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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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How does this image demonstrate the fundamental theory of projective geometry?

https://en.wikipedia.org/wiki/Projective_geometry https://en.wikipedia.org/wiki/Projective_geometry#/media/File:Theoreme_fondamental_geometrie_projective.PNG I can certainly see that projection is used in the image, but it goes completely…
John P
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Projective completion of an affine plane

I know that a infinite point in a projective space can be defined as a equivalence class of $||$ relation in affine space. So if i have an affine space $(\mathcal{A},\mathcal{D}, \phi)$, let $\mathcal{D}\subset\mathcal{P(\mathcal{A})}$ a set of…
MathLearner
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Intersection of projective space

I have to prove the following: Let $S^k\neq\varnothing$ be a projective space. It follows that: $$X(x)\in\cap_{i\in I}S_i^{k_i}\Leftrightarrow [x]\subset (\cap_{i\in I}S_i^{k_i+1})\subset P^{n+1}.$$ It seems to be obvious but I can't prove it. Could…
mathplayer
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Points on a line and lines through a point in projective geometry notation

Let $(P, L, I)$be a projective plane. For a line $l \in L$ let $p(l)= \{ p \in P | pIl \}$ be the points on $l$,and for a point $p \in P$ let $l(p)= \{ l \in L | pIl \}$ be the lines through $p$. From: Jürgen Richter-Gebert (auth.) - Perspectives…
tryst with freedom
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What is the relationship between $\mathbb{R}^3$ and the projective plane $P^2$?

I'm a computer science student and I am trying to understand the relationship between $\mathbb{R}^3$ and the projective plane $P^2$. I need to learn about the projective plane as it comes up in computer graphics. I know that $P^2$ can be thought of…
S. M.
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Represent point at infinity by homogeneous co-ordinates

Suppose I have an point $(3, 2)$ which I've seen has homogeneous coordinates $(3,2,1)$ and $k(3,2,1)$ where $k \neq 0$, and $(3,2,1)$ and $(3k,2k,k)$ both represent the same Cartesian point $(3, 2)$. My question is that could I write $(3,2)$ in…
S. M.
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Projective transformation at infinity

Suppose I have the line $2x+3y+w=0$ which contains the point $[1, -1, 1]$. After projective transformation given by multiplying by a matrix, it goes to infinite point $[1, -1, 0]$ which lies on the line $x+y+w =0$ at infinity. But the line at…
S. M.
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Proof of a circle defined by five points in projective geometry

I'm trying to understand how to prove that a circle can be defined in projective geometry by five points, where three are "classic points" and two are the circular points. For demostrating that is definable with three points I'm starting by the…
piClash
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Proving the concurrence of three lines.

Let $p_{1}, p_{2} \text{ and } p_{3}$ be three planes which intersect in a straight line (and not a point, which is generally the case). Let a fourth plane $p_{4}$ cut these planes (not at the line of intersection of the three planes). Evidently,…
user67803
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Homography in projective spaces

In $\mathbb{P}_\mathbb{R}^3$ the following quadric are given: $Q_1: x_0^2+x_2^2+x_3^2=0$, $Q_2: x_0^2+x_1^2-x_2^2-2x_0x_3+2x_2x_3=0.$ Say if there is a homography $\omega: \mathbb{P}_\mathbb{R}^3\to \mathbb{P}_\mathbb{R}^3$ such that $\omega(Q_1)…
Julio
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why the last rule of projective planes looking for 4 points instead of 3?

Looking at the rules of projective planes the rule indicates: There exists a set of four points, no three of which belong to the same line. But I'm wondering why should there be a set of 4 points? I'm asking what could have gone wrong if we had…
Ali1S232
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Projective equation of a circle given a centre

I feel like this problem given to me is much easier than I'm making it out to be. Basically I'm only asked to write the equation of a circle with a given centre $(a,b)$, then the projective equation of the circle with the same centre. How can I do…
twistedshell
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When is perspectivity of triangles transitive?

I am having a problem with Exercise 2.4.2 in the book Projective Geometry by H.S.M. Coxeter: If two quadrangles have the same quadrangular set, then their diagonal triangles are perspective. In my attempt I have drawn two quadrangles $ABCD$ and…
QED
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What's the cross ratio of 4 planes?

For example, $H_1 : 3X_0 + 2X_3 = 0\\H_2 : −X_0 − X_1 + X_2 = 0\\H_3 : 4X_0 + X_1 − X_2 + 2X_3 = 0\\H_4 : 5X_0 + 2X_1 − 2X_2 + 2X_3 = 0$
Íñigo
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Projective Maps

A projective map is a function which preserves cross-ratio. So consider a circle $\Gamma$ and a point $P$ not necessarily on $\Gamma$. Will the function which maps $\Gamma$ to the pencil of lines going through $P$ via $X \mapsto PX$ be…
gauri agrawal
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