Theory

Theory 1

Polar coordinates are pairs of numbers $(r,\theta )$ which identify points in the plane in terms of distance to origin and angle from $+x$-axis:

Converting $\mathrm{Polar}\leftrightarrow \mathrm{Cartesian}$

$$\begin{array}{c}\text{Polar}\to \text{Cartesian}\\ x=r\cos \theta \\ y=r\sin \theta \end{array}\begin{array}{c}\text{Cartesian}\to \text{Polar}\\ r=\sqrt{x^2+y^2}\\ \tan \theta ={\displaystyle \frac{y}{x}(x\ne 0)}\end{array}$$

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Polar coordinates have many redundancies: unlike Cartesian which are unique!

• For example: $(r,\theta )=(r,\theta +2\pi )$
  • And therefore also $(r,\theta )=(r,\theta -2\pi )$ (negative $\theta$ can happen)
• For example: $(-r,\theta )=(r,\theta +\pi )$ for every $r,\theta$
• For example: $(0,\theta )=(\mathrm{0,0})$ for any $\theta$

Polar coordinates cannot be added: they are not vector components!

• For example $(5,\pi /3)+(2,\pi /6)\ne (7,\pi /2)$
• Whereas Cartesian coordinates can be added: $(\mathrm{1,4})+(2,-2)=(\mathrm{3,2})$

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The transition formulas $\text{Cartesian}\to \text{Polar}$ require careful choice of $\theta$.

• The standard definition of ${\tan }^{-1}\left(\frac{y}{x}\right)$ sometimes gives wrong $\theta$
  • This is because it uses the restricted domain $\theta \in (-\pi /2,\pi /2)$; the polar interpretation is: only points in Quadrant I and Quadrant IV (SAFE QUADRANTS)
• Therefore: check signs of $x$ and $y$ to see which quadrant, maybe need $\pi$-correction!
  • Quadrant I or IV: polar angle is ${\tan }^{-1}\left(\frac{y}{x}\right)$
  • $\text{Q}\text{u}\text{a}\text{d}\text{r}\text{a}\text{n}\text{t}\text{I}\text{I}\text{o}\text{r}\text{I}\text{I}\text{I}\text{:}$ polar angle is ${\tan }^{-1}\left(\frac{y}{x}\right)+\pi$

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Equations (as well as points) can also be converted to polar.

For $\text{Cartesian}\to \text{Polar}$, look for cancellation from ${\cos }^2\theta +{\sin }^2\theta =1$.

For $\text{Polar}\to \text{Cartesian}$, try to keep $\theta$ inside of trig functions.

• For example:

$$r={\sin }^2\theta \gg \gg \sqrt{x^2+y^2}={\left(\frac{y}{\sqrt{x^2+y^2}}\right)}^2$$

Theory 2

To draw the polar graph of some function, it can help to first draw the Cartesian graph of the function. (In other words, set $y=r$ and $x=\theta$, and draw the usual graph.) By tracing through the points on the Cartesian graph, one can visualize the trajectory of the polar graph.

This Cartesian graph may be called a graphing tool for the polar graph.

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A limaçon is the polar graph of $r(\theta )=a+b\cos \theta$.

The shape of a limaçon is determined by the value of $c={\displaystyle \frac{b}{a}}$. Any limaçon can be rescaled to have this form:

$$r=1+c\cos \theta$$

$c=0$: Limaçon satisfying $r(\theta )=1$: unit circle.

$c=0.5$: Limaçon satisfying $r(\theta )=2+\cos \theta$: ‘outer loop’ circle with ‘flat spot’, not quite a ‘dimple’:

$c=1$: Limaçon satisfying $r(\theta )=1+\cos \theta$: ‘cardioid’ $=$ ‘outer loop’ circle with ‘dimple’ that creates a cusp:

$c=2$: Limaçon satisfying $r(\theta )=1+2\cos \theta$: ‘dimple’ pushes past cusp to create ‘inner loop’:

$c=\infty$: Limaçon satisfying $r(\theta )=\cos \theta$: ‘inner loop’ only, no outer loop exists:

$c=2$: Limaçon satisfying $r(\theta )=1+2\sin \theta$: ‘inner loop’ and ‘outer loop’ and rotated $\circlearrowleft {90}^{\circ }$:

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Transitions between limaçon types, $r(\theta )=1+c\sin \theta$:

Notice the transition points at $|c|=0.5$ and $|c|=1$:

The flat spot occurs when $c=\pm 0.5$

• Smaller $c$ gives convex shape

The cusp occurs when $c=\pm 1$

• Smaller $c$ gives dimple (assuming $|c|>0.5$)
• Larger $c$ gives inner loop

Theory 3

Roses are polar graphs of this form:

$$r=\cos (\theta ),r=\sin (2\theta ),r=\sin (3\theta ),r=\cos (4\theta )$$

The pattern of petals:

• $n=2k$ (even): obtain $2n$ petals
  • These petals traversed once
• $n=2k+1$ (odd): obtain $n$ petals
  • These petals traversed twice
• Either way: total-petal-traversals: always $2n$