I found any quadrangle in perspective view can be a projection of sqare.
I made a drawing using only ruler and compass to decide a squre from any quadrangle.
This drawing has Visio intelligence.
You can analyse how they change, when you change the quadrangle 1.
You can move quadrangle 1 with controls at the corner 1.
Square will move without rotation linked with the control
at another corner of quadrangle 1,
and will keep square,
but view point will not move.
With right click menu "Free Depth" at quadrangle 1,
enables you to move another 2 controls.
Aspect ratio of quadrangle will change,
then view point will move, and square rotate, but stay square.
Visio drawing was modified to move correctly in "Free Depth" mode.
Added a description of how to make this drawing in a step by step mode.
I made java applet version at
http://june.carrots.jp/perspective/cinderella/index.html
You can move red control point 1,2, and 3.
Also you can move red vanishing point 1 and 2.
But green View point is defined automatically.
Please, remember "Any quadrangle can be projection of a square,
if view point is properly selected".
I hope it will help young people studying perspective drafting and projective geometry.
Hope you enjoy.
I've collected my latest Visio drawings and stencils in one home page at
"STUDY of PERSPECTIVE VIEW with VISIO"
http://june.carrots.jp/perspective/index.html
Some of them are transformed into pdf files.