Here is some sage code to help you visualize coordinate transformations. I give three examples:
Polar coordinates:
# polar coordinates var('u','v') X(u,v) = v*cos(u) Y(u,v) = v*sin(u) # create plots where v is fixed and u ranges uplot = parametric_plot( (X(u,0),Y(u,0)), (u,-pi,pi)) for k in [0..4]: uplot = uplot+parametric_plot( (X(u,k),Y(u,k)), (u,-pi,pi)) # create plots where u is fixed and v ranges vplot = parametric_plot( (X(0,v),Y(0,v)), (v,0,4)) for k in [-10..10]: vplot = vplot+parametric_plot( (X(k*pi/10,v),Y(k*pi/10,v)), (v,0,4.5), color = "red") # plot all the plots mainplot = uplot + vplot mainplot.show(xmin=-5, xmax=5, ymin=-5, ymax =5, axes_labels = ["$x$","$y$"])
Hyperbolic coordinates:
# hyperboloidal coordinates var('u','v') X(u,v) = v*cosh(u) Y(u,v) = v*sinh(u) uplot = parametric_plot( (X(u,0),Y(u,0)), (u,-2,2)) for k in [0..4]: uplot = uplot+parametric_plot( (X(u,k),Y(u,k)), (u,-2,2)) vplot = parametric_plot( (X(0,v),Y(0,v)), (v,0,4)) for k in [-5..5]: vplot = vplot+parametric_plot( (X(k*.2,v),Y(k*.2,v)), (v,0,4), color = "red") mainplot = uplot + vplot mainplot.show(xmin=-5, xmax=5, ymin=-5, ymax =5, axes_labels = ["$x$","$y$"])
One more fun example:
# crazy example coordinates var('u','v') X(u,v) = u^2 - v^2 Y(u,v) = 2*u*v # create plots where v is fixed and u ranges uplot = parametric_plot( (X(u,0),Y(u,0)), (u,0,4)) for k in [-4..4]: uplot = uplot+parametric_plot( (X(u,k),Y(u,k)), (u,0,4)) # create plots where u is fixed and v ranges vplot = parametric_plot( (X(0,v),Y(0,v)), (v,0,4)) for k in [0..10]: vplot = vplot+parametric_plot( (X(k,v),Y(k,v)), (v,-4,4), color = "red") # plot all the plots max = 20 mainplot = uplot + vplot mainplot.show(xmin=-max, xmax=max, ymin=-max, ymax =max, axes_labels = ["$x$","$y$"])<span id="mce_SELREST_start" style="overflow:hidden;line-height:0;"></span>