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Desargues's theorem
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Desargues's theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
Corresponding sides of the triangles, when extended, meet at points on a line called the axis of perspectivity. The lines which run through corresponding vertices on the triangles meet at a point called the center of perspectivity. Desargues's theorem states that the truth of the first condition is necessary and sufficient for the truth of the second.
This intersection theorem is true in the usual Euclidean plane but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity. Commonly, to remove these exceptions, mathematicians "complete" the Euclidean plane by adding points at infinity, following Jean-Victor Poncelet. This results in a projective plane.
Desargues's theorem is true for the real projective plane and for any projective space defined arithmetically from a field or division ring; that includes any projective space of dimension greater than two or in which Pappus's theorem holds. However, there are many "non-Desarguesian planes", in which Desargues's theorem is false.
Denote the three vertices of one triangle by a, b and c, and those of the other triangle by A, B and C. Let the vertices of separate triangles be connected by lines. Then lines ab and AB meet in a first point, lines ac and AC meet in a second point, and lines bc and BC meet in a third point. Axial perspectivity means these three points all lie on a common line called the axis of perspectivity. Central perspectivity means that the three lines Aa, Bb and Cc are concurrent, at a point called the center of perspectivity.
Desargues never published this theorem, but it appeared in an appendix entitled Universal Method of M. Desargues for Using Perspective (Manière universelle de M. Desargues pour practiquer la perspective) to a practical book on the use of perspective published in 1648 by his friend and pupil Abraham Bosse (1602–1676).
The importance of Desargues's theorem in abstract projective geometry is due especially to the fact that a projective space satisfies that theorem if and only if it is isomorphic to a projective space defined over a field or division ring.
In an affine space such as the Euclidean plane a similar statement is true, but only if one lists various exceptions involving parallel lines. Desargues's theorem is therefore one of the simplest geometric theorems whose natural home is in projective rather than affine space.
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Desargues's theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
Corresponding sides of the triangles, when extended, meet at points on a line called the axis of perspectivity. The lines which run through corresponding vertices on the triangles meet at a point called the center of perspectivity. Desargues's theorem states that the truth of the first condition is necessary and sufficient for the truth of the second.
This intersection theorem is true in the usual Euclidean plane but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity. Commonly, to remove these exceptions, mathematicians "complete" the Euclidean plane by adding points at infinity, following Jean-Victor Poncelet. This results in a projective plane.
Desargues's theorem is true for the real projective plane and for any projective space defined arithmetically from a field or division ring; that includes any projective space of dimension greater than two or in which Pappus's theorem holds. However, there are many "non-Desarguesian planes", in which Desargues's theorem is false.
Denote the three vertices of one triangle by a, b and c, and those of the other triangle by A, B and C. Let the vertices of separate triangles be connected by lines. Then lines ab and AB meet in a first point, lines ac and AC meet in a second point, and lines bc and BC meet in a third point. Axial perspectivity means these three points all lie on a common line called the axis of perspectivity. Central perspectivity means that the three lines Aa, Bb and Cc are concurrent, at a point called the center of perspectivity.
Desargues never published this theorem, but it appeared in an appendix entitled Universal Method of M. Desargues for Using Perspective (Manière universelle de M. Desargues pour practiquer la perspective) to a practical book on the use of perspective published in 1648 by his friend and pupil Abraham Bosse (1602–1676).
The importance of Desargues's theorem in abstract projective geometry is due especially to the fact that a projective space satisfies that theorem if and only if it is isomorphic to a projective space defined over a field or division ring.
In an affine space such as the Euclidean plane a similar statement is true, but only if one lists various exceptions involving parallel lines. Desargues's theorem is therefore one of the simplest geometric theorems whose natural home is in projective rather than affine space.
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