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Projective Geometry

  Projective geometry is one of the few branches of mathematics which had its origins in the Middle Ages. It was developed to serve the needs of artists who wished to draw pictures in more realistic ways. The basic problem these artists faced was the difficulty of representing three-dimensional shapes in only two dimensions, putting the appearance of space onto a piece of canvas. As the Renaissance dawned, artists first studied the ways to solve this problem scientifically rather than relying on their instinct and experience.

Projective geometry is obtained from the usual Euclidean geometry by adding what is known as the ‘point at infinity’ to each line. The point at infinity of a given line is the place at which it meets the lines parallel to it, the equivalent of the vanishing point in perspective, or the apparent vanishing point on the horizon. This means that the geometry has no parallel lines, because every pair of lines has an intersection, and so the axioms of Euclid no longer hold. A projection of a figure onto a plane from a given point (which may be the point at infinity) is the intersection with the plane of those lines which join points in the figure to the given point. It is what you would see on a projection screen at the plane with the light at the given point. SMcL



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