The projection of a circle onto a plane is, in general, an ellipse.
Explanation:
When a circle is projected onto a plane that is not parallel to the circle's plane, the resulting shape will be an ellipse. The amount of distortion (i.e., how "squashed" the ellipse appears) depends on the angle between the circle's plane and the projection plane.
- If the plane is parallel to the circle's plane: The projection is a circle (congruent to the original if the projection is orthogonal).
- If the plane is perpendicular to the circle's plane: The projection is a line segment (a special case of an ellipse where the minor axis has length zero).
- For all other angles: The projection is an ellipse.
Why is it an Ellipse?
Think of the circle as being "tilted" relative to the projection plane. The diameter of the circle that's perpendicular to the axis of tilt will project to the major axis of the ellipse, while the diameter along the axis of tilt will project to the minor axis of the ellipse.
Exceptions:
- As mentioned above, when the projection plane is parallel to the plane of the circle, the projection is a circle. This is a special case where the ellipse becomes a circle.
- When projecting to a line (a 1-dimensional projection), it's a line segment.
In summary: The most general answer for the projection of a circle is an ellipse, with a circle and a line segment as special cases depending on the orientation of the plane onto which the circle is being projected.
