Polar Graphs - Summary
General Equation
Ι.
r = a + b cos θ
r = a + b sin θ
A. |a| = |b|
B. |a| < |b|
C. |a| > |b|
ΙΙ.
Description of Graph
Mathematical Name
LimaÇon (snail in French):
Cardioid (limacon)
LimaÇon with inner loop
Convex LimaÇon
Dimpled LimaÇon
Heart
Loop de Loop
i) Flatten Circle if a = 2b
ii) Dent or Dimple if a < 2b
Rose (rose curve)
Rose (flower/cloverleaf)
(if n is odd then n 'petals' ; if n is even then 2n 'petals')
r = a cos nθ
r = a sin nθ
o
('petals'
(n > 1)
360
n
apart)
o
(for sin nθ: 1st 'petal' at
ΙΙΙ.
IV.
r=a
a
center at ( 2 , 0) radius of 2
r = a sin θ
center at (0 , 2 ) radius of 2
a
horizonal intercepting at (0 , a)
(Note: θ must be a radian
decimal number)
r2 = a2sin 2θ
Line
vertical intercepting at (a , 0)
r=aθ
r2 = a2cos 2θ
a
Lines: diagonal at angle of ‘a’
or
a
cos θ
a
r=
sin θ
VI.
a
r = a cos θ
θ=a
; for cos nθ: 1st 'petal' at 0o)
Circles: center at pole, radius of ‘a’ Circle
or
r=
V.
90
n
Spiral
(counterclockwise if θ > 0)
(clockwise if θ < 0)
Spiral of Archimedes
Figure 8 (bow tie/propeller)
('petals' at 0o and 180o)
('petals' at 45o and 225o)
Lemniscate
(ribbon in Greek)
Note the symmetry of the graph with the horizontal (0o) polar axis or the vertical (90o) axis and the
type of trigonometric function the equation has in it. For all categories above, except part VI., if the
equation has in it the:
a) cos θ then the graph is symmetrical with the polar axis (horizontal, 0o axis)
b) sin θ then the graph is symmetrical with the vertical 90o axis
* R. Greenlee Wheaton Warrenville South High School, Wheaton Il. 03/09/2001*