Proving Ellipse When Viewing Circle with Non-Perpendicular Line of Sight

[redoxes]

Draw a circle in a paper, if the line of sight perpendular to the paper , we see a circle ,but if the line of sight is not perpendular to the paper, must we see an ellipse ? how to prove it ? how to find out the major axis or minor axis ? It seem that when we observe the circle ,it should be wide in near place and narrow in far place and should look like a chicken egg ,but it is not true ,how to explain the contradiction. When we view by eyes,we make an affine transformation or a projective transformation ? Whether the relation is true below : parallel projection = affine transformation, central projection = Perspective projection?

 

[tiny-tim]

Welcome to PF!

Draw a circle in a paper, if the line of sight perpendular to the paper , we see a circle ,but if the line of sight is not perpendular to the paper, must we see an ellipse ? how to prove it ? how to find out the major axis or minor axis ? It seem that when we observe the circle ,it should be wide in near place and narrow in far place and should look like a chicken egg ,but it is not true ,how to explain the contradiction.

Hi redoxes! Welcome to PF!

Proof that a circle viewed obliquely is an ellipse depends on how you define an ellipse.

The easiest way is to define it as a conic section, that is as the intersection of a plane and an upright circular cone …

then you can prove it using Dandelin spheres …

see the PF LIbrary on https://www.physicsforums.com/library.php?do=view_item&itemid=98"