Скачать или смотреть FPG4 Projections Preserve Harmonic Sets

Harmonic sets are collections of points, or lines, with very special properties that are abundant in projective geometry. One reason for this is that harmonic sets are preserved under projections, and so they survive some rather disruptive transformations. Harmonic sets lie close to the heart of projective geometry, and interrelate with a myriad of other concepts.In this lecture we define harmonic sets, as well as the more general, quadrangular sets. We prove that the last member of such a set has its position predetermined by the other members using Desague''s Theorem, and we exploit line-point duality to prove that projections preserve harmonic sets.

Projective geometry is more basic and important than Euclidean geometry, because it uses less assumptions, and in concerned with statements which remain true for a much wider range of different geometric setups. In fact, with this algebraic approach, we do not even define a metric. We shall see how such ideas, as well as those of polarity, harmony and conic curves arise as natural consequences of our small set of initial axioms.

I also created a less algebraic more visual, course in projective geometry, which begins with the video:

   • Projective Geometry 1 Without Equations, C...  


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