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

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Space of inscribed $n$-gons modulo projective transformations.

Say $P \sim Q$ ($P$ and $Q$ are «projectively equivalent») iff there is a projective transformation $f$ such that $f(P) = Q$. Then $\sim$ is an equivalence relation. I read that the space of ...
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Proof that Every collineation of $FP^2$ is semilinear?

A collineation of a projective plane is a permutation of the points and lines that preserves incidence. If $F$ is a field, then $FP^2$ is the projective plane of lines through the origin and planes ...
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9 views

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 ...
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20 views

Projection of a Triangle into a Tetrahedron

I was referring to a paper to implement an algorithm in which one of the step was to project the triangle into the ...
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21 views

Cross of two n-dimensional planes.

As you all know, there is geometric place of points of cross of two planes (given as plane vectors) explicitly written simply as ...
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106 views

Is a line just an infinitely large circle?

A line is infinite, right? Well, if $-\infty = \infty$, then a line is an infinitely large circle. (Does this have something to do with $1/0$?) It seems wrong, is it? Could I disprove it? How ...
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7 views

mat3x3 for orthographic projection [closed]

Is it possible to have 3x3 matrix, instead of 4x4 for orthographic projection? Assuming all my vectors 2d, and I don't use "projection offset" (left=0, bottom =0). Here is code from glm lib: ...
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13 views

Why is the projection of a vector V onto a span W, independent of the orthogonal basis of W.

Very straightforward question. I have read time and again in my book that it is independent but I don't understand why? Wouldn't changing the basis mean changing the length of the projection?
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15 views

Intersection fo Projective Lines

I think I've gone wrong with my reasoning somewere here but I'm not sure why. We embed $\mathbb{R}$ into the projective plane by $(x,y)\to[1,x,y]$, and consider the projective lines corresponding to ...
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53 views

A point $\in \mathbb{P}^2(\mathbb{C})$

What is the property that should be satisfied so that a point $\in \mathbb{P}^2(\mathbb{C})$ ? I am looking at an exercise where I have to find the flexes of a curve and this information is needed.
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63 views

How does the Möbius group act on circlines?

This is a continuation of my earlier, rather vague question. I am interested in studying the action of the Möbius group $PGL(2,\mathbb{C})$, on the circlines in the extended complex plane $\mathbb{C} ...
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243 views

What do we lose in Projective Spaces?

We can think of the Complex Numbers as an extension of the Real Numbers, similarly we can think of the Projective Plane naturally as a nice extension of the Euclidean Plane. But, when we go from real ...
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Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
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22 views

Central Projection - Project point $X$ on plane $\pi$

We consider the projection of three-dimensional projective space from a center $Z$ onto image plane $\pi$: \begin{align} X \longmapsto \alpha(X) = (Z \lor X) \cap \pi \end{align} since $\pi$ ...
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56 views

How do you work with the space of circles on the sphere considered as the projective line?

I'm trying to prove some things about the action of the Möbius group on the "circlines" in the extended complex plane, ie. circles on $\mathbb{C}P^1$. I find that while I have a good grip on Möbius ...
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23 views

Algebraic computation and interpretation of X(X^T) - I

Just stuck on the last part of a problem, and the solution gives that: If $[C(X)]^2$ = $X(X^T)-I$, X a 3x1 column vector, unit length, then I know that $XX^T$ is the orthogonal projection of vectors ...
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32 views

action of $GL_3$ on $P^2$

Find the action of $GL_3(K)$ on $\mathbb P_k^2 $, and compute its orbits and also the isotropy groups for all its orbits. ($K$ is an algebraically closed field) I know that $GL_3$ acts on ...
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231 views

Are morphisms between projective spaces required to be injective?

