# 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. Dec 2, 2016 at 10:00

mat = {{1, 1}, {1, 0}, {0, 1}};