 InverseTransformation - Maple Help

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DifferentialGeometry

 InverseTransformation
 find the inverse of a transformation Calling Sequence InverseTransformation(Phi, options) Parameters

 Phi - a transformation mapping one manifold M to another manifold N options - branch = "all" or branch = [pt1, pt2], where pt2 is a list of coordinates (Maple expressions) defining a point in M and Phi(pt2) = pt1 Description

 • The InverseTransformation command uses the Maple solve command to find a (local) inverse transformation Psi: N -> M, that is, Psi o Phi = identity on M and Phi o Psi = identity on N.
 • Use the Maple environment variable _EnvExplicit = true to obtain explicit formulas for the inverse.
 • In the case where there are multiple local inverses, the first one in the list returned by solve is returned by InverseTransformation.  This may vary from one Maple session to another.
 • With branch = "all", InverseTransformation returns a list of all the inverse transformations.
 • With branch = [pt1, pt2], InverseTransformation returns the particular inverse transformation Psi satisfying Psi(pt1) = pt2.
 • This command is part of the DifferentialGeometry package, and so can be used in the form InverseTransformation(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-InverseTransformation. Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$

Define a pair of 2-dimensional manifolds.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right):$$\mathrm{DGsetup}\left(\left[u,v\right],N\right):$

Example 1.

Define a simple transformation Phi1: M -> N with a unique global inverse.

 > $\mathrm{Φ1}≔\mathrm{Transformation}\left(M,N,\left[u=2x-3y,v=-x+2y\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{N}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{rr}2& -3\\ -1& 2\end{array}\right]\right]\right]{,}\left[\left[{2}{}{x}{-}{3}{}{y}{,}{u}\right]{,}\left[{-}{x}{+}{2}{}{y}{,}{v}\right]\right]\right]\right)$ (1)
 > $\mathrm{Ψ1}≔\mathrm{InverseTransformation}\left(\mathrm{Φ1}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{N}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{rr}2& 3\\ 1& 2\end{array}\right]\right]\right]{,}\left[\left[{2}{}{u}{+}{3}{}{v}{,}{x}\right]{,}\left[{2}{}{v}{+}{u}{,}{y}\right]\right]\right]\right)$ (2)

Use ComposeTransformations to checks the result of InverseTransformation.

 > $\mathrm{ComposeTransformations}\left(\mathrm{Ψ1},\mathrm{Φ1}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\right]\right]{,}\left[\left[{x}{,}{x}\right]{,}\left[{y}{,}{y}\right]\right]\right]\right)$ (3)
 > $\mathrm{ComposeTransformations}\left(\mathrm{Φ1},\mathrm{Ψ1}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{N}{,}{0}\right]{,}\left[{N}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\right]\right]{,}\left[\left[{u}{,}{u}\right]{,}\left[{v}{,}{v}\right]\right]\right]\right)$ (4)

Example 2.

Define a transformation  Phi2: M -> N  with multiple local inverses.

 > $\mathrm{Φ2}≔\mathrm{Transformation}\left(M,N,\left[u={x}^{2},v={y}^{2}\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{M}{,}{0}\right]{,}\left[{N}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}2{}x& 0\\ 0& 2{}y\end{array}\right]\right]\right]{,}\left[\left[{{x}}^{{2}}{,}{u}\right]{,}\left[{{y}}^{{2}}{,}{v}\right]\right]\right]\right)$ (5)
 > $\mathrm{Ψ2}≔\mathrm{InverseTransformation}\left(\mathrm{Φ2}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{N}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}\frac{1}{2{}\mathrm{RootOf}{}\left({\mathrm{_Z}}^{2}-u\right)}& 0\\ 0& \frac{1}{2{}\mathrm{RootOf}{}\left({\mathrm{_Z}}^{2}-v\right)}\end{array}\right]\right]\right]{,}\left[\left[{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{u}\right){,}{x}\right]{,}\left[{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{v}\right){,}{y}\right]\right]\right]\right)$ (6)

To get explicit solutions:

 > $\mathrm{_EnvExplicit}≔\mathrm{true}$
 ${\mathrm{_EnvExplicit}}{≔}{\mathrm{true}}$ (7)
 > $\mathrm{Ψ2}≔\mathrm{InverseTransformation}\left(\mathrm{Φ2}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{N}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}\frac{1}{2{}\sqrt{u}}& 0\\ 0& \frac{1}{2{}\sqrt{v}}\end{array}\right]\right]\right]{,}\left[\left[\sqrt{{u}}{,}{x}\right]{,}\left[\sqrt{{v}}{,}{y}\right]\right]\right]\right)$ (8)

To get all possible inverses:

 > $\mathrm{AllInverses}≔\mathrm{InverseTransformation}\left(\mathrm{Φ2},\mathrm{branch}="all"\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{N}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}\frac{1}{2{}\sqrt{u}}& 0\\ 0& \frac{1}{2{}\sqrt{v}}\end{array}\right]\right]\right]{,}\left[\left[\sqrt{{u}}{,}{x}\right]{,}\left[\sqrt{{v}}{,}{y}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{N}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}\frac{1}{2{}\sqrt{u}}& 0\\ 0& -\frac{1}{2{}\sqrt{v}}\end{array}\right]\right]\right]{,}\left[\left[\sqrt{{u}}{,}{x}\right]{,}\left[{-}\sqrt{{v}}{,}{y}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{N}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}-\frac{1}{2{}\sqrt{u}}& 0\\ 0& \frac{1}{2{}\sqrt{v}}\end{array}\right]\right]\right]{,}\left[\left[{-}\sqrt{{u}}{,}{x}\right]{,}\left[\sqrt{{v}}{,}{y}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{N}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}-\frac{1}{2{}\sqrt{u}}& 0\\ 0& -\frac{1}{2{}\sqrt{v}}\end{array}\right]\right]\right]{,}\left[\left[{-}\sqrt{{u}}{,}{x}\right]{,}\left[{-}\sqrt{{v}}{,}{y}\right]\right]\right]\right)\right]$ (9)

Since Phi2([- 1, - 1]) = [1, 1], we can ask for that particular inverse which maps [1, 1] to [- 1, - 1].  We can use either [1, 1] or [u = 1, v = 1] as arguments in the command InverseTransformation to indicate the coordinates of the point.

 > $\mathrm{InverseTransformation}\left(\mathrm{Φ2},\mathrm{branch}=\left[\left[1,1\right],\left[-1,-1\right]\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{N}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}-\frac{1}{2{}\sqrt{u}}& 0\\ 0& -\frac{1}{2{}\sqrt{v}}\end{array}\right]\right]\right]{,}\left[\left[{-}\sqrt{{u}}{,}{x}\right]{,}\left[{-}\sqrt{{v}}{,}{y}\right]\right]\right]\right)$ (10)
 > $\mathrm{InverseTransformation}\left(\mathrm{Φ2},\mathrm{branch}=\left[\left[u=1,v=1\right],\left[x=-1,y=-1\right]\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{N}{,}{0}\right]{,}\left[{M}{,}{0}\right]\right]{,}\left[{}\right]{,}\left[\left[\begin{array}{cc}-\frac{1}{2{}\sqrt{u}}& 0\\ 0& -\frac{1}{2{}\sqrt{v}}\end{array}\right]\right]\right]{,}\left[\left[{-}\sqrt{{u}}{,}{x}\right]{,}\left[{-}\sqrt{{v}}{,}{y}\right]\right]\right]\right)$ (11)