Take a piece of rigid cardboard. Draw a perfect circle on it. Hold it up, and take a picture, with the cardboard held perpendicular to the direction we're looking. You get a photo that looks like this:
Notice: it looks like a perfect circle in the photograph.
Now tilt the cardboard to the right, or tilt it back, so we're no longer viewing it straight on:
Notice that in the photograph the black ink has the shape of an ellipse now, instead of a circle.
What if we tilt it to the right and then tilt it back?
Visually, it still looks like an ellipse to me. Is it?
Conjecture. The shape of the black in in the photograph will always be a perfect ellipse, no matter what orientation the cardboard is held in.
Is this conjecture true? Can we prove it?
I think I can prove it is true if the cardboard is tilted to the left/right or front/back. However, I can't see how to prove it for a combination of those two operations.
If you don't to think about how cameras work, you can think of the problem like this: We stand facing a wall (which is perpendicular to the direction we're looking). We hold the cardboard in front of us in some orientation. Then, we project each speck of black ink onto the wall behind us, by tracing a line from our eye to the speck of ink and continuing until it hits the wall; then we draw a dot there on the wall. Consider the locus of points on the wall obtained in this way. What shape does this locus have? Is it always an ellipse?
Or, if you prefer: hold up a coin in a dark room. Shine a flashlight towards the coin. What is the shape of the shadow on the wall? Is it always an ellipse, no matter what orientation we hold the coin?