- #1

- 1,095

- 67

## Summary:

- Polar coordinates, plotting polar equations, scale invariance

Hello,

In the plane, using Cartesian coordinates ##x## and ##y##, an equation (or a function) is a relationship between the ##x## and ##y## variables expressed as ##y=f(x)##. The variable ##y## is usually the dependent variable while ##x## is the independent variable.

The polar coordinates ##r## and ##\theta## can be used instead of ##x## and ##y## to describe the same mathematical object. The variable ##r## is the dependent variable and a general polar equation has the form ##r=g(\theta)##. In general, when plotting a polar function like ##r=g(\theta)##, we plot it over the Cartesian axes instead of labeling the axes ##r## and ##\theta##. Why? For example, the polar equation of a circle of radius 4 centered at the origin is ##r=4## and the graph looks like a circle if plotted in the x-y plane. However, if plotted with the axes labeled ##r## and ##\theta##, the graph of ##r=4## would just be a horizontal straight line. Same goes for other coordinate systems: we always plot using the corresponding ##x=rcos\theta## and ##y=rsin\theta## values that derive from the specific ##r## and ##\theta##.

Scale Invariance: I read about polar equations describing shapes that are scale invariant: if scaled up by a constant, the enlarged shape is represented by the same parent equation, just rotated. I don't think all polar equations behave like this. What makes certain polar functions be scale invariant while others are not? Example: ##r=e^{\theta}##, if scaled up by a constant##A##, ##Ae^{(\theta)}=e^{(\theta+\theta_0)}## with the scale parameter ##A=e^{\theta_0}##. I guess only polar functions involving exponential functions have this property, correct? These shapes are said to be specified only in terms of

Thanks!

In the plane, using Cartesian coordinates ##x## and ##y##, an equation (or a function) is a relationship between the ##x## and ##y## variables expressed as ##y=f(x)##. The variable ##y## is usually the dependent variable while ##x## is the independent variable.

The polar coordinates ##r## and ##\theta## can be used instead of ##x## and ##y## to describe the same mathematical object. The variable ##r## is the dependent variable and a general polar equation has the form ##r=g(\theta)##. In general, when plotting a polar function like ##r=g(\theta)##, we plot it over the Cartesian axes instead of labeling the axes ##r## and ##\theta##. Why? For example, the polar equation of a circle of radius 4 centered at the origin is ##r=4## and the graph looks like a circle if plotted in the x-y plane. However, if plotted with the axes labeled ##r## and ##\theta##, the graph of ##r=4## would just be a horizontal straight line. Same goes for other coordinate systems: we always plot using the corresponding ##x=rcos\theta## and ##y=rsin\theta## values that derive from the specific ##r## and ##\theta##.

Scale Invariance: I read about polar equations describing shapes that are scale invariant: if scaled up by a constant, the enlarged shape is represented by the same parent equation, just rotated. I don't think all polar equations behave like this. What makes certain polar functions be scale invariant while others are not? Example: ##r=e^{\theta}##, if scaled up by a constant##A##, ##Ae^{(\theta)}=e^{(\theta+\theta_0)}## with the scale parameter ##A=e^{\theta_0}##. I guess only polar functions involving exponential functions have this property, correct? These shapes are said to be specified only in terms of

*angles*. Isn't that true for any polar equation since ##r=g(\theta)## and we can always express ##r## in terms of ##\theta##?Thanks!