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I have matrix $$ A=\begin{bmatrix} 1 & 1\\1 & 0\\0 & 1 \end{bmatrix}$$ The question, similar to the discussion at Transform sphere to an ellipse in $\mathbb{R}^2$, is to find the parametric equation of the transformation of the unit circle with $T(x)=Ax$.

I showed that the transformation maps $R^2$ to a plane in $R^3$. The plane will be the column space of matrix $A$. Because the null space of the $A^T$ is the orthogonal complement of the column space of matrix $A$, I can do this:

A = {{1, 1}, {1, 0}, {0, 1}};
NullSpace[Transpose[A]]

Which gives this basis for the null space of $A^T$. $$\beta=\left\{ \begin{bmatrix} -1\\1\\1 \end{bmatrix}\right\}$$

Now I can do this:

Clear[x, y, z]
{x, y, z}.First[NullSpace[Transpose[A]]]

Which gives me the equation of the plane as $-x+y+z=0$. Solving for $z$ gives me $z=x-y$, so I can draw this transformation of $R^2$.

Plot3D[x - y, {x, -2, 2}, {y, -2, 2}, Mesh -> None]

Which gives this image:

enter image description here

Next, I selected 100 random points from the unit circle (corey979's cool idea), then applied $T(x)=Ax$ to the set of points, using Michael E2's cool move.

pts = RandomPoint[Circle[], 100];
tpts = pts.Transpose[A];

Then I added these points to my image.

Show[

 Plot3D[x - y, {x, -2, 2}, {y, -2, 2}, Mesh -> None],
 Graphics3D[{
   Red, PointSize[Large],
   Point[tpts]
   }]
 ]

enter image description here

So I have excellent visualization that the transformation maps the unit circle to an ellipse on the plane $-x+y+z=0$ in $R^3$.

But, now my question. How can I find a parametric equation representing this ellipse in the plane so I can use ParametricPlot3D to graph it?

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  • $\begingroup$ Have you considered accepting an answer to the question you refered to? They are all excellent, so the choice might be hard, but I'm sure it can be done. $\endgroup$ – corey979 Dec 2 '16 at 10:00
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Just transform your favourite parametrization of the cirle,e.g.:

mat = {{1, 1}, {1, 0}, {0, 1}};
Show[
 Plot3D[x - y, {x, -2, 2}, {y, -2, 2}, Mesh -> None, 
  PlotStyle -> None], 
 ParametricPlot3D[mat.{Cos[u], Sin[u]}, {u, 0, 2 Pi}
  ], Graphics3D[{Red, PointSize[0.02], 
   Point[mat.# & /@ RandomPoint[Circle[], 10]]}]]

enter image description here

| improve this answer | |
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  • $\begingroup$ Nice answer. Thanks so much for the help. $\endgroup$ – David Dec 2 '16 at 7:26

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