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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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Perspective projection plane, calculating squares on the plane

Let's say I have a road I'm looking at from the top, have a square on it. Then I have a different location from which I look at the road, the square now is a convex quadrilateral. https://i.stack.imgur.com/KRljZ.png I have a square shaped texture I…
Krend
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Project point onto surface using yaw, pitch, roll rotation and standard trig formulas

Given the known coordinates of location A (0,0,50), at which we have a laser pointer aimed downward, I need a series of equations that calculate the projected point onto the flat surface below (x,y,0) using and in the order yaw, pitch, roll.…
user3324138
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Motivation for 2D homogeneous coordinates: ratio always determinate

I'm working through an old textbook called Algebraic Projective Geometry, by Semple and Kneebone. Early in the text, the authors write: When we introduced complex points (on p. 12) we explained that they are to be regarded as ideal points adjoined…
mannyglover
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Verify construction of cube using vanishing point perspective

Suppose we are a creative individual, and during our math exam would like to draw a picture of a cube using the vanishing point perspective. Let $A$ and $B$ be two adjacent vertices of a square in the plane, and $V$ the vanishing point. We extend…
user519413
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Intersection of ray with plane given in homogeneous coordinates

Given a plane $p\in\mathbb{P}^3$ expressed in homogeneous coordinates ($p_x$, $p_y$, $p_z$, $p_w$) a ray expressed as a source point $s\in\mathbb{P^3}$ expressed in homogeneous coordinates ($s_x$, $s_y$, $s_z$, $s_w$) a direction $d\in S^2$…
Museful
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Solution verification for finding parametric equation of a line

I have the following problem in $E_3^*$ I have to find a parametric equation of a line passing through the point $M(3,1,2,1)$ and the infinite point of of a line $l$ formed from two planes $\alpha$ and $\beta$. $$ \alpha: x+y=0\\\beta:…
Nikola
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Finding the distance of points after projective transformation

This is more of a follow-up question to the previous: Recreating a line after projective transformation Referring to the below diagram (adapted from the wiki page on vanishing points): Say if I have a line $L$ that: Lies exactly on the plane…
John Tan
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Recreating a line after projective transformation

Say if I have 2 lines $L_1, L_2$ in 3D space, which are parallel to each other, and the perpendicular distance between any 2 points are known. Imagine if I take a photograph of the 2 lines. $L_1, L_2$ will be projected onto the image, resulting in…
John Tan
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Obtaining hyperbola in projective geometry (Courant)

Reading "Courant's – What is Mathematics", where a definition has just been introduced stating "[a] conic is the locus of intersection of corresponding lines in two projectively related pencils". A little later "If the pencil O and O' are congruent…
Max
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Polarity of lines in $\mathbb{RP}^3$

I am a bit struggling with the following exercise. Let Q be a non-degenerate quadric in $\mathbb{RP}^3$, $l \subset \mathbb{RP}^3$ a line and l' $\subset \mathbb{RP}^3$ its polar. Show that all polar planes of points on l pass throught l'. I…
Polymorph
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How can I get square in projective geometry?

Suppose I have two points which are vanishing points. I want to draw a square using vanishing points. I can easily draw a rectangle, but I couldn't find how to find two orthogonal line with same length. Two vanishing points have enough information…
nugad
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How to find perspectivity of the ranges on a line

Question: Complete the perspectivity of the ranges on the lines $AB$ and $DF$ with centre $H$: $ABF$ $\barwedge$ ? I got $F$ as my first point but is lost after that. Need help. Also I was told to label my own lines on this picture
behold
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Pair of elliptic involution on a projective line

In a projective plane $\alpha$ let $u$ and $v$ be two lines and $\Theta$, $\Psi$ be elliptic involution on $u$, respectively, $v$. Prove that there exists two pencil of rays on which $\Theta$ and $\Psi$ induces the same involution.
Fabio Lucchini
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common point at infinity of two parallel planes

How do I find the common points at infinity of two parallel planes: $\alpha: x-y+2z-t=0$ $\beta: 3x-3y+6z-7t=0$ My understanding is that the planes are not only going to have common points, but they are going to have infinitely many common points…
Nikola
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Maps between projective spaces

Let $K$ be a field, let $V$ and $V'$ be two vector spaces on $K$ and let $f : V \to V'$ an isomorphism. If I consider the projective spaces $P(V)$ and $P(V')$, the map ${\sigma}_f : P(V) \to P(V')$, given by ${\sigma}_f(\langle u \rangle) = \langle…
joseabp91
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