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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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The sum of two projective spaces

Let $K$ be a field and let $V$ be a vector space on $K$. We consider the projective space $P(V)$ on $V$ (the set of the vector subespaces $A$ in $V$ such that $\dim A = 1$). Let $S , T$ be two vector subespaces in $V$. By definition $P(S)$ and…
joseabp91
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Conic in the projective plane

An imaginary conic belongs to the projective plane? Because the projective plane has only elements of $\mathbb{R^3}$ and then I can not understand that classification of conics. There are imaginary elipes as parallel imaginary lines (degenerate…
user411479
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constant Cross Ratio of pyramidoid edge projections

EDIT1: Can we define a (non-trivial) rigid pyramidoid in 3 space where every projection of a set of four edges $ (OA,OB,OC,OD)$ meeting at a vertex $O$ have the same constant Cross-Ratio by arbitrary rotations? Vertices $(A,B,C,D)$ and base sides…
Narasimham
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In the real projective plane, what is the equation of the line passing through a point and the point at infinity

Let P=(a,b) in the real projective plane, then what is the equation of the line passing through P and the point at infinity?
DpS
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In the projective plane P2(K), where K is field, is the line passing through two distinct points not unique?

So, we form the coefficient matrix, which is a and the rank of this matrix is two which leaves one free variable, which means the coefficients of the line are not unique. But this is in direct contradiction with the axioms laid out for projective…
DpS
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Show that every line in the projective completion contains at least 3 points.

We know that P is a model of incidence geometry that satisfies the Euclidean parallel property and that R is its projective completion I'm trying to show that every line in R contains at least 3 points of R. I'm thinking there are two cases for when…
James Banner
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Drawing the reciprocal of a circle through the circle of inversion.

I have a general question about drawing the reciprocals of circles through the circle of reciprocation. I understand inversion and reciprocation are two entirely different things yet somehow connected. When reciprocating a circle(alpha) outside of…
user338365
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Point Selection for Cross Ratio

The cross ratio relates the positions of four co-linear points in 3d space. I understand definitions a published online, such as that from wikipedia: https://en.wikipedia.org/wiki/Cross-ratio However, I'm am unsure what the general rule is for…
Austin
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Why doesn't an isosceles triangle belong to the projective plane?

I am reading Coxeter's projective geometry book and he says that the isosceles triangle doesn't belong to the projective plane. I see how in projective geometry we solely use the compass as opposed to in Euclidean geometry we use both the compass…
user19405892
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Half-space membership in projective geometry?

Given a plane normal $n=(n_x,n_y,n_z)$ and a point on that plane $p_0\in\mathbb{R}^3$, testing whether another point $p\in\mathbb{R}^3$ is "above" the plane is easy: $n\cdot(p-p_0) > 0$ Is there an equivalent in projective geometry?
user1071136
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Preserving shapes in perspective projection

Perspective projection is very simple to perform, but when I tried to prove that certain geometric elements preserve their identity when projected I faced many difficulties though intuitively it seems obvious. I managed to prove that a 3D line is…
Controller
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How is it that dot product does projection but so does division?

Can anyone explain how it is that dot products can perform projection but so can division, and how the processes are related (or not)? With dot product, you can project one vector onto another vector: $\hat{v} = \hat{a} \cdot \hat{b}$ Where $b$ is…
Alan Wolfe
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Intersection of lines in projective space

I'm given the lines tu+sv and rw+kz where t,s,r,k are constants and u=(-5,0,1/4,0) v=(0,1,1/3,0), w=(4/3,-1,1/2,0), z=(7,1/2,-1/3,0). How can I find their point of intersection? Thanks for all the help.
Annnnnet
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Show that the real projective line, P1, is orientable

I'm asked to show that the real projective line, P1, is orientable. I'm not quite sure how to define orientable to prove this. Thanks.
Annnnnet
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Why is the cross ratio of $(u,v,\lambda, \omega(\lambda))$ a constant?

In the book by Beltrametti, he states that if $\omega$ is a projectivity of a line $r$, it has two fixed points, $u$ and $v$. If $u$ and $v$ are distinct, then the cross ratio $(u,v,\lambda, \omega(\lambda))$ does not depend on $\lambda$. I tried…
KittyL
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