Projective geometry is closely related to perspective geometry. These types of geometry originated with artists around the 14th century.

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Finding the Transform matrix from 4 projected points (with Javascript)

I'm working on a project using Chrome - JS and Webkit 3D CSS3 transform matrix. The final goal is to create a tool for artistic projects using projectors and animation - somewhat far away from using ...
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119 views

Cross-ratio relations

The way I define the cross-ratio in projectve geometry: Let $P_0,P_1,P_2,P_3$ being four points on a projective line G, such that $P_0,P_1,P_2$ are pairwise distinct. Let $\pi:\mathbb KP^1\rightarrow ...
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27 views

Projective Co-ordinate Geometry

I am learning projective geometry in my computer vision course. So, we represent a co-ordinate point in an image as a homogeneous co-ordinate as $(x,y,1)$. My professor says that if we are given two ...
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Plücker Relations

Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Plücker relations are quadratic forms on $\wedge^d V$ whose zero set is exactly the set of decomposable ...
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The intersection of two parallel lines

My teacher told me that two parallel lines have a point of intersection, it is called Point at infinity. But I I can't understand how it can be true or how was it proved, can someone explain this to ...
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Problem in Deducing Perspective Projection Matrix

I understand the traditional way(use similar triangle and make depth value linear) to deduce the perspective projection matrix. But I want to try another approach after I read this text: Fundamentals ...
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902 views

Augmented Reality Transformation Matrix Optimization

i am a software developer, i'm working on an Augmented Reality system. I'd like to receive some advice in order to optimize my math model. My program has to be slim and fast. Here's the situation: ...
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1answer
526 views

Rectify image from congruent planar shape objects

I am implementing an algorithm to remove projective distortions on the following image. I understand this is possible by applying the following transformation: $$ \begin{matrix} 1 ...
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145 views

Sample Code to Generate Points on the Rim of a Randomly Rotated Cone : What's Going On Here?

Related to this question: http://math.stackexchange.com/questions/407897/randomly-generate-point-on-shell-from-3-points-2-angles-with-uniform-angle-dis I'm trying to reverse engineer the ...
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5answers
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Help understanding Algebraic Geometry

I while ago I started reading Hartshorne's Algebraic Geometry and it almost immediately felt like I hit a brick wall. I have some experience with category theory and abstract algebra but not with ...
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406 views

topologies of spaces in escher games

There have been a couple of games released (or in development) in the past couple of years which do some weird topological tricks: Echochrome (video), Crush (video), and Fez (video). Do the spaces ...
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342 views

A modern textbook on affine and projective spaces

Requirements: Scalar fields other than $\mathbb{R}$ and $\mathbb{C}$. Precise. Visual explanations are good, but they must complement definitions and proofs, not replace them. No repetition of text. ...
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357 views

Learning projective geometry

My ultimate goal is to learn some algebraic geometry with the more concrete immediate goal of understanding things like how $\mathbb R\mathrm P^2$ is embedded in $\mathbb C\mathrm P^2$ or how $\mathbb ...
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169 views

Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
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0answers
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Projection and Pseudocontraction on Hilbert space

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
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Prerequisite of Projective Geometry for Algebraic Geometry

I studied Euclidean Geometry in high-school, and I have not studied anything relates to geometry since I started studying in university. I am now intending to study Algebraic Geometry, however, I ...
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321 views

Perspective problem - trapezium turned square

True or false: If you draw a trapezium on the ground, there always exists a point above (but not necessarily directly above) the trapezium such that the trapezium looks like a square from that point. ...
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388 views

Analytically flavored book in projective geometry

I am looking for a book in projective geometry, using the apparatus of linear algebra, complex analysis, and, perhaps, modern algebra, in full. The counterexample is the Hartshorne's book on ...
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Description of varieties in $\mathbb{P}^2\times \mathbb{P}^1$

If $[x:y]$ are coordinates of $\mathbb{P}^1$ and $[X:Y:Z]$ are coordinates of $\mathbb{P}^2$, what do the following varieties look like? $V(x^2X+y^2Y+xyZ)\subset \mathbb{P}^2\times \mathbb{P}^1$ ...
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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 ...
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1answer
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Equation for non-orthogonal projection of a point onto two vectors representing the isometric axis?

