Take a circle and a line. Shear the entire picture. Does the shortest distance between the circle and the line get mapped by the shearing transformation to the shortest distance between the ellipse and line?

Question

thomas5267 · Answer

How do you transform the shortest distance, which is a scalar, under a linear transformation, which only applies to vectors?

kainui · Answer

I was really just too lazy to write out "path of shortest distance"...

thomas5267 · Answer

I am really tempted to say no.  Consider a circle centered in the origin.  Any vector from the origin to the circle is a shortest path but after shearing most of them are not thr shortest anymore.  I know origin is not a line but it is really hard to conceive that it doesn't work for a point but it does for a line.

thomas5267 · Answer

Consider the circle $x^2+y^2=1$ and line $\lambda\begin{pmatrix}1\-1\end{pmatrix}+\begin{pmatrix}1.5\1.5\end{pmatrix}$.  The matrix representation of the circle is$$
\mathbf{x}^TA\mathbf{x}=0\
A=
\begin{pmatrix}
1&0&0\
0&1&0\
0&0&1
\end{pmatrix}
$$in homogeneous coordinates.  Under the shear transformation $M=\begin{pmatrix}1&2\0&1\end{pmatrix}$, the coefficient matrix $A$ transform as $M'^TAM'=\begin{pmatrix}1&2&0\2&5&0\0&0&-1\end{pmatrix}$, corresponding to $x^2+4xy+5y^2-1=0$, $M'=\begin{pmatrix}1&2&0\0&1&0\0&0&1\end{pmatrix}$.  The line transform to $\lambda\begin{pmatrix}-1\-1\end{pmatrix}+\begin{pmatrix}4.5\1.5\end{pmatrix}$.  The transformed line and the ellipse intersects despite the circle and the original line did not.  Furthermore, the closest point on the original line to the circle is $\begin{pmatrix}1.5\1.5\end{pmatrix}$ by construction but it transforms to $\begin{pmatrix}4.5\1.5\end{pmatrix}$, which is not one of the two points in which the transformed line intersects the ellipse.

thomas5267 · Answer

Nope, this is incorrect.  The shear transformation from circle to ellipse in homogeneous coordinate is not this.

thomas5267 · Answer

Somehow the matrix $\begin{pmatrix}1&2&0\0&1&0\0&0&1\end{pmatrix}$ reflected the ellipse upside down.

thomas5267 · Answer

Apparently the coefficient matrix transform by $A\mapsto \left(M^{-1}\right)^TAM^{-1}$.  The graph from the correct transformation looks like your assertion is true...