10-14-08, 10:35 AM #1
If an inﬁnitesimal segment of a curve (dy,dx) in a 2-D space, having a slope p, is transformed to (dY,dX) with a new slope P, ﬁnd the transformation to do this. Such a transformation should be called a ‘contact transformation’.
I assumed X=X(x,y) and Y=Y(x,y)
Then, dY=(∂Y/∂x)dx+(∂Y/∂y)dy and dX=(∂X/∂x)dx+(∂X/∂y)dy with (dY/dX)=P
Again, dy=(∂y/∂X)dX+(∂y/∂Y)dY and dx=(∂x/∂X)dX+(∂x/∂Y)dY with (dy/dx)=p
To specify the transformation,I need to specify the matrix elements: a11,a12,a21,a22
which are respectively,[the matrix is (dX,dY)=Matrix(dx,dy)]
The above equations gave me four relations:
Do these relations (I notice 4 equations and 4 unknowns) give unique solutions?...Somehow, I am making mess with the solution part.Actually, though there are 4 equations,I am not sure if I can write the inverse of C and D as
Can anyone please suggest the significance of the "contact transformation" term?
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