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Questions tagged [projective-geometry]

Projective geometry is closely related to perspective geometry. These types of geometry originated with artists around the 14th century.

Projective Geometry is the study of the descriptive properties of geometric figures. It deals with objects/shapes that have been distorted/skewed by perspective transformations.

The Projective Plane:

1.) Homogeneous coordinates

2.) The Principle of Duality

3.) Pencil of lines

4.) Cross Ratio

5.) Conics

6.) Absolute Point

7.) Collineations

8.) Laguerre formula

Howard Eves and Carroll V. Newsom. An Introduction to the Foundations and Fundamental Concepts of Mathematics. Holt, Rinehart and Winston, New York, rev. ed. edition, 1965.

H. S. M. Coxeter. Projective Geometry. Blaisdell Publishing Company, 1964.

H. S. M. Coxeter. The Real Projective Plane. McGraw Hill Book Company, Inc. 1949.

William P. Berlinghoff and Fernando Q. Gouvea. Math through the Ages: A Gentle History for Teachers and Others. Oxton House Publ. and Mathematical Association of America, expanded edition, 2004.

Birchfield, Stanley.1998. http://vision.stanford.edu/~birch/projective/node2.html

C. D. H. Cooper. 2010. Geometry: Projective Geometry Symmetry Ruler and Compass. http://web.science.mq.edu.au/~chris/geometry/chap00.pdf

Joseph L. Mundy and Andrew Zisserman. Appendix – Projective Geometry for Machine Vision. (pg. 463 – 518). http://www.cs.drexel.edu/~kon/introcompvis/reading/zisserman- mundy.pdf

Snuoht. Basic Projective Geometry (Aug 2009). http://www.youtube.com/watch?v=tnvqT0OUStw&NR=1&feature=fvwp

See here for more.

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Cross of two n-dimensional planes.

As you all know, there is geometric place of points of cross of two planes (given as plane vectors) explicitly written simply as $\mathbb{r}=c_1\mathbb{n_1}+c_2\mathbb{n_2}+\lambda\mathbb{n_1}\times\mathbb{n_2}$ More general is $x^i = c_1 n_1^i +…
sanaris
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Intersection fo Projective Lines

I think I've gone wrong with my reasoning somewere here but I'm not sure why. We embed $\mathbb{R}$ into the projective plane by $(x,y)\to[1,x,y]$, and consider the projective lines corresponding to $y=mx$ and $y=mx+c, c\neq 0$ and find there…
Tim
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A point $\in \mathbb{P}^2(\mathbb{C})$

What is the property that should be satisfied so that a point $\in \mathbb{P}^2(\mathbb{C})$ ? I am looking at an exercise where I have to find the flexes of a curve and this information is needed.
evinda
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Central Projection - Project point $X$ on plane $\pi$

We consider the projection of three-dimensional projective space from a center $Z$ onto image plane $\pi$: \begin{align} X \longmapsto \alpha(X) = (Z \lor X) \cap \pi \end{align} since $\pi$ is a plane and given by its homogeneous coordinates…
Stefan Falk
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Construct the 4th point if we know the cross ratio!

Given 3 collinear points A, B,C. Find D (by constructing) if R(ABCD)=1.7/3 (projective geometry). Can someone help me with this problem.
user571977
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Projective real line is homeomorphic to $S$.

Let $S = S^1 / \sim$, being $x \sim y$ if $x = - y$. To obtain the compactness and the connectess of the projective real line, $\mathbb{R} P^1$, I need to prove that $S$ is homeomorphic to $\mathbb{R} P^1$. I can show that exists a biyection $h :…
joseabp91
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