I am not a mathematician (yet), but rather more of an artist and have a question about perspective which can be described with projective geometry.
The diameter of a sphere and a circle are equal. A station point and picture plane are perpendicular to the center of each shape. Each shape is equal distance from the picture plane and station point.
When we make the projection will the shapes from the sphere and circle on the picture plane be congruent?
Consider this, you cannot see 50% of a sphere unless the projection point is infinitely far away from the sphere (orthographic projection). So the most we can see in finite space is upwards of 49% but not quite 50%, which is where the hemisphere and diameter of the sphere are. If the circular contour of the sphere being projected isn't from the hemisphere, but very close to it, how can the two shapes appear congruent?
Best Answer
In the above figure you see that there are three kinds of rays emanating from the station point: rays that hit the surface of the sphere and illuminate it there, rays that miss the sphere and illuminate the projection plane, and finally the rays that just touch the sphere. These special rays form a circular cone and touch the sphere along the red circle, which is the shadow limit on the sphere. The image of the red circle is a conic section on the projection plane. This conic section is again a circle in my disposition.
Now you have in addition a wire circle of the same radius as the sphere has. If you place this circle properly, which is not on the sphere, you can have an image coinciding with the shadow boundary of the sphere on the projection plane. This is the green circle in my figure.