we get the following equations relating the
new coordinates to the old:
r' = sqrt(r^{2}+r_{0}^{2}2rr_{0}cos(thetatheta_{0}),
theta' = arctan([r sin(theta)r_{0}sin(theta_{0})]/[r cos(theta)r_{0}cos(theta_{0})]),
r = sqrt(r'^{2}+r_{0}^{2}+2r'r_{0}cos(theta'theta_{0}),
theta = arctan([r'sin(theta')+r_{0}sin(theta_{0})]/[r'cos(theta')+r_{0}cos(theta_{0})]).
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Rotations About the Pole
If we wish to rotate the polar axis through an angle of
theta_{0} while keeping the pole fixed, the transformation of
coordinates is
r' = r,
theta' = theta  theta_{0},
r = r',
theta = theta' + theta_{0}.
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Lines
Let the slope of the line be m, its intersection with the polar axis be
(a,0), its perpendicular distance from the pole be p, its inclination
be alpha, and the inclination of any line perpendicular to it be omega.
Then omega = alpha ± Pi/2. If the line is through the pole O, so that
p = 0, the equation reduces to sin(thetaalpha) = 0. If the line
passes through the pole O and point P_{1}, then the equation is
sin(thetatheta_{1}) = 0.
The equation of a line not through the pole O can have any of several
forms:
 Polar normal form:
r = p sec(thetaomega).
 Slope rintercept form:
r = a sin(alpha)/sin(alphatheta),
if the rintercept is (a,0), a > 0, or
r = a sin(alpha)/sin(thetaalpha),
if the rintercept is (a,Pi), a > 0.
 Two point form:
r = r_{1}r_{2}sin(theta_{2}theta_{1})/(r_{1}sin[thetatheta_{1}]r_{2}sin[thetatheta_{2}]).
 Point slope form:
r = (mr_{1}cos[theta_{1}]r_{1}sin[theta_{1}])/(m cos[t]sin[t]),
if m is finite.
 Point inclination form:
r = r_{1}sin[theta_{1}alpha]/(sin[thetaalpha]),
if m is finite.
 Parametric form:
r = sqrt(t^{2}+r_{1}^{2}+2tr_{1}cos[theta_{1}alpha]),
theta = Arccos([t cos(alpha)+r_{1}cos(theta_{1})]/[sqrt(t^{2}+r_{1}^{2}+2tr_{1}cos[theta_{1}alpha])]),
where t is any real number.
 General form:
A cos(theta) + B sin(theta) + C/r = 0,
where A, B, and C are real numbers, not both A and B are zero, and C is
nonzero.
The distance from r = p sec(thetaomega) to P_{1} is
d = r_{1}cos(theta_{1}omega)  p.
If r = p_{1}sec(thetaomega_{1}) and
r = p_{2}sec(thetaomega_{2}) are two intersecting
lines, then their intersection point has coordinates
r = sqrt(p_{1}^{2}+p_{2}^{2}2p_{1}p_{2}cos[omega_{1}omega_{2}])/sin(omega_{1}omega_{2}),
theta = Arccos([p_{2}sin(omega_{1})p_{1}sin(omega_{2})]/[sqrt(p_{1}^{2}+p_{2}^{2}2p_{1}p_{2}cos[omega_{1}omega_{2}]).
Three lines
A_{1}cos(theta) + B_{1}sin(theta) + C_{1}/r = 0,
A_{2}cos(theta) + B_{2}sin(theta) + C_{2}/r = 0,
A_{3}cos(theta) + B_{3}sin(theta) + C_{3}/r = 0,
are concurrent (that is, all pass through a single point) if and only
if the determinant

A_{1} 
B_{1} 
C_{1} 

= 0. 
A_{2} 
B_{2} 
C_{2} 
A_{3} 
B_{3} 
C_{3} 
Three lines
r = p_{1}sec(thetaomega_{1}),
r = p_{2}sec(thetaomega_{2}),
r = p_{3}sec(thetaomega_{3}),
are concurrent (that is, all pass through a single point) if and only
if
p_{1}sin(omega_{3}omega_{2}) +
p_{2}sin(omega_{1}omega_{3}) +
p_{3}sin(omega_{2}omega_{1}) = 0.
The perpendicular bisector of the line segment
P_{1}P_{2} has the equation
r = (r_{1}cos[theta_{1}alpha] +
r_{2}cos[theta_{2}alpha])/(2 cos[thetaalpha]).
