Let $A$ be the " moving" point $(a,b)$ with $a$ and $b$ two variable coordinates ranging from $-10$ to $+10$, with equal rhythm of change.
When I create this point on a graphing calculator, I observe that it follows a rectangular path.
https://www.desmos.com/calculator/my5f8bwjs7
When I prevent one coordinate from varying for a while, and then let it change again, I observe that the path changes, but remains rectangular.
In case this observation is not accidental, how to explain it analytically?
I understand intuitively that the simultaneous change in coordinates causes a translation. But I can't see why, at a given moment, the movement follows a square angle.
Also, I think I could define a given rectangle as the union of line segments ( each with a definite equation) considered as sets of points.
But What analytic definition or " rectangle" should be used to prove that the path has to be rectangular.
How to define a rectangle somewhat as a geometrical " locus"?
You have selected $|v_x|=|v_y|$. You have several cases. When both are positive, $v_x=v_y>0$, then $A$ is moving along a path with slope $+1$ from the low values of $x$ and $y$ to high values of $x$ and $y$. If both values for velocity are negative, you are once again moving on a path with slope $+1$, but in the opposite direction. That means you move either on the same line, or on parallel lines. Similarly, $v_x=-v_y>0$ and $v_x=-v_y<0$ are parallel trajectories. If the time spent on each pair of parallel lines is identical in both direction, you are moving on a parallelogram. Since the slope for $v_x=v_y$ is $+1$ and for $v_x=-v_y$ is $-1$, the trajectories are perpendicular. So you get a rectangle.