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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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Scaling axes to reflect perspective plane of an image.

Suppose I have an image which contains a road in it, and I want to be able to pinpoint the locations of cars on that road using pre-calibrated distances. How to do that using formulas that map pixel coordinates to the calibrated distances? A method…
Hemkesh Molla
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Prove two parallel lines intersect at infinity in $\mathbb{RP}^3$

I have to prove two parallel lines intersect at infinity in $\mathbb{RP}^3$. I have to use the direction vectors and that points at infinity have last coordinate $0$. I tried solving a system of equations but it didn't work. What I wanted to do was…
Annnnnet
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Book for real projective space

I need a book that analytically studies the real projective space, one that is very clear and has examples. Thanks
Annnnnet
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Finding two missing card corners under perspective

If I take a photo of a rectangular playing card, I can reconstruct the screen/photo-space position of one missing corner which lies outside the photo (outside the camera frustum): assuming simple perspective projection and no lens distortion, I just…
leander
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A problem in projective geometry...

I have the following projectivity: $$ f[x_1,x_2,x_3]=[4x_1+2x_2-x_3,2x_2,x_3,-x_2-x_3]. $$ I have to find all the lines $L$ such that $f(L) \subset L$. I've found the eigenvalues of this matrix, which are three distinct real values. So the lines…
TheWanderer
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Proving a theorem using Pappus' theorem

I need some help. I want to prove Desargues' theorem via using Pappus' theorem. And I don't know how. Please, help me!
Antonio Riveira
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A doubt in the proof of Desargues' Theorem.

I have a question regarding the proof of Desargues' Theorem. When the traingles $ABC$ and $A'B'C'$ are assumed to be lying on the same plane. A point $X$ is taken outside that plane, and the lines $XA,XB,XC,XA',XB'$ and $XC'$ are drawn. Then $D$…
user67803
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Showing there are 6 possible values for the cross ratio

If we look at the cross ratio $(x_0 x_1:x_2 x_3) = \lambda$ of 4 points in projective space, I can see that by looking at all possible permutations (24 of them) of the points we can see that only 6 of them give the same cross ratio. I.e: $\lambda,…
Wooster
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Applications of Finite Projective Planes

Can someone point me towards some applications of finite projective planes that are approachable without too much background knowledge? So far, I have vector spaces, Latin Squares, and Sudoku, but I was wondering if there were any others?
Nishant
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cross ratio - how to calculate values with it?

I have this scenario here. It is a projective image. What would be the cross-ratio formula for these points?: $$v,c_{r},r_{2},r_{1}$$ 2.And let be $$dst(r_{2},r_{1})=30cm$$ in the real world. How can I calculate the other values (distances)…
gartenabfall
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Show that three pairwise non-intersecting lines in $\mathbb{R}\mathbb{P}^{3}$ have a transversal.

Let $\mathbb{P}(U1)$ and $\mathbb{P}(U2)$ be two non-intersecting lines in the 3-dimensional projective space $\mathbb{R}\mathbb{P}^{3}$ = $\mathbb{P}(\mathbb{R}^{4})$. Show that $\mathbb{R}^{4}$ equals the direct sum U1 $\oplus$ U2. Deduce that…
Greg
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Beginning Incidence Geometry: Flags & types

So far, I understand that type means something like a line or a point but what does this notation mean?: type(x) such that x is an element of F? Or what does type(x) in general mean? For flags, I have that they are defined as pairwise incident…
cakey
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Question 1.1 in Projective Geometry by Coxeter

Thus projective geometry deals with triangles, quadrangles, and so on, but not with right-angled triangles, paralleograms, and so on. -Projective Geometry, Coxeter pg 3. The first question of the section is: Which of the following figures…
yiyi
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Equality of Projective Subspaces Not Contained in a Hyperplane Based on Intersection

Let $S$ be a projective space of dimension $n$, and $H$ a hyperplane in $S$ with dimension $n-1$. Consider two projective subspaces, $P_1$ and $P_2$, not contained in $H$. We question if $P_1$ and $P_2$ are identical given their intersections with…
Octavai Ji
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cross ratio and harmonic points

Let A, B, C, D be 4 an ordered 4 tuple of different points on a line, assume cross ratio [A, B, C, D]=[B, A, C, D], then [A, B, C, D, ]=-1. This seems to be an easy question but I cannot figure out. By definition of cross…
diiiiiklllllll
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