perspectives of a circle with LiveGraphics3D

A perspective is a central projection : the transformation of points and lines in the 3D space onto a plane by connecting corresponding points with conccurent lines (the common point is the perspective centre (if this point is at the infiniy, all the lines become parallel and we have a projection. The branch of geometry dealing with the properties and invariants of geometric figures under perspective is called projective geometry.

The perspectives of a circle (in red, in the light green plane) on the light blue plane (using the red point which can be moved with the mouse) is a conic (in green).
The intersection line of the circle's plane with the plane parallel to the perspective plane and going through the perspective centre is "send to the infinity". The tangents at the point(s) of the circle in that plane (in yellow) become the "tangents at infinity" :
- a parabola is tangent to the "line at infinity,
- the asymptotes (in blue) of an hyperbola are the tangents at infinity.