By Eisenhart L.P.

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During this booklet the main points of many calculations are supplied for entry to paintings in quantum teams, algebraic differential calculus, noncommutative geometry, fuzzy physics, discrete geometry, gauge conception, quantum integrable platforms, braiding, finite topological areas, a few facets of geometry and quantum mechanics and gravity.

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In doing this, it is convenient to consider the plane perspective to be defined by its center, O, the image, G′ of some particular point, G, and its vanishing line, v, rather than its axis, l. As we have seen (Theorem 2, Sec. 3) with these data given, the image, P', of an arbitrary point, P, is found by first determining the intersection, V, of v and PG, and then locating P' as the intersection of OP and the line through G' which is parallel to OV. Specifically, let us investigate the image of a circle, Γ, under a plane perspective.

Into another parabola? 12. How can a hyperbola be transformed into an ellipse by a plane perspective ? Into a parabola ? Into another hyperbola? 13. What special property, if any, does the image of a circle have when the center of the plane perspective is the center of the circle? 14. If a circle Γ is tangent at V to the vanishing line of a plane perspective, what line through V is transformed into the axis of the parabola which is the image of Γ? What point on Γ is the preimage of the vertex of the image parabola?

The first of these was Brunelleschi (1379-1446), who by 1425 had developed a system of perspective which he used in his own work and taught to other painters. The first text on perspective, a treatise by Alberti (1404-1472), appeared in 1435. Later Piero della Francesca (c. 1418-1492), a gifted mathematician as well as an outstanding painter, extended considerably the work of Alberti. Still later, both Leonardo da Vinci (1452-1519) and Albrecht Diirer (1471-1528) wrote treatises on perspective which not only presented the mathematical theory of perspective but insisted on its fundamental importance in all painting.