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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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In the affine plane, I am having trouble with these definitions

If the number of points in an affine plane is finite, then if one line of the plane contains $n$ points then: all lines contain $n$ points, every point is contained in $n + 1$ lines, there are $n^2$ points in all, and there are a total of $n^2 + n$…
cakey
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In projective geometry and curved lines

For example, Fano's Geometry exemplifies a triangle with a circle within it. How is this possible? Are lines not defined to be straight? Is this geometry projective geometry(or is projective geometry a different set of axioms)?
cakey
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In projective geometry, what do we need to show an isomorphism(or that two geometries are the same)

We have X(set), I(type), and *(incidence relation). I believe my teacher said we need to only check that we have a bijection in terms of X(elements) and I(type), but what about the incidence relation?
cakey
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Actions of Diff(S^1) and Vect(S^1) on the space of projective structures on S^1

I'm reading the book "Projective differential geometry: Old and new" and encounter this problem. Given a projective structure on $S^1$ (or $\mathbb{R}$), we have a developing map $\phi: \mathbb{R} \to \mathbb{RP}^1$ (as in that of a geometric…
Quang-Nhat Le
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Projective transformation that fixes a circle and sends an interior point to the center

The question is related to this question: In this handout theorem 2.3(Homographies that exist) item two claims For some circle ω and interior points P, Q, we can send ω to itself and send P to Q. (Note that Q is usually taken to be the center of…
hbghlyj
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How to computer camera FOV given corresponding points in world space and view space

The context might be an old photograph of (say) soldiers in a trench or a picnic in a forest, but without a horizon or any vanishing points. My conjecture is that 4 known points in world space mapping to 4 points in projected image space are enough,…
david.pfx
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Confused about center of projection in projective geometry

I'm new to projective geometry. As of my understanding, when we want to map 3d scene to a plane, we can use homogeneous coordinates to imitate the center of projection as eyes and a projective plane that is 1 unit far away from the eyes. Therefore a…
footycrazy
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Questions about configurations in projective geometry?

I have several questions relating the definition and property of configuration in geometry: What are the intuitions and purposes behind the fact that each point is incident to the same number of lines and each line is incident to the same number of…
duck
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Show collinear points in 3d project to collinear points in 2d

Let $A,B,C$ be collinear points in 3d. Show that their projections $a,b,c$ respectively, onto 2d image plane are also collinear. If $A=[x_1,y_1,z_1], B=[x_2,y_2,z_2], C=[x_3,y_3,z_3]$ then $a=[x_1/z_1, y_1/z_1,1], b=[x_2/z_2, y_2/z_2,1], c=[x_3/z_3,…
jroc
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Determine and classify the projective conics

Determine and classify the projective conic that contain the points: $\langle 0,0,1\rangle,\langle 0,1,1\rangle,\langle 1,0,1\rangle,\langle 1,1,1\rangle,\langle 1 / 2,2,1 \rangle$. I used this projective conic equation but then I can't reach any…
Teresa Rosa
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Perspective sequence that maps collinear points A,B,C,D to D,C,B,A

Find the perspective sequence that maps collinear points A,B,C,D to D,C,B,A. Attempt: If we need to find a sequence of three perspectives that (A,B,C)->(A,C,B), where A, B, C are collinear, then first we mark p is a line through A, B, C, let S be…
GENERAL123
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Perspective warp evaluation as a convex combination of 4 sample points

Suppose we know that $F:\mathbb{R}^2 \to \mathbb{R}^2$ is a perspective warp function; that is, there exist real parameters $a,b,c,d,e,f,g,h,i$ such that, for all $(x,y)$: $$ F((x,y)) = ((ax + by + c), (dx + ey + f)) / (gx + hy + i). $$ And…
Don Hatch
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Basic Projective Plane Question

So the axioms for a projective plane are given by: Any two “points” are contained in a unique “line.” Any two “lines” contain a unique “point.” There exist four “points”, no three of which are in a “line.” Consider now this question: Suppose that…
Vasting
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When a point belongs to a line in the real projective plane

I was reading this blog about how to compute intersections in the real projective plane and I'm struggling with this sentence: Notice that the POINT (a, b, c) is contained in the LINE [x, y, z] exactly if ax + by + cz = 0 Can you explain this a…
riccardoventrella
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Showing eye point is along same line as the vanishing point of the rails and perpendicular to the image plane

Context-1 Consider a picture of rails being taken from a camera(and our eyes is at position as shown in the figure ) and consider the picture formed in image plane to be as shown in figure. Questions: I know that image plane will be…
ProblemDestroyer
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