Projective Geometry: Proving Existence of Centers S_1, S_2 & S_3

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In summary, The conversation discusses a question about Projective Geometry, specifically regarding points and lines. It mentions the existence of certain points and lines, and the concept of a fixed point in a projection, which the asker will try out.
  • #1
Norman.Galois
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I'm doing reading course on Projective Geometry.

I was presented this question (in the textbook, not homework):

In [itex]P_2 R[/itex], let A, B, and C be points on a line L and let A', B', and C' be points on a line L'. Prove there exists points [itex]S_1[/itex], [itex]S_2[/itex] and [itex]S_3[/itex], and lines [itex]l_1[/itex] and [itex]l_2[/itex] such that projection from L to [itex]l_1[/itex] with center [itex]S_1[/itex], ...

And it continues. The remainder is not important. What do they mean by center [itex]S_1[/itex]?

Thank you.
 
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  • #2
Isn't it a fixed point in some projection?
 
  • #3
radou said:
Isn't it a fixed point in some projection?

That seems to make more sense. I'll try that out. Thanks!
 

1. What is projective geometry?

Projective geometry is a branch of mathematics that studies geometric properties that are invariant under projective transformations. These transformations involve mapping points from one plane to another, while preserving straight lines and ratios of distances.

2. What is the significance of proving the existence of centers S_1, S_2, and S_3 in projective geometry?

The centers S_1, S_2, and S_3 refer to the circumcenter, orthocenter, and centroid of a triangle in projective geometry. Proving their existence not only helps in solving geometric problems, but also provides a deeper understanding of the concepts and properties of projective geometry.

3. How is the existence of centers S_1, S_2, and S_3 proven in projective geometry?

The existence of these centers is proven by using the properties of projective transformations and the fact that certain geometric configurations are invariant under these transformations. By demonstrating that these centers remain the same after a projective transformation, their existence is proven.

4. What are some real-life applications of projective geometry?

Projective geometry has various applications in fields such as computer graphics, computer vision, architecture, and engineering. It is used to create realistic 3D images, design structures and buildings, and solve geometric problems in robotics and navigation systems.

5. Can projective geometry be applied to non-Euclidean spaces?

Yes, projective geometry can be applied to non-Euclidean spaces, such as hyperbolic and elliptic geometry. In fact, it is often used to study and compare different types of geometries and their properties. However, the concepts and theorems may differ in these non-Euclidean spaces.

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