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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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3D curve reconstruction from (mostly) uncorrelated points on multiple 2D projections

We use Neural Network tracking to attempt to track a complex and chaotically changing edge among 6 camera perspectives. This chaos is what we want to capture, but we run into a few issues. The tracking points stay on the line very well, but tend to…
9sven6
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Project picture frame on a wall with known height but unknow length

I'm trying to project a picture frame on a wall using using the css property matrix3D. The idea is: we draw the four corners of the wall on any room photo and the picture frame is projected on the wall with its proper dimensions. projection on wall…
Raphael
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Find a pixel within a transform matrix

I have been using this answer as a guide to find specific point within a shape. I am using a camera to read the area that is projected on a wall. This first drawing shows the boundary of the image as detected by the camera. Using the coordinates p1,…
fizch
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Understanding whether the image on a camera is a projective plane or not

I know that projective geometry began to develop from art at the start . So suppose if we take a picture of a rails of a station we would get a picture which (i think maybe wrong) is a projective transformation of a $3D$ structure into a $2D$ one…
ProblemDestroyer
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Hilbert's definition of plane configurations

We define a plane configuration as a system of $p$ points and $l$ straight lines arranged in a plane in such a way that every point of the system is incident with a fixed number $\lambda$ of straight lines of the system and every straight line of…
tryst with freedom
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How can I find a proportional length on a perpendicular line from a perspective?

I am working on a project where I am mapping images of buildings onto 3D meshes and I want the dimensions of the mesh to be proportional to the building in the image. I have six points for a given image that describe two sets of vanishing lines and…
Tommy
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Understanding projective plane conceptually (page-342, Road to Reality by Roger Penrose)

In the above picture, I am a bit confused how it turns out parallel lines seems to meet in the artist's potrait. Could someone explain in simple words why the roads which don't intersect in the ambient world do intersect in the painting? So far,…
tryst with freedom
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Confusion about an example in Miles Reid Undergraduate Algebraic Geometry

He is giving examples of lines at infinity and how they correspond to asymptotes (pg. 14). So he says: "The hyperbola $(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1)$ in $\mathbb{R}^2$ corresponds in $\mathbb{P}^2\mathbb{R}$ to…
Daniel
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Show that $\det (M) =0$ is the equation of the line joining two distinct points $[a_1:b_1:c_1], [a_2: b_2: c_2] \in \mathbb{P}^2$.

Show that the $\det(M) = \det \begin{bmatrix} x & y & z \\ a_1& b_1 & c_1 \\ a_2& b_2 & c_2 \\ \end{bmatrix}=0$ is a equation of the line joining two distinct points $[a_1:b_1:c_1], [a_2: b_2: c_2] \in \mathbb{P}^2$. I've just started studying…
john
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How to understand projective maps

I started learning projective geometry recently and i know just the basics how cross ratios are defined and some applications as given in Alexander Remorov's projective geometry note. But recently i came across a note which is titled "The method of…
green_blue
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Is every collineation of the real projective plane given by some linear transformations?

Suppose we have some collineation of the real projective plane $\alpha$. Is it possible that $\alpha$ does not use linear transformations? My thoughts are that it isn't possible. This is because we can just add lines and planes normally (points and…
Ook
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Monge's viewing method to prove Desargues' theorem

For instance, consider the triangle in space (see Figure 2.4). Assume that a triangle $A,B, C$ is projected to two different mutually perpendicular projection planes. The vertices of the triangle are mapped to points $A',B',C'$ and $A'',B'',C''$ in…
tryst with freedom
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What are the affine points induced by a non-standard choice of a given line at infinity in $RP^2$?

We are working in the projective space $RP^2$. For a general element $x \in RP^2$ we use the notation $x := [(x_0, x_1, x_2)].$ In $RP^2$ we typically choose the line $x_2 = 0$ to be the 'line at infinity'. Then all points in $RP^2$, which are not…
LotrFanAndMath
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Why do different parallel lines when projected from a point onto a plane lose their parallism?

How can one prove that when two lines are parallel in space ($\mathbb R^3$) and they are projected from a point onto a plane (not parallel to the lines), they are no longer parallel? It seems something easy to visualize but I have not been able to…
Lucas Cruz
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Projectivity that maps $r$ to $r'$, $s$ to $s'$, $P$ to $P'$

Find the projectivity $f$ in $\Bbb P^2(\Bbb R)$ that maps: · Line $r: X_0=X_1$ to line $r': X_0+X_1=0$ · Line $s: X_0+X_1+X_2=0$ to line $s': X_1+X_2=0$ · Point $P[1:2:1]$ to point $P'[1:0:0]$ My thoughts: To define a projectivity I need two sets of…
pink frog
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