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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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Calculate area with cross ratio

Suppose that I am given a drawing of a table, on which a book lies in one of the corners. The measures of the book are known, how can I find the measures of the table using cross ratios?
harajm
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Construct sum of points on projective line

In a course in Geometry we where asked to geometrically construct the sum of two points $x$ and $y$ on a projective line by help of "the theorem on complete quadrilaterals". This theorem (as stated in our lecture notes) says that if $A,B,C,D$ are…
harajm
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Projection is a projective map

Let $l,m$ be two distinct lines in the projective plane and let $P$ be a point that is not on either of the lines. Prove that the projection $P$ of $l$ onto $m$ is a projective map. My idea was to find a linear injective map in the underlying…
user178468
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is it possible to project any triangle on a plane as a right triangle on another plane?

I scratching my head over this problem from my projective geometry book (C. R. Wylie, Jr). Given a triangle in the plane $z = 0$, is it possible to find a viewing point, $C$, from which the triangle will appear on the plane $y = 0$ as a right…
Mark
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In projective geometry the dual of the cross ratio dual is an angle measurement?

I am trying to get my head around angles in projective geometry. I understand (more or less) the cross ratio and that it can be seen as an distance measurement. (for example in the Beltrami Cayley Klein model of hyperbolic geometry) But then there…
Willemien
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Construction of Projective Plane Up to Order 5

How does one construct projective planes of order 5? What are the references of projective geometry that describe the construction of projective planes of order at least up to 5?
Hans
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A question about lines in the projective space.

Let $ax+by+cz=0$ be a line in projective space. Let the line be satisfied by two points $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$. Then we have $$a_1x+a_2y+a_3z=0$$ $$b_1x+b_2y+b_3z=0$$ This implies that $\begin{vmatrix}…
algebraically_speaking
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A question about three collinear points

This video (at 44:00) says that in a projective space if three points are collinear, and two of those points lie at infinity, then the third point will also have to lie at infinity. I wonder why that has to be true. Why can I not have three…
user123
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Chirality of a Möbius band without boundary?

In this answer it is remarked that the real projective plane minus one point is homeomorphic to the Möbius strip without boundary. A normal Möbius strip is topologically equivalent to a real projective plane with the whole inside of a conic section…
Gerard
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If I have some function F that fixes two points in a line

How do I show that this function is the identity function? We are currently studying collineations and I do not know how to know the other points within the line are not permuting(or moving around). Basically, how can I show they are fixed…
cakey
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What does the closure of a set mean?

In my book for projective geometry, this symbol: < x > means a subspace containing points x. But my teacher calls it "the closure of x". Does this mean the same thing. He also described "closure operations". Is the closure just talking about the…
cakeyone
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:= the intersection of subspace U such that U is a subspace of L containing M

L is a linear space and set M is the set of points of L. The definition I put above is "the smallest subspace of L generated by M". The thing I don't understand in this definition is why do we need the brackets around M; how come we can't just say M…
cakeyone
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cross ratio - is it ordering invariant

I want to know if the cross Ratio depends upon the ordering of the points around a particular point . I am calculating the cross ratio as : CR = A(1,2,3)*A(1,4,5)/(A(1,2,4)*A(1,3,5) where A is the area of the triangle formed using the given…
Deepesh
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Desargues Configuration

Section 3.2 of Coxeter's Projective Geometery discusses the self-dual $10_3$ configuration, ten points and ten lines, with three pointson each line and three lines through each point. I am looking at the Figure 2.3a, but I am having trouble trying…
yiyi
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Incidence geometry: Prove that there is only 1 unique configuration 7_3_

$7_3$ also know as Fano plane. How can I prove that only 1 configuration exists for 7 points and 7 lines with 3 points on every line and 3 lines at every point. I would think an incidence matrix would help, but I'm not sure how to begin. I know a…
cakeyone
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