# Math Insight

### Applet: Nonlinear 2D map

This applet illustrates the mapping of a rectangle by the transformation $(x,y)=\cvarf(\cvarfv,\cvarsv)=(\cvarfv^2-\cvarsv^2,2\cvarfv\cvarsv)$. You can change the function $\cvarf$ by typing in new formulas for its components using the boxes in right panel. The left panel shows a rectangle $\dlr^*$ in the $\cvarfv\cvarsv$-plane, which you can change by dragging the yellow points. The mapping of $\dlr^*$ by $\cvarf$ is illustrated in the right panel, where the region $\dlr^*$ is stretched into the region $\dlr$, which is the image of $\dlr^*$ under the mapping $\cvarf$. Use the + and - buttons of each panel to zoom in and out.

For the transformation $(x,y)=\cvarf(\cvarfv,\cvarsv)=(\cvarfv^2-\cvarsv^2,2\cvarfv\cvarsv)$, you must keep the rectangle $\dlr^*$ completely in one half-plane for $\cvarf$ to be a one-to-one map. If both $\cvarfv$ and $\cvarsv$ can change sign in $\dlr^*$, then the mapping onto $\dlr$ overlaps itself. If you change the transformation, then there may be other restrictions on $\dlr^*$ to keep the mapping of $\dlr^*$ by $\cvarf$ one-to-one.

Applet file: nonlinear_2d_map.ggb