Contents:

Coordinates ||  Points ||  Directions ||  Translations ||  Rotations ||  Lines ||  Triangles ||  Conic Sections  [hyperbolas | parabolas | ellipses | circles]

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(x²+y²),
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 P₁ be (r₁,theta₁), those of P₂ be (r₂,theta₂), and those of P₃ be (r₃,theta₃).

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Points

The distance from P₁ to P₂ is

d = sqrt(r₁²+r₂²-2r₁r₂cos[theta₂-theta₁]).

The coordinates of the point dividing the line segment P₁P₂ in the ratio a/b are:

(sqrt[b²r₁²+a²r₂²+2abr₁r₂cos(theta₂-theta₁)]/[a+b], arctan([br₁sin(theta₁)+ar₂sin(theta₂)]/[br₁cos(theta₁)+ar₂cos(theta₂)]).

As a special case, when a = b, the midpoint of the line segment has coordinates

(sqrt[r₁²+r₂²+2r₁r₂cos(theta₂-theta₁)]/2, arctan([r₁sin(theta₁)+r₂sin(theta₂)]/[r₁cos(theta₁)+r₂cos(theta₂)]).

P₁, P₂, and P₃ are collinear if and only if the determinant

	r1cos(theta1)	r1sin(theta1)	1		= 0,
	r2cos(theta2)	r2sin(theta2)	1
	r3cos(theta3)	r3sin(theta3)	1

or else

r₁r₂sin(theta₂-theta₁) + r₂r₃sin(theta₃-theta₂) + r₃r₁sin(theta₁-theta₃) = 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:

alpha₁ = alpha₂,
m₁ = m₂.

The angle between two directions is given by

alpha₂ - alpha₁ = arctan([m₂-m₁]/[1+m₁m₂]).

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Translations

If we wish to move the pole of our polar coordinate system from (0,0) to (r₀,theta₀), 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(r²+r₀²-2rr₀cos(theta-theta₀),
theta' = arctan([r sin(theta)-r₀sin(theta₀)]/[r cos(theta)-r₀cos(theta₀)]),

r = sqrt(r'²+r₀²+2r'r₀cos(theta'-theta₀),
theta = arctan([r'sin(theta')+r₀sin(theta₀)]/[r'cos(theta')+r₀cos(theta₀)]).

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Rotations About the Pole

If we wish to rotate the polar axis through an angle of theta₀ while keeping the pole fixed, the transformation of coordinates is

r' = r,
theta' = theta - theta₀,

r = r',
theta = theta' + theta₀.

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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(theta-alpha) = 0. If the line passes through the pole O and point P₁, then the equation is sin(theta-theta₁) = 0.

The equation of a line not through the pole O can have any of several forms:

• Polar normal form:

  r = p sec(theta-omega).

• Slope r-intercept form:

  r = a sin(alpha)/sin(alpha-theta),

  if the r-intercept is (a,0), a > 0, or

  r = a sin(alpha)/sin(theta-alpha),

  if the r-intercept is (a,Pi), a > 0.

• Two point form:

  r = r₁r₂sin(theta₂-theta₁)/(r₁sin[theta-theta₁]-r₂sin[theta-theta₂]).

• Point slope form:

  r = (mr₁cos[theta₁]-r₁sin[theta₁])/(m cos[t]-sin[t]),

  if m is finite.

• Point inclination form:

  r = r₁sin[theta₁-alpha]/(sin[theta-alpha]),

  if m is finite.

• Parametric form:

  r = sqrt(t²+r₁²+2tr₁cos[theta₁-alpha]),
  theta = Arccos([t cos(alpha)+r₁cos(theta₁)]/[sqrt(t²+r₁²+2tr₁cos[theta₁-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(theta-omega) to P₁ is

d = r₁cos(theta₁-omega) - p.

If r = p₁sec(theta-omega₁) and r = p₂sec(theta-omega₂) are two intersecting lines, then their intersection point has coordinates

r = sqrt(p₁²+p₂²-2p₁p₂cos[omega₁-omega₂])/sin(omega₁-omega₂),
theta = Arccos([p₂sin(omega₁)-p₁sin(omega₂)]/[sqrt(p₁²+p₂²-2p₁p₂cos[omega₁-omega₂]).

