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.

       

An important class of transformations are the linear transformations, which are of the form

T( u,v) = á au+bv,cu+dv ñ

Linear transformations are so named because they map straight lines to straight lines, and in particular, map lines through the origin in the uv-plane to lines through the origin in the xy-plane.

If points ( u,v) in the plane are associated with column matrices, [ u,v] ^t, then the linear transformation T(u,v) = á au+bv,cu+dv ñ can be written in matrix form as

T æ ç è u v ö ÷ ø = é ê ë a b c d ù ú û é ê ë u v ù ú û

The matrix of coefficients a,b,c,d is called the matrix of the transformation.       

EXAMPLE 2    Find the image of the unit square under the linear transformation

T æ ç è u v ö ÷ ø = é ê ë 2 1 1 1 ù ú û é ê ë u v ù ú û

Solution: Since linear transformations map straight lines to straight lines, we need only find the image of the 4 vertices of the unit square. To begin with, the point ( 0,0) is mapped to (0,0) . Associating the point ( 1,0) to the column vector [ 1,0] ^t yields

T æ ç è 1 0 ö ÷ ø = é ê ë 2 1 1 1 ù ú û é ê ë 1 0 ù ú û = é ê ë 2 1 ù ú û

Thus, the point ( 1,0) is mapped to the point (3,-2) . Likewise, associating ( 0,1) with [0,1] ^t leads to

T æ ç è 0 1 ö ÷ ø = é ê ë 2 1 1 1 ù ú û é ê ë 0 1 ù ú û = é ê ë 1 1 ù ú û

and associating ( 1,1) with [ 1,1] ^t leads to

T æ ç è 1 1 ö ÷ ø = é ê ë 2 1 1 1 ù ú û é ê ë 1 1 ù ú û = é ê ë 3 2 ù ú û

That is, ( 0,1) is mapped to ( 1,1) and (1,1) is mapped to ( 3,2) . Thus, the unit square in the uv-plane is mapped to the parallelogram in the xy-plane with vertices ( 0,0) , ( 2,1) , ( 1,1) , and (3,2) .

       

  Check your Reading: Is the transformation in example 1 a linear transformation? Explain.