How to reflect an object without changing its geometry

Question

I have an object in which has its orientation defined by 3 vectors and a point and I need to reflect it across a plane without changing the geometry of the object (For example if there was text on the object it would not be mirrored.

EX. https://i.stack.imgur.com/JyaLm.png

In this image the 3 axis vectors are represented by the blue red and green lines, the point is represented by where those 3 lines intercept on that part and the green plane is the plane I am mirroring over.

I can not find out what type of transformation I would need to do on these vectors and points to get the end result.

Answer 1

What you have, according to the picture in your link, is a half-circle rotation about the Green axis,followed by a translation. Let v=(r,b,g) where r,b,g are the co-ordinates with respect to the Red,Blue, Green axes. A half-circle rotation sends v to (-r,-b,g). For a translation,choose fixed values p,q,r and send each V to (r+p,b+q ,g+r). So altogether you need f((r,b,g))=(-r+p,-b+q,g+r). To determine p,q,r it will suffice that the formula gives the desired value for just one point.

Answer 2

In ${\mathbb R}^3$ there are three kinds of special transformations called "reflections":

(i) "Point-reflection": Any point ${\bf x}$ is sent to its mirror point with repect to some fixed point ${\bf p}$: $$T_{\bf p}:\quad {\bf x}\mapsto 2{\bf p}-{\bf x}\ .$$ When ${\bf p}={\bf 0}$ this is just $$T_{\bf 0}:\quad {\bf x}\mapsto -{\bf x}\ .$$ The corresponding matrix is $=-I$ and has determinant $-1$. It follows that text painted on an object $A\subset {\mathbb R}^3$ is mirrored under $T_{\bf p}$.

(ii) Reflection with respect to a line $g\subset{\mathbb R}^3$: When $g$ is the $z$-axis then the corresponding reflection is given by the matrix $$\bigl[T_g]=\left[\matrix{-1&0&0\cr 0&-1&0\cr 0&0&1\cr}\right]$$ with determinant $1$. It follows that text painted on an object $A\subset {\mathbb R}^3$ is not mirrored under $T_g$. Note that $T_g$ can be viewed as a rotation by $180^\circ$ around the axis $g$.

(iii) Reflection with respect to a plane $\pi\subset{\mathbb R}^3$: When $\pi$ is the plane $z=0$ then the corresponding reflection is given by the matrix $$\bigl[T_\pi]=\left[\matrix{1&0&0\cr 0&1&0\cr 0&0&-1\cr}\right]$$ with determinant $-1$. It follows that text painted on an object $A\subset {\mathbb R}^3$ is mirrored under $T_\pi$. This is the kind of reflection we are accustomed to when looking at a mirror.