# Transformation of Unit Circle to a Plane in $R^3$

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: 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]
}]
] 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?

• 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. – corey979 Dec 2 '16 at 10:00

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]]}]] • Nice answer. Thanks so much for the help. – David Dec 2 '16 at 7:26