# Crazy orientation of solid oblong

1. Oct 12, 2012

### julian

I got these people trying to say that you can orientate a solid oblong so that it looks half the length and width with the only residule effect that it looks like it is slanted sideways a bit. Does anyone know what it is or are they crazy. I think they might be crazy. How do you explain this to them in simple terms?

2. Oct 12, 2012

### Simon Bridge

Erg - it may make sense in terms of perspective and projection.
Do you have an example?

I could tilt an oblong block (a book say) back so it's surface makes an angle 30deg to horizontal ... to an observer looking along the horizontal, it will kind-of have half it's full height. (projection to the vertical would be sin(30)=1/2=0.5) This would also bring the bottom into view - the bottom face would be tilted 60deg to the horizontal and therefore the width will not appear to be half (sin(60)=√3/2=0.87).

Last edited: Oct 12, 2012
3. Oct 15, 2012

### julian

Something like a book viewed from a camera. But they want to angle the book so that the book looks half the height AND half the width (not talking about halving the width of the bottom face but the book itself) at the same time and while keeping the angle between the top edge and side edge close to 90 degrees. when image is viewed from the computer screen. You cant do that can you?

Last edited: Oct 15, 2012
4. Oct 15, 2012

### Simon Bridge

If all you are interested in is the projected dimensions of the front pane then I would say that is kinda possible ... the resulting projection would be a diamond.

If I put the book so it is face-on ... center it on (0,0,0) with the x-axis to the left, y-axis up, and z pointing at the observer.
Height h is along y, width w is along x, and thickness t is along z. But we are only concerned with height and width - so ignore thickness.
We are interested in projections h' and w' in the x-y plane... so we are not worried about perspective that a camera would introduce.

If I rotate the book 60deg about the x axis, it's projection in the x-y plane will be w'=w, and h'=h/2

Can you rotate the already rotated figure so that w'=w/2 now, without changing h'?

You can certainly can just rotate it by 60deg about the y-axis to get w'=w/2 if you define w' to be the horizontal distance across the projection.
(The projection of the projection onto the x-axis.)
Defining w' to be the perpendicular distance between opposite sloping sides (like you'd normally do for a rhombus) - you'd need a different rotation to make w'=w/2.

But does it still make sense to call it a book of half width? Did "they" intend the book to remain rectangular (all angles at the vertices to be 90deg)?

Have you asked "these people" to tell you what transformation does what they claim?

Last edited: Oct 15, 2012