Hi. Say we want to parametrize the plane R^2.

[daudaudaudau]

Hi. Say we want to parametrize the plane R^2. This can be done for example using (x,y) cartesian, i.e. a pair of intersecting lines, OR (r,theta) polar coordinates, i.e. a half line intersecting a circle. But it cannot be done using (x,r) coordinates, i.e. a line intersecting a circle, because sometimes the line will not intersect the circle, sometimes it will intersect it once and sometimes it will intersect the circle twice! How can I know whether a parametrization is any good? I.e. on what mathematical grounds can I reject the (x,r) parametrization?

 

[LCKurtz]

Hi. Say we want to parametrize the plane R^2. This can be done for example using (x,y) cartesian, i.e. a pair of intersecting lines, OR (r,theta) polar coordinates, i.e. a half line intersecting a circle. But it cannot be done using (x,r) coordinates, i.e. a line intersecting a circle, because sometimes the line will not intersect the circle, sometimes it will intersect it once and sometimes it will intersect the circle twice! How can I know whether a parametrization is any good? I.e. on what mathematical grounds can I reject the (x,r) parametrization?

You at least need a function from your parameter space (u,v) onto the (x,y) plane. And a function must be single valued. Your example fails because the (x,r) = (1,2) would map to two points: (1,sqrt(3)) and (1,-sqrt(3)).