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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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Deducing that a Mobius tranformation fixes three points

A projective tranformation on $\mathbb{P}^{1 }$ is a map $P(f): P(V) \to P(V)$ induced by a bijective linear map $f: V-\{0\} \to V-\{0\}$, where $V$ is a two dimensional vector space. Any linear map $f: V \to V$ is entirely described by where it…
user7090
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Number of double points of a projectivity

Let $K$ be a field and let $V$ be a vector space on $K$, with $\dim V = 3$. Let $r$ be a projective line in $P(V)$ and let $\{A , B ; C\}$ be a reference system in $r$, and we'll consider also $\{A' , B' ; C'\}$, being $A' = \sigma(A)$, $B' =…
joseabp91
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Projective characterization of isometries

An homography on a projective plane is a similarity if and only if it commutes with the absolute involution on the line at infinity. There is a similar characterization for isometries?
Fabio Lucchini
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Understanding the notion of lines in general position

A set of lines is said to be in general position if no $3$ of them meet in the same point. What is the reason of this definition? In $\mathbb{P}^2$, i can understand that two generic lines meet in a single point. But, for example, in $\mathbb{P}^n$…
un umile appassionato
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Shadow of an edge of a surface of revolution

A surface of revolution meridian $ r= f(z) $ has laterally incident sunshine. What should be $f,$ if the shadow cast by edge $ z= a $ on its inner surface has a terminator always remaining in a plane? EDIT1: I had verified paraboloid as a particular…
Narasimham
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Cross Ratios of an irregular octahedron

An irregular octahedron projects on a plane and associated with each of the 6 vertices is a Cross Ratio. Are these related? Does an invariance relation exist among the six by arbitrary rotations of the octahedron?
Narasimham
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Is the projective plane of $F^2$ isomorphic to the set of $1$-dimensional subspaces of $F^3$?

This is a little question inspired from Hartshorne's Geometry, which I've been juggling around for a while. Suppose that $\Pi$ is the Cartesian plane $F^2$ for some field $F$, with the set of ordered pairs of elements of $F$ being the points and…
yunone
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How are the following two projective maps equivalent?

Any element of $PGL(2,\Bbb{C})$ determines a map $$\Bbb{P}^1\to\Bbb{P}^1$$ $$[z:w]\to [az+bw:cz+dw]$$ in other words, a Mobius map $$[z:1]\to [\frac{az+b}{cz+d}:1]$$ How are the two maps equivalent? How can we just assume that $w=1$ and that…
user67803
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Projective Geometry : preservation of alignment

Show that any three points on a line can be sent to any other three points on a line by projection. Logically this makes sense. However, when trying to show that this works I am not sure where to begin. I thought about trying to use cross-ratio's…
Jacob Rodgers
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Are there any mathematical equations or a math procedure by which you can define a perspective drawing?

I have seen so many mathematical procedures including matrix. But all don't seem to plot a perspective view. Why is it hard to establish simple vector equations by which you can input the true x,y,z coordinates of any point in the object and output…
Dhirgham Murran
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Projection of an oblique circle on XZ plane

While going through an exercise of surface integration, I got confused in this problem.The surface is the intersection of sphere $S:x^2+y^2+z^2-1=0$ and the plane $P:y-x=0$. Clearly, the curve of intersection is the circle $S=0,P=0$ whose…
Nitin Uniyal
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Method for Visualizing Projective Space

I've been trying to understand projective space as follows: Consider the plane at z=1 as the 2D affine plane, and for any curve in this affine plane, let the inclusion in projective space be the set of all lines formed by a point on the curve and…
kfriend
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coordinates depending on perspection

The diagram below shows a perspective drawing of two squares, with coordinates given-on the drawing-for some of the corners of the squares(the line a the top is the horizon) The diagram below shows the same two squares, now seen from above, with…
Elizabeth Hill
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Do four projected lines from a square?

I've detected a few lines in an image. Four of them are the edges of a square depicted in the image. The projective transformation of the camera has of course been applied to all the lines, making it impossible to use regular metrics such as…
vidstige
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lines through a point of the projective plane

I'm having difficulty understanding a particular example of Mumford's "Red Book". In exemple D, first chapter, he considers the set of lines passing through a point of $\mathbb{P}_2$ (do we call it pencil?), and says that we identify it with…
Andrei.B
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