Coordinate Transformations

If a parametric surface is of the form r(u,v) = á f( u,v) ,g( u,v),0 ñ , then it maps the uv-plane to the xy-plane. Moreover, r( u,v) induces a chart on some part of the xy-plane and correspondingly, it maps sets in the uv-plane to sets in the xy-plane.

Thus, parametric surfaces of the form r( u,v) = á f( u,v) ,g( u,v) ,0 ñ are called coordinate transformations. That is, a coordinate transformation is a mapping of the form

T( u,v) = á f( u,v) ,g( u,v) ñ

and f,g are called the components of the transformation.

It follows that T maps a set S in the uv-plane to a set T(S) in the xy-plane:

If S is a region, then we use the fact that x = f( u,v) and y = g( u,v) to find the image of the boundary of S under T( u,v) .       

EXAMPLE 1    Find T( S) when T( u,v) = á uv,u²-v² ñ and S is the unit square in the uv-plane (i.e., S = [ 0,1] ×[0,1] ).

Solution: To find the boundary of T( S) in the xy-plane, we find the images under  x = uv and y = u²-v²  of the lines bounding the unit square in the uv-plane.

Side of Unit Square		Result of T(u,v)		Image in xy-plane
u = 0		x = 0 and y = -v2 < 0		negative y axis
u = 1		x = v and y = 1-v2		,y = 1-x2
v = 1		x = u and y = u2-1		y = x2-1
v = 0		x = 0 and y = u2 > 0		positive y axis

 As a result, T( S) is the region in the xy-plane bounded by x = 0, y = x²-1, and y = 1-x².

       

EXAMPLE 2    Find T( S) when T( u,v) = á vcos( u) ,vsin( u) ñ and S is the region given by

S: v = 0 v = sin( u)

Solution: When v = 0, then T( u,v) = á0,0 ñ . Thus, the u-axis is mapped to the origin. When v = sin( u) , then

T( u,v) = á sin( u) cos( u),sin2( u) ñ

However, x = sin( u) cos( u) , y = sin²(u) implies that

x2  =  sin2( u) cos2( u)  =  sin2( u) ( 1-sin2( u) )  =  y( 1-y)

That is, x² = y-y², which is the same as x²+y²-y = 0. Completing the square leads to

x2+y2-y+ 1 4  =   1 4 x2+ æ è y-  1 2 ö ø 2    =   1 4

Thus, the image of v = sin( u) is the circle of radius 1/2 centered at ( 0,1/2) .