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Triangles
The area of a triangle whose vertices are P_{1}, P_{2},
and P_{3} is given by
K = [r_{1}r_{2}sin(theta_{2}theta_{1}) + r_{2}r_{3}sin(theta_{3}theta_{2}) + r_{3}r_{1}sin(theta_{1}theta_{3})]/2.
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Conic Sections
A conic section is the set of points P in a plane determined by a line
D (a directrix) and a point F (a focus) not on D, such
that the ratio of distances PF/PD = e (the eccentricity, which
is nonnegative). A vertex is a point where the distances PF and
PD are least. An axis is the line through F perpendicular to D.
The latus rectum is the distance between those two points on the
curve which also lie on a line through F parallel to D.
The value of e determines what kind of curve the conic section forms:
 e > 1, an hyperbola.
 e = sqrt(2), an equilateral hyperbola.
 e = 1, a parabola.
 e < 1, an ellipse.
 e = 0, a circle.
If F is at the pole O, and D has equation r = a sec(theta), then the
equation of the conic section is
r = ae/(1e cos[theta]).
A vertex has coordinates (ae/[1+e],Pi), an axis has equation
sin(theta) = 0, and the latus rectum has length 2ae/(1+e).
If F is at the pole O, and D has equation r = a sec(thetaomega), then
the focus form of the equation of the conic section is
r = ae/(1e cos[thetaomega]).
If F is at (ae,0) and D has equation r = a sec(theta), so the pole O
is a vertex, then the vertex form of the equation of the conic
section is
r = 4ae cos[theta]/(1e^{2}cos^{2}[theta]).
We need not consider conic sections other than those with the pole at
a focus, a vertex, or a center, since we can always move the pole to
this location. Similarly, we need not consider those whose axis is
neither parallel to nor perpendicular to the polar axis, since we can
always move the polar axis to one of those orientations.
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Hyperbolas
The line segment connecting the two vertices, which lies on the axis,
is called the transverse axis, and has length 2a. Its midpoint
is the center of the hyperbola. Perpendicular to the transverse
axis at the midpoint is the conjugate axis, whose length is 2b.
The eccentricity is e = sqrt(a^{2}+b^{2})/a. The
distance from the center to either of the two foci is ae. The distance
from a vertex to the nearest focus is a(e1). The distance from the
center to either of the two directrices is a/e. The length of the
latus rectum is 2b^{2}/a.
The absolute value of the difference of the distances from any point
on the hyperbola to the two foci is 2a.
The equation of the hyperbola has one of the following forms:
 Standard form:
r^{2}(b^{2}cos^{2}[theta]a^{2}sin^{2}[theta])
= a^{2}b^{2},
r = ab/sqrt(a^{2}+[a^{2}+b^{2}]cos^{2}[theta]),
r = a sqrt(e^{2}1)/sqrt(1+e^{2}cos^{2}[theta]).
The center is at the pole O, the foci have coordinates (ae,0) and
(ae,Pi), the vertices have coordinates (a,0) and (a,Pi), and the
directrices have equations r = ±(a/e)sec(theta). The asymptotes
have equations tan(theta) = ±b/a. The tangent at point
P_{2} has equation
r = a^{2}b^{2}/(a^{2}r_{2}sin[theta]sin[theta_{2}]b^{2}r_{2}cos[theta]cos[theta_{2}])
 Asymptotic form:
r^{2} = a^{2}/sin(2 theta).
In this form, the center is the pole O, a = b, e = sqrt(2), the foci
have coordinates (a,Pi/4) and (a,5 Pi/4), the transverse axis has
equation theta = Pi/4, the conjugate axis has equation
theta = 3 Pi/4,
and the asymptotes have equations theta = 0 and theta = Pi/2.
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Parabolas
The distance from the vertex to F is p = a/2. The distance from the
vertex to D is p. The length of the latus rectum is 4p.
The equation of the parabola can be put into any of the following
forms:
 Standard form (vertical):
r = 4p sin(theta)/cos^{2}(theta).
In this form the vertex is the pole O, the axis is vertical, F has
coordinates (p,Pi/2), and D has equation r = p/sin(theta).
 Standard form (horizontal):
r = 4p cos(theta)/sin^{2}(theta).
In this form the vertex is the pole O, the axis is horizontal, F has
coordinates (p,0), and D has equation r = p/cos(theta).
