Polar Graph Plotter — Theory
1. Polar equations
A polar equation gives the radius \(r\) as a function of the angle \(\theta\):
\[
r=f(\theta).
\]
For each value of \(\theta\), the equation gives a distance \(r\) from the origin.
The point is then plotted in that direction.
2. Converting polar points to Cartesian points
A graphing screen uses Cartesian coordinates, so each polar point is converted using
\[
x=r\cos\theta,
\qquad
y=r\sin\theta.
\]
Therefore the polar curve \(r=f(\theta)\) is plotted as the parametric curve
\[
x=f(\theta)\cos\theta,
\qquad
y=f(\theta)\sin\theta.
\]
3. Negative radius values
In polar coordinates, \(r\) may be negative. A negative radius means the point is plotted in the
opposite direction from the angle.
\[
(r,\theta)\equiv(-r,\theta+\pi).
\]
This is why polar curves can form loops and petals even when the formula looks simple.
4. Limaçons
A common polar family is the limaçon:
\[
r=a+b\cos\theta
\qquad\text{or}\qquad
r=a+b\sin\theta.
\]
The shape depends on the relative sizes of \(|a|\) and \(|b|\).
5. Example: \(r=3+2\cos\theta\)
This has the form
\[
r=a+b\cos\theta
\]
with
\[
a=3,\qquad b=2.
\]
Since \(|b|<|a|<2|b|\), the graph is a dimpled limaçon.
6. Cardioids
A cardioid is a special limaçon. It occurs when \(|a|=|b|\):
\[
r=a+a\cos\theta
\qquad\text{or}\qquad
r=a+a\sin\theta.
\]
Examples include
\[
r=2+2\cos\theta
\]
and
\[
r=4-4\sin\theta.
\]
7. Rose curves
Rose curves usually have the form
\[
r=a\cos(n\theta)
\qquad\text{or}\qquad
r=a\sin(n\theta).
\]
When \(n\) is a positive integer:
- If \(n\) is odd, the rose usually has \(n\) petals.
- If \(n\) is even, the rose usually has \(2n\) petals.
For example,
\[
r=4\sin(3\theta)
\]
has \(3\) petals, while
\[
r=3\cos(4\theta)
\]
has \(8\) petals.
8. Circles in polar form
Some circles have simple polar equations. For example,
\[
r=2a\cos\theta
\]
represents a circle. Multiplying by \(r\),
\[
r^2=2ar\cos\theta.
\]
Since \(r^2=x^2+y^2\) and \(r\cos\theta=x\), we get
\[
x^2+y^2=2ax.
\]
This can be rewritten as
\[
(x-a)^2+y^2=a^2.
\]
9. Spirals
A basic Archimedean spiral has form
\[
r=a\theta.
\]
As \(\theta\) increases, \(r\) increases linearly, so the point moves farther from the origin.
Spirals often need a larger interval, such as
\[
0\le\theta\le6\pi,
\]
to show several turns.
10. Symmetry checks
Polar graphs often have symmetry. Common tests include:
These tests are useful, but polar graphs can have more subtle symmetry than these simple checks show.
11. Tracing a polar graph
The best way to understand a polar graph is to trace it as \(\theta\) increases.
For each angle:
- Compute \(r=f(\theta)\).
- Move distance \(r\) from the pole in the direction \(\theta\).
- If \(r<0\), move in the opposite direction.
- Convert to \((x,y)\) for plotting.
This is exactly what the \(\theta\)-slider and Play mode show.
13. Common mistakes
- Using degrees inside a radian formula: most graphing formulas assume radians.
- Forgetting negative radius: negative \(r\) values are valid and often create loops or petals.
- Choosing too small a \(\theta\)-interval: spirals and roses may need more than one full turn.
- Using too few samples: polar curves with petals or loops need enough plot samples.
- Assuming every curve is one-to-one: many polar curves trace the same point more than once.
- Ignoring symmetry: symmetry helps predict the shape before plotting.
