Curves in Polar Coordinates

by
K. D.  Stroyan
University of Iowa

copyright 1997 by Academic Press, Inc. - All rights reserved.

Notebook Overview

This NoteBook plots some curves given in "polar coordinates."  This simply means that  x  and  y  can be specified by saying that they lie at angle  t  on a circle of radius  r.  In the Cirlecs NoteBook, we saw that this means

x = r Cos[t]
	y = r Sin[t]

An equation in "polar coordinates" means that we specify a relation between  r  and  t.

References to the Text

This NoteBook goes with Chapter 16.3 of the text.

Step-by-step

Roses

We know that the equations  x = r Cos[t], y = r Sin[t]  point out a vector  {x,y}  at angel  t  a distance  r.  If we make  r   oscillate as  t  goes around, we get loops like pedals of a rose:

Clear[x, y, r, t]; r = Cos[n t]x = r Cos[t]; \(y = r\ Sin[t]; 
n = 2\)ParametricPlot[{x, y}, {t, 0, 2 π}, 
      AspectRatio -> Automatic]; n = 3ParametricPlot[{x, y}, {t, 0, 2 π}, 
      AspectRatio -> Automatic]; n = 4ParametricPlot[{x, y}, {t, 0, 2 π}, 
      AspectRatio -> Automatic];

Spirals

We know that the equations  x = r Cos[t], y = r Sin[t]  point out a vector  {x,y}  at angle  t  a distance  r.  If we make  r   grow as  t  goes around, we get a spiral:

Clear[x, y, r, t]; r = t; x = r Cos[t]; y = r Sin[t]ParametricPlot[{x, y}, {t, 0, 2 π}, 
      AspectRatio -> Automatic];

These graphs are easy to make if someone tells you the polar equation.  The pictures are interesting.  Go fo it:

Exercises

Exercise 16.3.1: Loops and Spirals

Mathematica Flower

The parametric equations for a Mathematica flower are

r = (1/2) (3 Cos[t]^2 - 1); 
	x = r Cos[t] ;
	y = r Sin[t]

Enter these in the appropriate places in the next cell and run.

\(Clear[x, y, r, t]; 

    r\  = \(?? ???\ \ (*Polar\ equation,\ r\ =\ fct[t]*)
x\  = \ 
          \(?? ????????
y\  = \ 
              \(?? ????????\ 

                  \(ParametricPlot[{x, y}, {t, \ 0, \ 2  Pi}, \ 
                    AspectRatio\  -> \ Automatic]; \)\)\)\)\)

Logarithmic Spiral

The parametric equations for a Logarithmic Spiral are

r = Exp[(t/π)] ;
 	x = r Cos[t];
 	y = r Sin[t]

Enter these in the appropriate places in the next cell and run.

\(Clear[x, y, r, t]; 

    r\  = \(?? ???\ \ (*Polar\ equation,\ r\ =\ fct[t]*)
x\  = \ 
          \(?? ????????
y\  = \ 
              \(?? ????????\ 

                  \(ParametricPlot[{x, y}, {t, \ 0, \ 2  Pi}, \ 
                    AspectRatio\  -> \ Automatic]; \)\)\)\)\)

Polar Lines

Plot the parametric curve with

r = a/Cos[t]
	x = r Cos[t]
	y = r Sin[t]

Try  a = 1, 2, 3.  Explain the result.  (HINT: Sybstitute  r  in the x and y equations.)

\(Clear[x, y, r, a, t]; 
r = a\/Cos[t]; 
x = r\ Cos[t]; 

    y = r\ Sin[t]; 
a = 1\)\(ParametricPlot[{x, y}, {t, \(-\(π\/3\)\), π\/3}, 
      PlotPoints -> 333, AspectRatio -> 1, 
      AxesOrigin -> {0, 0}]; 
a = 2\)ParametricPlot[{x, y}, {t, -(π/3), (π/3)}, 
      AspectRatio -> 1, AxesOrigin -> {0, 0}];

Plot the parametric curve with

r = a/Sin[t]
	x = r Cos[t]
	y = r Sin[t]

Try  a = 1, 2, 3.  Explain the result.

Polar Circles

Plot the parametric curve with

r = 2 a Cos[t]
	x = r Cos[t]
	y = r Sin[t]

Try  a = 1, 2, 3.  Explain by drawing a triangle inscribed in half of the circle with the {x,y} vector of length  r  as one leg and a diameter as another.

Clear[x,y,r,t,a];
r = 2 a Cos[t]
x = r Cos[t];
y = r Sin[t];
a = 1
circ1 = ParametricPlot[{x,y},{t, -Pi/2,Pi/2},
	AspectRatio -> Automatic, AxesOrigin -> {0,0}]; 
a = 2
circ2 = ParametricPlot[{x,y},{t, -Pi/2,Pi/2},
	AspectRatio -> Automatic, AxesOrigin -> {0,0}];
	Show[circ1,circ2];

Plot the parametric curve with

r = -2 a Cos[t]
	x = r Cos[t]
	y = r Sin[t]

Plot the parametric curve with

r = 2 a Sin[t]
	x = r Cos[t]
	y = r Sin[t]

Plot the parametric curve with

r = -2 a Sin[t]
	x = r Cos[t]
	y = r Sin[t]

Conics in Polar Coordiantes

The conic with ecentricity  e, focus point  (0,0)  and vertical directrix  k  units out on the positive  x-axis has polar equation

r = ((e k)/(1 + e Cos[t]))

When  0 < e < 1, the curve is an ellipse, when  e = 1, the curve is a parabola (and has a singularity at ±π), and when  e > 1, the curve is an hyperbola.  Check it out and try some of your own.

Clear[x, y, r, t, e]; r = ((e k)/(1 + e Cos[t]))x = r Cos[t]; y = r Sin[t]; k = 2; e = (3/4)ParametricPlot[{x, y}, {t, -π, π}, 
      AspectRatio -> Automatic, AxesOrigin -> {0, 0}]; e = 1ParametricPlot[{x, y}, {t, -π, π}, AspectRatio -> 1, 
      AxesOrigin -> {0, 0}]; e = 2ParametricPlot[{x, y}, {t, -π, π}, 
      AspectRatio -> Automatic, AxesOrigin -> {0, 0}];

Cardioids

A cardioid is given by

r = 1 + Sin[t]
or
	r = 1 + Cos[t]
and
	x = r Cos[t]
	y = r Sin[t]

Draw these heart-shaped curves.  How could you expand the shape to make it  3  times bigger?

Limicons

A "limicon" is given by

r = a +  Sin[t]
or
	r = a +  Cos[t]
and
	x = r Cos[t]
	y = r Sin[t]

where  a  is different from  1.
Draw some examples in the cases where  a < 1,  1 < a < 2, and  a >= 2.
How could you expand one of the shapes to make it  3  times bigger?

Lemniscates

Lemniscates are curves where  r  satisfies one of:

(r^2) = (a^2) Cos[2 t](r^2) = -(a^2) Cos[2 t](r^2) = (a^2) Sin[2 t](r^2) = -(a^2) Sin[2 t]

Sketch half of one with one square root and the other half with the negative root, then show them together.