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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.

2534 questions
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Rectify image from congruent planar shape objects

I am implementing an algorithm to remove projective distortions on the following image. I understand this is possible by applying the following transformation: $$ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ l_1 & l_2 &…
Andres Felipe
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Degrees of freedom in a Homography matrix

Homograpy transformation maps a point in one plane into a point in another plane, $$\begin{pmatrix}x'\\ y'\\ 1\end{pmatrix} = H_{3\times 3}\begin{pmatrix}x\\ y\\ 1\end{pmatrix},$$ where the points $X'$ and $X$ are represented using homogeneous…
avocado
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Understanding Projective Geometry; images of circles becoming ellipses, parabolas, or hyperbolas

Projective Geometry quote from Mathematics: Its Content, Methods, and Meaning This question is specifically in an attempt to understand Mathematics: Its Content, Methods, and Meaning. At the linked point in the book, it says: The difference in them…
Mike Williamson
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A question from "Foundations of Projective Geometry" by Hartshorne.

"Foundations of Projective Geometry" by Hartshorne says the following: The completion of the affine plane of four points is a projective plane with 7 points. The affine plane of $4$ points is essentially a paralellogram $ABCD$. The completion…
user67803
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Is ''isomorphic'' between projective planes actually equivalence relation?

I'm new to projective geometry and I just read a few pages of Hartshone's Foundation of Projective Geometry. On page 5 he defines two projective planes to be isomorphic if there exists, a bijection from one to another that takes collinear points…
Zichen Gao
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Is there a projection matrix for 2D to 1D perspective projection?

I was wondering, if there is a projection matrix for a perspective projection of a 2D point to a line. E.g. a random point being projected to the line at $x=1$, parallel to the y axis in the direction of the origin $(0,0)$. I know that the easiest…
Corbie
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Prove that $f$ is not a projectivity

Definition (projectivity). Let $V,W$ vector spaces over some field $k$ and $V\neq\{0\}$. Let $F\::V\to W$ an injective linear map and let $v \in V\ \setminus\{0\}$. A projective map $\mathbb{P}(F\ ):\mathbb{P}(V\ )\to\mathbb{P}(W\ ):kv\mapsto kF(v)$…
user12205
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line at infinity

I tried solving the following question, could you have a look at my answer and tell me whether it's right or wrong? All input is appreciated. Question: Let $ABCD$ be the vertexs of a parallelogram in the affine plane immersed in its projective…
notacat
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Projective Geometry Question.

I want to learn projective gemetry. I have the coxeter book and found some youtube videos. I have just started. I have tried answering this Challege question from the video, and so far. I have made many lines on paper, but haven't gotten…
yiyi
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What are projection transformations?

As much as I know a projection transformation is the mathematical conversion of a map from one projected coordinate system to another, generally used to integrate maps from two or more projected coordinate systems into a GIS. But I lack the…
F. A. Mala
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Why do projective bases need $n+2$ points?

I am a little bit confused about why we define a basis of an $n$-dimensional projective space $P(V)$ to be a set of $n+2$ points in general position, rather than a set of $n+1$ points in general position. For vector spaces a basis is supposed to…
Cubi73
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Parallel and perpendicular line through a point

I inherited some code that deals with what I now know are homogeneous coordinates in projective geometry (probably not exactly the right terms, but hopefully close enough that you know what I mean). It takes as input points in 2D space, converts…
Eric Simonton
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Projectivity on a line and harmonic groups

On a projective line, we have pairwise distinct points $A,B,U,P,O,A',B',Q$ such that the following groups are harmonic: $ABUO$; $APUA'$; $BPUB'$; $A'B'PQ$. Let consider the following…
Fabio Lucchini
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Find coordinate infinity points from unity point using synthetic geometric constructions

A common way to put coordinates on $\mathbb P^k\mathbb R$ is to choose $k+2$ points (such that no one of them lies on the hyperplane generated by any $k$ of the others) and interpret them as the origin $(0, 0, \dots, 0, 1)$, the unit versors $(1, 0,…
Giovanni Mascellani
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Basic Projective Geometry Question

Can someone help me to see why any two points in $\mathbb P^1$ are linearly equivalent as divisors? If this is true, how come two points on a smooth projective cubic curve are not linearly equivalent?
Connor James
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