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Projective Geometry

Desargues' Theorem, Coordinate Projective Geometry, Cross Ratio

Projective geometry is the study of geometric properties which are not changed by a projective transformation. A projective transformation is one that occurs when: points on one line are projected onto another line; points in a plane are projected onto another plane; or points in space are projected onto a plane, etc. Projections can be parallel or central.

For example, the Sun shining behind a person projects his or her shadow onto the ground. Since the Sun's rays are for all practical purposes parallel, it is a parallel projection.

A slide projector projects a picture onto a screen. Since the rays of light pass through the slide, through the lens, and onto the screen, and since the lens acts like a point through which all the rays pass, it is a central projection. The lens is the center of the projection.

Some of the things that are not changed by a projection are collinearity, intersection, and order. If three points lie on a line in the slide, they will lie on a line on the screen. If two lines intersect on the slide, they will intersect on the screen. If one person is between two others on the slide, he or she will be between them on the screen.

Some of the things that are or can be changed by a projection are size and angles. One's shadow is short in the middle of the day but very long toward sunset. A pair of sticks which are crossed at right angles can cast shadows which are not at right angles.

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