How do you express x and y in terms of u and v so that the region $\{(u,v): 0\le u, v\le 1\}$ is mapped to the triangular region in the $xy$-plane with vertices $(0,0)$, $(1,0)$, and $(0,1)$?
Now, this is similar to a question I've asked before, but I still don't understand the answer I got on a similar question.
Also, can anyone suggest tutorials that address how to perform these type of questions?
Thanks.
On
According to this transformation from triangle $0<s<1,0<t<1-s$ to square $-1<\xi,\mu<1$ is done by:
$$ s = \frac{1 + \xi}{2} \\ t = \frac{1}{4}(1-\xi)(1+\mu) $$
$$f(u,v)=(u,u*v) $$ $$x:=u,y:=u*v$$ is such a function witch is continous.
you can think of it as shrinking every vertical line in the squere to a line(or a point) from zero to the line $u=v$,witch means,until the u coordinate of that line.
an other function can be a reflection through the line $v=-u+1$:
$$ f(u,v)=(u-|u-v|,v-|u-v|)$$ $$x:=u-|u-v|,y:=v-|u-v|$$(i'm not sure this is the function.can someone check?)
that is,switching the coordinates if we are above the line,and no change otherwise.
of course,the shapes are homeomorphic, so you can also find a function that is a homeomorphism,but it is less geometricly intuitive.