Fun with Math!: polar

Showing posts with label polar. Show all posts

Monday, January 16, 2012

Spiral

A spiral is formed with a simple equation $r=a\theta, a\neq 0$ . The graph below set 'a' as positive, therefore there are two equations for you to tick. Choose whichever equation you prefer, you may choose both as well, then drag the value of 'a' according to your preference. Then, drag the value of $\theta$ to view the forming of spiral.

Betty, Created with GeoGebra

Rose

For the equation $r=k\cos(a\theta)$ or $r=k\sin(a\theta)$ , a rose is formed. The number of petal depends on the value of 'a'. Try to drag on the value of 'a' and compare the results when 'a' is even and when 'a' is odd.

Betty, Created with GeoGebra

Saturday, January 14, 2012

Lemniscate

The equations $r^2= a^2\cos(b\theta)$ and $r^2= a^2\sin(b\theta)$ form the shape of lemniscate. If you take a look at the graph (in default setting) below, you might find out that there is no graph formed for $\theta = 90^{o}$ till $180^o$ , and from $270^o$ to $360^o$ . Can you explain why? You may drag the value of 'a' to see the difference of graphs and $\theta$ for the forming of the curve. Then, choose another equation to display its graph and compare. What can you say about these two graphs?

Betty, Created with GeoGebra

Friday, January 13, 2012

Polar

A coordinate (x,y) in xy-plane can be written in polar form $(r,\theta)$ , where r is the directed distance from (0,0) to (x, y), and $\theta$ is the directed angle formed by the vector $\left \langle x,y \right \rangle$ ; from the positive side of x-axis, where
$r=\pm \sqrt{x^2 + y^2}$
$\theta=\tan^{-1} \frac{y}{x}$

You may try to drag the value of the sliders in the graph to look at Point A and the coordinates in $(x,y)$ and $(r,\theta)$ . Take note that the value of $\theta$ is in radian mode.

There are some interesting graphs can be obtained through polar equations, such as

Limaçon : $r = a\pm b\cos(\theta)$ or $r = a\pm b\sin(\theta)$ , where $a\neq b$
Cardioid : $r= a\pm a\cos(\theta)$ or $r= a\pm a\sin(\theta)$
Rose : $r= a\cos(b\theta)$ or $r= a\sin(b\theta)$
Lemniscate : $r^2= a^2\cos(b\theta)$ or $r^2= a^2\sin(b\theta)$
Spiral : $r=a\theta, a\neq 0$

etc.

Click on the links above to explore the shapes mentioned.

Betty, Created with GeoGebra

Limaçon & Cardioid

A limaçon is a snail curve. The equations $r = a \pm b\cos(\theta)$ and $r = a \pm b\sin(\theta)$ give to the shape of it when $a \neq b$ .

When $a=b$ , a cardioid is formed. Try to change the values of a and b by dragging it. You may also drag the value of $\theta$ to see the forming of the curve.

Betty, Created with GeoGebra

Pages

Monday, January 16, 2012

Spiral

Rose

Saturday, January 14, 2012

Lemniscate

Friday, January 13, 2012

Polar

Limaçon & Cardioid