Applet: Polar coordinates map of rectangle
The transformation from polar coordinates to Cartesian coordinates $(x,y)=\cvarf(r,\theta) = (r \cos\theta, r \sin \theta)$ can be viewed as a map from the polar coordinate $(r,\theta)$ plane (left panel) to the Cartesian coordinate $(x,y)$ plane (right panel). This transformation maps a rectangle $\dlr^*$ in the $(r,\theta)$ plane into a region $\dlr$ in the $(x,y)$ plane that is the part of an angular sector inside an annulus. You can change the regions $\dlr^*$ and $\dlr$ by dragging the purple or cyan points in either panel. To further visualize the action of the map $(x,y)=\cvarf(r,\theta)$, you can drag the labeled red and blue points anywhere inside the large rectangle $0 \le r \le 6$, $0 \le \theta <2\pi$ and corresponding disk $x^2+y^2 \le 6^2$.
Applet file: polar_coordinates_map_rectangle.ggb
Applet links
This applet is found in the pages
General information about Geogebra Web applets
This applet was created using Geogebra. In most Geogebra applets, you can move objects by dragging them with the mouse. In some, you can enter values with the keyboard. To reset the applet to its original view, click the icon in the upper right hand corner.
You can download the applet onto your own computer so you can use it outside this web page or even modify it to improve it. You simply need to download the above applet file and download the Geogebra program onto your own computer.