Suppose I have two vectors that are not orthogonal (let's say, an isometric grid) representing the new axis. Suppose I want to project a point onto these two vectors, how would I do it? Dot product ...
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Textbook for Projective Geometry

So while not actually a specific problem I'm struggling with, I was hoping for some of your insight! For a course, I'm currently reading Stillwell's Four Pillars of Geometry. While it does a nice ...
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1answer
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Distinguished open sets in $\mathbb{P}^n$

I have given a homogenous polynomial $F\in\mathbb{C}[x_0,\ldots,x_n]$ of degree $d>0$ and consider the open set $U_F=\{p\in\mathbb{P}^n; F(p)\neq 0\}$. I'd like to prove that $U_F$ is isomorphic to ...
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1answer
50 views

Finding the inverse of a map from $CP^1$ to $S^2$

Given the map: $$f:CP^1 \to S^2\ ,\ f[z:w] = \left(\frac{2\mbox{Re}(w\bar{z})}{|w|^2+|z|^2},\frac{2\mbox{Im}(w\bar{z})}{|w|^2+|z|^2}, \frac{|w|^2-|z|^2}{|w|^2+|z|^2}\right)$$ How would I go about ...
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The Process of Choosing Projective Axes to Put an Elliptic Curve into Weierstrass Normal Form

I'm reading the book "Rational Points on Elliptic Curves" and on page 23 the author takes an arbitrary (non-singular) elliptic curve in the projective plane and finds a rational point $O$, referring ...
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38 views

Show that three pairwise non-intersecting lines in $\mathbb{R}\mathbb{P}^{3}$ have a transversal.

Let $\mathbb{P}(U1)$ and $\mathbb{P}(U2)$ be two non-intersecting lines in the 3-dimensional projective space $\mathbb{R}\mathbb{P}^{3}$ = $\mathbb{P}(\mathbb{R}^{4})$. Show that $\mathbb{R}^{4}$ ...
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1answer
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Birational isomorphism $\mathbb{P}^n\times \mathbb{P}^m\to \mathbb{P}^{n+m}$

One can show that $\mathbb{P}^n\times \mathbb{P}^m$ is birational to $\mathbb{P}^{n+m}$ by making note of the Zariski topologies and the canonical isomorphism between affine spaces $\mathbb{A}^n\times ...
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1answer
101 views

projective geometry hyperplane

For $j=0,\ldots,n$ consider the affine hyperplane $A_j:=e_j+\langle e_0,\ldots,e_{j-1},e_{j+1},\ldots,e_n\rangle$ in $\mathbb K^{n+1}$ and the associated embedding $\tau_j:\mathbb ...
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1answer
115 views

Find the line closest to 2 2D points passing through a 3rd point

Given 3 homogeneous 2D points, $p=[p_x,p_y,1]$, $q=[q_x,q_y,1]$, and $v=[v_x,v_y,v_z]$, with $p$ and $q$ finite (last coordinate is $1$), but $v_z$ maybe either $1$ - finite or $0$ - at infinity: ...
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556 views

Finding a 3D transformation matrix based on the 2D coordinates

I have a square with the length of the sides being 1. This square is transformed by an unknown transformation matrix in the 3D space and then projected back to the plane (the projection is known). I ...
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1answer
108 views

Rank of a degenerate conic

This question comes from projective geometry. A degenerate conic $C$ is defined as $$C=lm^T+ml^T,$$ where $l$ and $m$ are different lines. It can be easily shown, that all points on $l$ and m lie on ...
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66 views

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 ...
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What Projections preserve Pseudocontractiveness?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a pseudocontraction, i.e., $$ \left\| f(x) - f(y) \right\|^2 \leq \left\| x-y\right\|^2 + \left\| f(x) - f(y) - (x-y) \right\|^2 $$ for all $x,y \in ...
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In the affine plane, I am having trouble with these definitions

If the number of points in an affine plane is finite, then if one line of the plane contains $n$ points then: all lines contain $n$ points, every point is contained in $n + 1$ lines, there are $n^2$ ...
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39 views

In projective geometry and curved lines

For example, Fano's Geometry exemplifies a triangle with a circle within it. How is this possible? Are lines not defined to be straight? Is this geometry projective geometry(or is projective geometry ...
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Projective transformation by mapping fixed points to ideal circular points

I am trying to compute a projective transformation out of four pairs of points following the procedure described in this post, namely: Two circular points. A similarity rotation center. An arbitrary ...
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1answer
64 views

Find the line closest (SSD) to 2 2D points passing through a 3rd point

Given 3 homogeneous 2D points, $p=[p_x,p_y,1]$, $q=[q_x,q_y,1]$, and $v=[v_x,v_y,v_z]$, with $p$ and $q$ finite (last coordinate is $1$), but $v_z$ maybe either $1$ - finite or $0$ - at infinity: ...
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1answer
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Geodesics (3): A projective space based on the torus (instead of the sphere)

From the Wikipedia article on projective planes: [...] consider the unit sphere centered at the origin in $\mathbb{R}^3$. Each of the $\mathbb{R}^3$ lines in this construction intersects the ...
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Geodesics (2): Is the real projective plane intended to make shortest paths unique?

From the Wikipedia article on geodesics: In Riemannian geometry geodesics are not the same as “shortest curves” between two points, though the two concepts are closely related. The ...