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  Analytic Geometry: Polar Coordinates  

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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(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.


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Compiled by Robert L. Ward.

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