Parabolas are geometric figures that appear often in the real world.
A ball thrown up at an angle travels along a parabolic arc, a main cable in a suspension forms an arc of a parabola, and the familiar "dish antennas" have parabolic cross sections.
In space geometry, parabola can be obtained by sectioning (or cutting) a right circular conical surface with a suitable plane - parallel to one line on the conical surface. Therefore parabola can be defined as an image of a circle under a specific collinear mapping - perspective collineation, relating two planar cuttings on the conical surface through its vertex V as the centre S of the perspective collineation (Figure 1). The two cutting planes, the circle plane and the ellipse plane, intersect in the axis of the perspective collineation, which is the set of all points in the space that are invariant under the collinear mapping.
Figure 1: Parabola kยด as a collinear image of circle k. |
Being a plane curve, parabola can be determined also in other ways:
Figure 2: Parabola as image of circle under central collineation |
Central collineation is determined by the centre S and axis o (which are all invariant points of the mapping), and a pair of corresponding points , where O is the centre of circle k, but its image O' is not the centre of the corresponding parabola. Diameter r of the circle k that is perpendicular to the axis o of collineation is mapped to the diameter r' of the parabola, which is parallel to the parabola axis. Diameter AB parallel to the axis and parallel tangent lines to circle in its endpoints are mapped to the parabola chord A'B' also parallel to the axis and tangents to parabola in its endpoints meeting in the point R'. Centre of line segment O'R' is point on parabola Q', it is the image of the circle point Q on diameter r. The other endpoint T is mapped to infinity, to the parabola ideal point .
Planar cutting of a right circular conical surface by plane intersecting all surface lines but one is a parabola with the axis paralell to this surface line. Construction is illustrated in the Figure 3.
Figure 3: Parabolic cut on conical surface of revolution |