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Ellipses
The line segment connecting the two vertices, which lies on the axis,
is called the major axis, and has length 2a. Its midpoint is
the center of the ellipse. Perpendicular to the major axis at
the center is the minor axis, whose length is 2b. The
eccentricity is e = sqrt(a^{2}b^{2})/a < 1. The
distance from the center to either of the two foci is ae. The distance
from a vertex to the nearest focus is a(1e). The distance from the
center to either of the two directrices is a/e. The length of the
latus rectum is 2b^{2}/a.
The sum of the distances from any point on the ellipse to the two foci
is 2a.
The equation of the ellipse has the following form:
 Standard form:
r^{2}(a^{2}sin^{2}[theta]+b^{2}cos^{2}[theta])
= a^{2}b^{2},
r = ab/sqrt(a^{2}[a^{2}b^{2}]cos^{2}[theta]),
r = a sqrt(1e^{2})/sqrt(1e^{2}cos^{2}[theta]).
The center is at the pole O, the foci have coordinates (ae,0) and
(ae,Pi), the vertices have coordinates (a,0) and (a,Pi), and the
directrices have equations r = ±(a/e)sec(theta).
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Circles
A circle is an ellipse with e = 0, a = b = R, coincident foci, and
directrices at infinity. Its equation can take any one of the
following forms:
 General form:
r^{2}  2r(h cos[theta] + k sin[theta]) = R^{2} 
h^{2}  k^{2}.
This has center at (r_{1}, theta_{1}) and radius R,
where
r_{1} = sqrt[h^{2}+k^{2}],
theta_{1} = arctan[k/h].
The tangent to this circle at P_{2} has equation
r = p sec(thetaomega),
p = r_{2} sqrt(R^{2}r_{1}^{2}sin^{2}[theta_{2}theta_{1}])/R,
omega = theta_{2} + arcsin(r_{1}sin[theta_{2}theta_{1}]/R).
 A circle passing through the pole O with center
(r_{1},theta_{1}) has radius R = r_{1} and
equation
r = 2r_{1} cos(thetatheta_{1}).
 A circle with center at the pole O has equation
r = R.
 Diameter form, where P_{1} and P_{2} are the
endpoints of a diameter:
r^{2} 
r(r_{1}cos[thetatheta_{1}]+r_{2}cos[thetatheta_{2}])
+ r_{1}r_{2}cos(theta_{2}theta_{1})
= 0.
 Three point form, where P_{1}, P_{2}, and
P_{3} lie on the circle:

r^{2} 
r cos(theta) 
r sin(theta) 
1 

= 0, 
r_{1}^{2} 
r_{1}cos(theta_{1}) 
r_{1}sin(theta_{1}) 
1 
r_{2}^{2} 
r_{2}cos(theta_{2}) 
r_{2}sin(theta_{2}) 
1 
r_{3}^{2} 
r_{3}cos(theta_{3}) 
r_{3}sin(theta_{3}) 
1 
r^{2}(r_{1}r_{2}sin[theta_{1}theta_{2}] + r_{2}r_{3}sin[theta_{2}theta_{3}]
+ r_{3}r_{1}sin[theta_{3}theta_{1}])
+ r(r_{1}[r_{2}^{2}r_{3}^{2}]sin[thetatheta_{1}]
+ r_{2}[r_{3}^{2}r_{1}^{2}]sin[thetatheta_{2}]
+ r_{3}[r_{1}^{2}r_{2}^{2}]sin[thetatheta_{3}])
 r_{1}r_{2}r_{3}(r_{1}sin[theta_{2}theta_{3}]
+ r_{2}sin[theta_{3}theta_{1}] + r_{3}sin[theta_{1}theta_{2}])
= 0.
A circle with center P_{1} and radius R is tangent to the line
r = p sec(thetaomega) if and only if
p  r_{1}cos(theta_{1}omega) = R.
The line and circle do not intersect if
p  r_{1}cos(theta_{1}omega) > R,
and they intersect in two points if
p  r_{1}cos(theta_{1}omega) < R.
Two circles with centers at P_{1} and P_{2} and radii
R_{1} and R_{2} are externally tangent if and only if
r_{1}^{2} + r_{2}^{2} 
2r_{1}r_{2}cos(theta_{1}theta_{2}) =
(R_{1}+R_{2})^{2}.
The same two circles are internally tangent if and only if
r_{1}^{2} + r_{2}^{2} 
2r_{1}r_{2}cos(theta_{1}theta_{2}) =
(R_{1}R_{2})^{2}.
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Compiled by Robert L. Ward.
