Perspective (geometry)

Two figures in a plane are perspective from a point O, called the center of perspectivity, if the lines joining corresponding points of the figures all meet at O. Dually, the figures are said to be perspective from a line if the points of intersection of corresponding lines all lie on one line. The proper setting for this concept is in projective geometry where there will be no special cases due to parallel lines since all lines meet. Although stated here for figures in a plane, the concept is easily extended to higher dimensions. The line which goes through the points where the figure's corresponding sides intersect is known as the axis of perspectivity, perspective axis, homology axis, or archaically, perspectrix. The figures are said to be perspective from this axis. The point at which the lines joining the corresponding vertices of the perspective figures intersect is called the center of perspectivity, perspective center, homology center, pole, or archaically perspector. The figures are said to be perspective from this center. Perspectivity If each of the perspective figures consists of all the points on a line (a range) then transformation of the points of one range to the other is called a central perspectivity. A dual transformation, taking all the lines through a point (a pencil) to another pencil by means of an axis of perspectivity is called an axial perspectivity. An important special case occurs when the figures are triangles. Two triangles that are perspective from a point are called a central couple and two triangles that are perspective from a line are called an axial couple. Karl von Staudt introduced the notation to indicate that triangles ABC and abc are perspective. Desargues' theorem states that a central couple of triangles is axial. The converse statement, that an axial couple of triangles is central, is equivalent (either can be used to prove the other). Desargues' theorem can be proved in the real projective plane, and with suitable modifications for special cases, in the Euclidean plane.