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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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Conic representation in projective plane P2

Suppose we have Conic described by 2nd-degree equation in the plane : y² + 4x² = 4 Find the Conic representation in projective plane P2
Ivone Oswandi
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The Pappus Line

The text which is unclear to me: Let the range of points $(A_i)$ be projectively related to the range of points $(B_i)$ under the correspondence $A_i \leftrightarrow B_i$. Fix $i$: then the pencil of lines through $A_i$ is projectively related to…
O K
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Restricting a form in a projective variety

If I have a symmetrical bilinear form and a projective variety. How do I restrict that form to my variety? Let's say I'm studying a hyper-quadric. I get its singular, and I want to classify it. It occurred to me, as I obtained two planes, to cut…
d010xonu
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Plane projective transformations that preserve a unit circle at the origin

From the system $H^{-T}CH^{-1} = kC$ where $C = \begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{pmatrix}$, I can obtain 6 constraints on the 10 unknown elements, 9 from $H$ and one $k$. Then I can deduce that this system has (10 - 6) four…
ImmenselyHappy
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Determine 4th vertex of a square in a perspective projection given 3 other vertices

I have a picture of three vertices of a square of edge of length 1, that has one vertex covered. Is it possible to recover the 2d coordinates (in the picture) of the 4th vertex? Or is it not enough information? For now I came up with a system of…
Golob
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Proving that the angle between the main diagonal of a cube and a skew diagonal of a face of the cube is 90 degrees

I need to prove that the angle between the main diagonal of a cube and a skew diagonal of the face of the cube is 90 degrees. I can do this with vectors, but I have to use applications from projective geometry to prove this. Note that the skew…
Roger
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Continuity of central projections

I was wondering if central projections between planes are continuous functions. By central projection I mean the following: let $\Gamma$ and $\Omega$ be planes in $\mathbb{R}^3$ and let $p$ be a point of neither. A central projection $f$ maps a…
NamelessCuriosity
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Parallel Projection Of An Ellipse

I am trying to figure out how a formula I am looking at was derived. Given a 2-D function $f(x,y)$ that has a constant value of $\rho$ within an ellipse given by $$ \frac {x^2} {A^2} + \frac {y^2} {B^2} = 1,$$ and is zero outside of this ellipse,…
user9651
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How to interpret triangle centers in projective geometry?

I'm talking about the real projective plane. Let's say we have a triangle, and we do a projective transformation, taking the vertices of the triangle to $(1,0,0), (0,1,0)$ and $(0,0,1)$, does this transformation take centroid of our original…
user572457
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Projective geometry - How to compute a Plücker matrix from a camera center, direction vector and rotation matrix?

for a homework assignment I am supposed to compute the intersection between two lines in 3D. One of them is defined by the camera center, a direction vector and rotation matrix. Unfortunately, I don't understand how I can get the Plücker matrix line…
chibi03
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Find the equation of line d such that H(a,b;c,d)

In the Extended Projective Plane, let a be the line with equation x=0, b the line with equation y=x-4, and c with equation y=-x+4. Find the equation of the line d such that H(a,b;c,d). I understand the process to determine this line d. I would…
lj_growl
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Proving that X,Y,Z in Quadrangle is collinear using axiom P5

Let P be a projective plane satisfying P5 and P6. Let ABCD be a complete quadrangle with diagonal points P,Q, and R. So, P=AB (int)CD, Q=AC(int)BD, and R=AD(int)BC. Let X,Y,Z be the points given by H(A,B;P,X), H(A,C;Q,Y), and H(B,C;R,Z). Use P5 to…
lj_growl
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Fixing the cross ration between four points places the fourth point in a strange position.

I am using the formula for the cross ratio found here: http://en.wikipedia.org/wiki/Cross-ratio#Definition Let us focus our attention on the cross ratio of points which are in an arithmetic sequence. For example the real-line points…
wircho
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Unique line in $\mathbb{P}^4$ intersecting three pairwise non-intersecting lines not in a hyperplane.

I need to show that there is a unique line (in $\mathbb{P}^4$ I assume, or could they also mean in $\mathbb{R}^5$?) that intersects three lines $L,M,N$ which are pairwise non-intersecting and not in the same hyperplane. In a previous exercise I…
The Coding Wombat
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Given the projectivity $\textbf x'=H\textbf x$, why is $\textbf x' \times H\textbf x = 0$?

I'm referring to section 4.1 in Multiple View Geometry by Hartley, where the Direct Linear Transformation (DLT) algorithm is explained. I have the intuition that since the points $\textbf x_i'$ and $\textbf x_i$ are correspondences of two different…
Makondo
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