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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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Prove there's a unique projective transformation that maps four points to four points

As the title states, the question is to prove that there's a unique Projective Transformation that maps four points of $\mathbb{R^2}$ to the projective plane. I tried defining the projective transformation as $\bf{x'} $$= M\cdot $$\bf x$ where $M$…
Alex Matt
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Coordinates for line in projective geometry with given points

The points $P = (1: -2: 3), Q = (2: 2: -1)$ and $R = (3: 0: 2)$ lie on one Line g in $\mathbb{P}^{2}\mathbb{R}$. Choose a coordinate for g such that ${(P, Q, R)}$ has coordinates in ${(0, 1, ∞)}$. How many possibilities are there?
Semi
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Show point is the harmonic conjugate of intersections of tangents on non-degenerate conic section

I'm trying to prove this statement: Let $\mathcal{K}$ be a non-degenerate conic section in $\mathbb{R}P^2$, with on it three distinct points $A,B$ and $C$. Let $a$ be the tangent line to $\mathcal{K}$ in $A$, $b$ the tangent in $B$ and $c$ the…
Jarne Renders
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A demonstration with Cross Ratio

In the Extended Euclidean Plane, H, let A=(0,0), B=(5,0), and C=(4,0). Show that there exists a point D=(x,0) for some real number x, such that Rx(A,B;C,D)=pi. I have the formulas for these things. However, I am having trouble applying them…
lj_growl
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Dimension in $\mathbb{P}^4$ of $\langle L,M \rangle \cap N$ with $L,M,N$ pairwise non-intersecting and not in one hyperplane

Given three lines, $L, M, N \in\mathbb{P}^4$, not in one hyperplane and not pairwise intersecting, I need to calculate $$\dim(\langle L,M\rangle\cap N).$$ By the dimension of intersection theorem for projective spaces we have $$\dim(\langle L,M…
The Coding Wombat
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What would be a geometric Euclidean interpretation of the homogeneous plane (0,0,0,8)?

Homogeneous coordinates have one dimension more than the corresponding Euclidean coordinates. The Euclidean origin can be described with projective coordinates as (0,0,0,1). So, geometrically, what would be an interpretation of the homogeneous…
Luk
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Projective Geometry: Prove that the mapping in P2R is not well-defined.

Prove that the mapping F: P2(R) to P2(R) given by F(x1,x2,x3) = (x1x2, x2, x3) is not well-defined. I know that to determine whether a mapping is well-defined, you should pick two points that are the same in P2(R) and show that the mapping…
lj_growl
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What is a projective range?

Wikipedia says a projective range is the dual of a pencil of lines on a point. Surely that means: all points on a line. But Wikipedia also says a projective range may be a projective line or a conic. This definition and this property seem to be at…
pdmclean
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How to prove that in PG(2,R), if a projectivity takes 3 points to themselves, then the projectivity takes all points to themselves?

If I have $(P,Q,R,X,Y,Z,...)\barwedge(P,Q,R,X',Y',Z',...)$ and all points lie on line $l$, How do I go about proving $(X,Y,Z,...)=(X',Y',Z',...)$? By a theorem, there exists a unique projectivity taking P,Q,R to itself. If I define it by $f$, such…
Neo
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How to visualize the pole/polar of a line/point in projective space PG(2, R) not in contact with the curvature?

So, I understand that given a curvature $\mathbb{C}$. The polar of a point on the conic is tangent to the conic at that point and the pole of a line touching the conic only once and tangent to the conic is the point where it touches. I'm failing to…
Neo
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Two triangles are inscribed in conic iff their edges are tangent to another conic.

In the list of recomended problems for my course "Projective Algebraic Geometry" I found following problem: Prove that two triangles are inscribed in some conic $C_1$ $\iff$ their edges are tangents of another conic $C_2$ . All geometry is over…
Gleb Chili
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Implicit formula of the projective completion of an affine plane

Let $E: (u,v) \mapsto (3,1,1)+u(1,0,2)+v(1,0,1)$ an affine plane. How do I get its completion in $\mathbb{P}_{3}(\mathbb{R})$. I started with projective geometry and have my difficulties with understanding. I know that $\mathbb{P}_{3}(\mathbb{R}) =…
Erdbeer0815
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Geometric construction of tangents to a conic

I'm stuck trying to solve the following exercise. Let C be a conic in $RP^2$ and P a point on C. Give a geometric construction for the tangnt line to C at P only using a straight edge. (That is, one can draw the line through any two given points,…
Polymorph
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Projective Transformation mapping lines

I am asked to find a projective transformation which maps the lines $L_1, L_2$ to $L^{'}_1, L^{'}_2$ respectively, where $$L_1 : X - 2Y = 0 \:, \quad L_2 : Y+Z=0 \:, \quad L^{'}_1 : 2X+Y+Z=0 \:, \quad L^{'}_2 : X+3Y =0$$ I've been trying to think of…
Killuminati
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How do you transform from affine space to projective space?

I have already seen this question but I still have enormous doubts: Difference between Projective Geometry and Affine Geometry I'm reading Multiple View Geometry by Hartley and Zisserman, chapter 1. First they say this: We can get around this by…
Foobarer
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