Three lines

A₁cos(theta) + B₁sin(theta) + C₁/r = 0,
A₂cos(theta) + B₂sin(theta) + C₂/r = 0,
A₃cos(theta) + B₃sin(theta) + C₃/r = 0,

are concurrent (that is, all pass through a single point) if and only if the determinant

	A1	B1	C1		= 0.
	A2	B2	C2
	A3	B3	C3

Three lines

r = p₁sec(theta-omega₁),
r = p₂sec(theta-omega₂),
r = p₃sec(theta-omega₃),

are concurrent (that is, all pass through a single point) if and only if

p₁sin(omega₃-omega₂) + p₂sin(omega₁-omega₃) + p₃sin(omega₂-omega₁) = 0.

The perpendicular bisector of the line segment P₁P₂ has the equation

r = (r₁cos[theta₁-alpha] + r₂cos[theta₂-alpha])/(2 cos[theta-alpha]).

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Triangles

The area of a triangle whose vertices are P₁, P₂, and P₃ is given by

K = [r₁r₂sin(theta₂-theta₁) + r₂r₃sin(theta₃-theta₂) + r₃r₁sin(theta₁-theta₃)]/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:

1. e > 1, an hyperbola.
   1. e = sqrt(2), an equilateral hyperbola.
2. e = 1, a parabola.
3. e < 1, an ellipse.
   1. 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/(1-e 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(theta-omega), then the focus form of the equation of the conic section is

r = ae/(1-e cos[theta-omega]).

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]/(1-e²cos²[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(a²+b²)/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(e-1). The distance from the center to either of the two directrices is a/e. The length of the latus rectum is 2b²/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:

  r²(b²cos²[theta]-a²sin²[theta]) = a²b²,
  r = ab/sqrt(-a²+[a²+b²]cos²[theta]),
  r = a sqrt(e²-1)/sqrt(-1+e²cos²[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 P₂ has equation

  r = a²b²/(a²r₂sin[theta]sin[theta₂]-b²r₂cos[theta]cos[theta₂])

• Asymptotic form:

  r² = a²/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)/cos²(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)/sin²(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(a²-b²)/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(1-e). The distance from the center to either of the two directrices is a/e. The length of the latus rectum is 2b²/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:

  r²(a²sin²[theta]+b²cos²[theta]) = a²b²,
  r = ab/sqrt(a²-[a²-b²]cos²[theta]),
  r = a sqrt(1-e²)/sqrt(1-e²cos²[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:

  r² - 2r(h cos[theta] + k sin[theta]) = R² - h² - k².

  This has center at (r₁, theta₁) and radius R, where

  r₁ = sqrt[h²+k²],
  theta₁ = arctan[k/h].

  The tangent to this circle at P₂ has equation

  r = p sec(theta-omega),
  p = r₂ sqrt(R²-r₁²sin²[theta₂-theta₁])/R, omega = theta₂ + arcsin(r₁sin[theta₂-theta₁]/R).

• A circle passing through the pole O with center (r₁,theta₁) has radius R = r₁ and equation

  r = 2r₁ cos(theta-theta₁).

• A circle with center at the pole O has equation

  r = R.

• Diameter form, where P₁ and P₂ are the endpoints of a diameter:

  r² - r(r₁cos[theta-theta₁]+r₂cos[theta-theta₂]) + r₁r₂cos(theta₂-theta₁) = 0.

• Three point form, where P₁, P₂, and P₃ lie on the circle:

  	r2	r cos(theta)	r sin(theta)	1		= 0,
  	r12	r1cos(theta1)	r1sin(theta1)	1
  	r22	r2cos(theta2)	r2sin(theta2)	1
  	r32	r3cos(theta3)	r3sin(theta3)	1

  
  r²(r₁r₂sin[theta₁-theta₂] + r₂r₃sin[theta₂-theta₃] + r₃r₁sin[theta₃-theta₁])  + r(r₁[r₂²-r₃²]sin[theta-theta₁] + r₂[r₃²-r₁²]sin[theta-theta₂]   + r₃[r₁²-r₂²]sin[theta-theta₃]) - r₁r₂r₃(r₁sin[theta₂-theta₃]   + r₂sin[theta₃-theta₁] + r₃sin[theta₁-theta₂]) = 0.

A circle with center P₁ and radius R is tangent to the line r = p sec(theta-omega) if and only if

|p - r₁cos(theta₁-omega)| = R.

The line and circle do not intersect if

|p - r₁cos(theta₁-omega)| > R,

and they intersect in two points if

|p - r₁cos(theta₁-omega)| < R.

Two circles with centers at P₁ and P₂ and radii R₁ and R₂ are externally tangent if and only if

r₁² + r₂² - 2r₁r₂cos(theta₁-theta₂) = (R₁+R₂)².

The same two circles are internally tangent if and only if

r₁² + r₂² - 2r₁r₂cos(theta₁-theta₂) = (R₁-R₂)².