the perspectives of a circle

A perspective is a central projection: the points of a plane are sent on an other plane, using concurrent lines connecting
corresponding points (the common point is the perspective centre; if this point is at the infinity, all the lines become parallel and we have a usual projection). The branch of geometry dealing with the properties and invariants of geometric figures under perspective is called projective geometry.

A perspective of a circle (in red, in the light green plane) on the light blue plane, using the red point as perspective centre, is a conic (in green).
The intersection line of the circle's plane (light green) with the plane parallel to the perspective plane (light blue) and going through the perspective centre is "send to the infinity". If this line and the circle are separate the image of the circle is an ellipse ; if not the tangents to the circle at the point(s) of intersection (in yellow) become the "tangents at infinity" to the conic:
  - a parabola is tangent to the "line at infinity",
  - the asymptotes (in blue) of an hyperbola are the "two tangents at infinity"; the centre of the hyperbola is the image of the
intersection point of the two tangents to the circle.