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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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Desargues Theorem of two triangles in perspective has symmetric order 120(why?)

10 points and 10 lines construct Desargues' theorem, but since the order of the symmetric group is 120 this means we are permuting 5 elements. But I am confused to what these elements are. In total there are 6 points on both triangles in the…
cakey
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Is the projective special unitary group a lie group?

Is the projective special unitary group a lie group? If so then what is its Lie algebra and where can I find out more about these objects?
Benjamin
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Euler's line : a projective geometry proof

In a given triangle $ABC$, let $G$ be the common point to the three medians, $H$ be the common point to the three altitudes, and $M$ be the common point to the perpendicular bissectors of the three sides. These three points lie on a line. It is…
DC75
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Projective transformation that fixes the unit circle and sends a point to the origin

I'd like to find a projective transformation that fixes the unit circle and sends some point on the $x$-axis within the unit circle to the origin (or I guess a random point in the unit circle, however, as I know that I can rotate, I thought this…
Sha Vuklia
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Does the axiom imply its dual?

Suppose we have the following axiom for the projective plane: Axiom: If a projectivity leaves invariant each of three distinct points on a line, it leaves invariant every point of the line. The dual of this axiom is the following statement: Dual:…
QED
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Prove Pascal's theorem by homogeneous coordinates

I was trying to prove Pascal's theorem by using homogeneous coordinates with the following configurations (interactive graph at Desmos): A,B,C,D,E,F (homogeneous coordinates) are on a conic. G,H,I are the intersection points of pairs of lines…
lochiwei
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Equation of a ray passing through the camera center and an image point

If C is the camera center in homogeneous world coordinates and x is a point on the image plane, the ray passing through C and x is said to be: $$ X(\lambda) = P^{+}x + \lambda C $$ where P is the camera matrix that maps world points to the image…
jc211
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projective plane in projective space

I read in an article that the collection of lines going through a point in RP3 form a projective plane. I can't understand why . we know that a point in RP3 is a line going through the origin of the vector space R4 and a line going through this…
david
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Homogeneous coordinate representation of a vertical line

Is there homogenous coordinate representation for a vertical line passing through an arbitrary point on the x axis (say C). Generally this is represented as: $x = C$ in euclidean geometry would it be $(-1/C, 0, 1)$ in homogenous coordinates (P2…
InKodeWeTrust
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Relationship between Euclidean lines and projective points

I'm learning the very basics of projective geometry, and I read the following in Stan Birchfield's notes, An Introduction to Projective Geometry: Why is it that "overall scaling is unimportant"? In his notes, Birchfield uses this to build the idea…
on-pasta
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The dimension of the space of lines in projective space.

For example in three dimensional projective space. I thought lines in projective space corresponded to planes in euclidian space. So the space is just the dimension of the Grassmanian $G(2,3)$. Which is $3$. But in a book I'm reading it is said that…
faridrb
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How can I learn more about projective geometry

I am here for a simple question for which I have done some search on this site and google. Considering I was successful on my search, there is not a coherent explanation about these. I would like to learn this geometry in a very detailed way. Could…
ozumsafa
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A projective geometry problem about a set of lines envelope a curve of class 2

Let $P \in RP^2$ fixed. Let C be a conic. Consider all lines $L_{AB}$ such that there exist $A,B \in C$ such that $\measuredangle APB = \frac{\pi}{2}$. Prove that the set of lines $L_{AB}$ envelope a curve of class 2. I think Laguerre's Formula can…
roger98
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Why is the cross product of two points a line and the cross product of two lines a point?

I am currently studying projective geometry. I have trouble understanding the point-line duality concept. Why is the cross product of two points a line and the cross product of two lines a point?
Kevin Wu
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Intersection of a line and line at infinity in projective space

I understand parallel lines in Euclidean space intersect at the line at infinity in terms of projective space. My question is for a single line. A single line if extended to infinity must intersect the line at infinity at some point (correct me if…
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