Perspective (geometry)
Two figures in a
Terminology
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.[1]
Perspectivity
If each of the perspective figures consists of all the points on a line (a
Triangles
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.[3]
Notation
Related theorems and configurations
There are ten points associated with these two kinds of perspective: six on the two triangles, three on the axis of perspectivity, and one at the center of perspectivity.
If two triangles are a central couple in at least two different ways (with two different associations of corresponding vertices, and two different centers of perspectivity) then they are perspective in three ways. This is one of the equivalent forms of Pappus's (hexagon) theorem.[5] When this happens, the nine associated points (six triangle vertices and three centers) and nine associated lines (three through each perspective center) form an instance of the Pappus configuration.
The Reye configuration is formed by four quadruply perspective tetrahedra in an analogous way to the Pappus configuration.
See also
Notes
- ^ Young 1930, p. 28
- ^ Young 1930, p. 29
- ^ Dembowski 1968, p. 26
- ISBN 0-88385-522-4. 21,2.
- ^ Coxeter 1969, p. 233 exercise 2
References
- MR 0123930
- Dembowski, Peter (1968), Finite geometries, MR 0233275
- Young, John Wesley (1930), Projective Geometry, The Carus Mathematical Monographs (#4), Mathematical Association of America