1915.sagews

Polar coordinates have the form $(r, \theta)$ where $r$ represents a distance from the origin or 'pole', and $\theta$ represents an angle of rotation.

The same point is specified by $(r, \theta)$ in polar coordinates, $(r \cdot cos \theta, r \cdot sin \theta)$ in rectangular coordinates, and $r(\cos \theta + i\sin \theta)$ in complex coordinates.

def polar(r, theta): return (r*cos(theta), r*sin(theta))

@interact
def _(r = (0..10), theta = (0..2*pi, step = pi/180)):
    P = polar(r, theta)
    radius = line([(0,0), P])
    arc = points([polar(r,x) for x in [0..theta, step = pi/180]], color = 'red')
    cosP = line([P, (r*cos(theta), 0)], linestyle = '--')
    sinP = line([P, (0, r*sin(theta))], linestyle = '--')
    print 'Polar:\t\t',(r, theta),'=',(r, theta.n(digits = 4))
    print 'Rectangular:\t','(',r,'*cos(',theta.n(digits = 4),'),',r,'*sin(',theta.n(digits = 4),'))','=',
    print (n(r*cos(theta), digits = 4), n(r*sin(theta), digits = 4))
    print 'Complex:\t',r,'*(cos(',theta,') + sin(',theta,')*I) =', r*n((cos(theta), sin(theta)), digits = 4)
    show(circle((0,0),r)+radius+arc+cosP+sinP, aspect_ratio = 1)

A polar equation $r = f(\theta)$ defines a distance $r$ as a function of an angle of rotation $\theta$:

var('theta')

def r(theta): return theta

domain = [0..2*pi, step = pi/180]

r(theta)

To understand the graph of a polar equation, think of the function $r(\theta)$ as computing the distance you travel in the direction of $\theta$.

Here is a graph of $r$ using polar coordinates $(r(\theta), \theta)$:

P = [(r(theta)*cos(theta), r(theta)*sin(theta)) for theta in domain]
show(points(P), aspect_ratio = 1)

For comparison, here is a graph of $r$ using rectangular coordinates in the way you have seen previously where the horizontal axis represents $\theta$ and the vertical axis represents $r(\theta)$,  $(\theta, r(\theta))$:

P = [(theta, r(theta)) for theta in domain]
show(points(P), aspect_ratio = 1)

Circles

$r = a \cos(\theta)$ or $r = a \sin(\theta)$

def polar(r, theta): return (r*cos(theta), r*sin(theta))
pole = (0, 0)

domain = [0..2*pi, step = pi/60]

@interact
def _(a = (-10..10), kind = ['cos', 'sin']):
    f = eval(kind)
    r(theta) = a*f(theta)
    print 'r =',a,kind,'(',theta,')'
    L = [line([pole, polar(r(x), x)]) for x in domain]
    show(sum(L), aspect_ratio = 1)

@interact
def _(a = (-10..10), kind = ['cos', 'sin']):
    f = eval(kind)
    r(theta) = a*f(theta)
    print 'r =',a,kind,'(',theta,')'
    show(polar_plot(r, 0, 2*pi), aspect_ratio = 1)

Rose Curves

$r = a \sin(n\theta)$  or  $r = a \cos(n\theta)$

@interact
def _(a = (0..20), n = (0..20), kind = ['sin', 'cos']):
    f = eval(kind)
    r(theta) = a*f(n*theta)
    print r(theta)
    show(polar_plot(r, 0, 2*pi), aspect_ratio = 1)

Limacons

$r = a + b \sin\theta$  or  $r = a + b \cos\theta$

@interact
def _(a = (-10..10), b = (-10..10), kind = ['sin', 'cos']):
    f = eval(kind)
    r(theta) = a + b*f(theta)
    print r(theta)
    show(polar_plot(r, 0, 2*pi), aspect_ratio = 1)

Lemniscates

$r^2 = a^2 \sin 2\theta$ or $r^2 = a^2\cos 2\theta$

@interact
def _(a = (-5..5), kind = ['sin', 'cos']):
    f = eval(kind)
    r(theta) = a*sqrt(f(2*theta))
    if f == sin: P = polar_plot(r, 0, pi/2) + polar_plot(-r, 0, pi/2)
    elif f == cos: P = polar_plot(r, -pi/4, pi/4) + polar_plot(-r, -pi/4, pi/4)
    print r^2
    show(P, aspect_ratio = 1)

def polar(r, theta): return (r*cos(theta), r*sin(theta))

@interact
def _(a = (-10..10), b = (-10..10), n = (1..10), kind = ['sin', 'cos'], start = 0, end = 2*pi, inc = pi/60):
    f = eval(kind)
    r(theta) = a + b*f(n*theta)
    print 'r =',a,'+',b,'*',kind,'(',n,'*',theta,')'
    pole = (0, 0)
    domain = [start..end, step = inc]
    L = [line([pole, polar(r(x), x)]) for x in domain]
    show(sum(L), aspect_ratio = 1)

@interact
def _(a = (-10..10), b = (-10..10), n = (1..10), kind = ['sin', 'cos']):
    f = eval(kind)
    r(theta) = a + b*f(n*theta)
    print r
    show(polar_plot(r, 0, 2*pi), aspect_ratio = 1)

In the following graph each point is determined by $(r(\theta), \theta)$:

Q1 = [(r(theta)*cos(theta), r(theta)*sin(theta)) for theta in [0..1/2*pi, step = pi/120]]
Q2 = [(r(theta)*cos(theta), r(theta)*sin(theta)) for theta in [1/2*pi..pi, step = pi/120]]
Q3 = [(r(theta)*cos(theta), r(theta)*sin(theta)) for theta in [pi..3/2*pi, step = pi/120]]
Q4 = [(r(theta)*cos(theta), r(theta)*sin(theta)) for theta in [3/2*pi..2*pi, step = pi/120]]
show(points(Q1)+points(Q2, color = 'red')+points(Q3, color='green')+points(Q4, color = 'black'), aspect_ratio = 1)

r(theta) = sqrt(theta)
polar_plot(r, 0, 2*pi).show(aspect_ratio = 1)

P = [(r(theta)*cos(theta), r(theta)*sin(theta)) for theta in [0..2*pi, step = pi/60]]
show(points(P), aspect_ratio=1)

r(theta) = cos(3*theta)
r

show(polar_plot(r, 0, 2*pi), aspect_ratio = 1)

P = [(r(theta)*cos(theta), r(theta)*sin(theta)) for theta in [0..2*pi, step = pi/120]]

show(points(P), aspect_ratio=1)

r(x) = 1-2*cos(x)

polar_plot(r, 0, 2*pi).show(aspect_ratio = 1)

P = [(r(theta)*cos(theta), r(theta)*sin(theta)) for theta in [0..2*pi, step = pi/120]]

show(points(P), aspect_ratio=1)

r(x) = 1 - x*sin(4*x)

polar_plot(r, 0, 2*pi).show(aspect_ratio = 1)

r(x) = 1 - x*sin(2*x)

domain = [0..4*pi, step = pi/180]

points([(r(x)*cos(x), r(x)*sin(x)) for x in domain]).show(aspect_ratio = 1)

r(pi/180).n()

r(x) = 1 - x*sin(2*x)

show(polar_plot(r, 0, pi), aspect_ratio = 1)

r(theta) = 1-2*cos(theta)

def polar_point(r, theta): return (r*cos(theta), r*sin(theta))

domain = [0..2*pi, step = pi/180]

L = sum([line([(0,0),polar_point(r(theta), theta)], color = (random(), random(), random())) for theta in domain])

show(L, aspect_ratio = 1)