From Wikipedia's morphisms between projective spaces: Injective linear maps $T \in L(V,W)$ between two vector spaces $V$ and $W$ over the same field $k$ induce mappings of the corresponding ...
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43 views

Questions about intersection of linear varieties in a projective space

Let $X, Y, Z$ be linear varieties of dimension $r, s, t$ respectively in $\mathbb{P}^n$. If $r+s\ge n$, then $X\cap Y\neq \varnothing$. Furthermore, if $X\cap Y\neq \varnothing$, then $X\cap Y$ is a ...
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32 views

affine and projective line are homeomorphic

Reading this post here Problem in proving that $\mathbb{A}^2$ is not homeomorphic to $\mathbb{P}^2$ I came up with the following question: Why are the $\mathbb A^1$ and $\mathbb P ^1 $ ...
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292 views

fixed point projective geometry

I am thinking about the following: Let $\sigma:\mathbb C P^n\rightarrow\mathbb C P^n$ be a projectivity with $\sigma\circ\sigma=id_{\mathbb C P^n}$. I define the set of all fix points by ...
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Ceva, Desargues and Pascal's theorems for conics

I was told in class today that these three theorems are valid in projective geometry and with conic sections (I'm taking a modern geometry class) but I can't seem to find proofs anywhere online, and ...
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41 views

Restrictions of maps between projective varieties.

Let $f\colon X\to Y$ be a surjective algebraic map between two projective $k$-varieties, where $k$ is algebraically closed. Let $n=\dim(X),\,m=\dim(Y)$. Suppose furthermore that X,Y are irreducible. ...
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15 views

projective space over finite fields

Let $A,B$ be sets non empty sets. Let say that if $p\in A$ then $p$ is said to be a point and if $l \in B$ then $l$ is said to be a line. Let $C$ be a set of the form $\{p,l \}$ with $p \in A$ and ...
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Problem in proving that $\mathbb{A}^2$ is not homeomorphic to $\mathbb{P}^2$

let $k$ be an algebraic closed field. All the spaces are equipped with the usual zariski topologies. All the proofs of this fact that I've seen rely on the fact that two lines in $\mathbb{P}^2$ ...
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There is some intuitive idea of Pascal's 's theorem in Projective Geometry?

In projective geometry, Pascal's theorem (formulated by Blaise Pascal when he was 16 years old) determines that a hexagon inscribed in a conic, the lines that contain the opposite sides intersect in ...
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24 views

2d to 3d projection problem

I am writing a software where user can add objects in the 3d space and I want to make the user to be able to drag those objects with the mouse. Whenever my mouse moves I have an event fired ...
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1answer
13 views

Calculating connecting line of two points using homogenouse coordinates

Having to points $A = (-1,2)$ and $B = (1,0)$ and their respective homogenouse coordinates $(1:-1:2)$ and $(1:1:0)$ the line $f$ connecting both points is given by $f = A \lor B$. In $\mathbb{R}^3$ ...
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47 views

Constructing a 2x2 matrix R which represents reflection in the x-y plane,

Construct a 2x2 matrix R which represents reflection in the x-y plane through the line $$(cos(\theta)x+(sin(\theta)y=0$$, where $\theta$ is any real number. (Let's call this line "L".) Write an ...
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Intersection of two hyperplanes

$G$ and $H$ are hyperplanes in $\mathbb{P}_n$ with coordinates $g=(g_0, \ldots, g_n)$, $h=(h_0, \ldots, h_n)$. How can I find a symmetric matrix $A_Q$ of a quadric $Q$ with $ Q = G \cap H$, where ...
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17 views

First and second projection, definition and/or motivation for name

I have read that in the ordered pair $z=(x_1,x_2)$, an element of a direct product $Z=X_1 \times X_2$ of sets $X_1$ and $X_2$, the element $x_1$ is called the first projection and $x_2$ is called the ...
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54 views

Is the cross ratio the unique invariant under projective transformations up to multiples?

I have been studying the actions of $PSL_2(\mathbb{R})$ on the hyperbolic plane recently, and the hyperbolic distance $d(z_1, z_2)$ is the absolute value of the log of absolute value of the cross ...
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48 views

Axioms vs Models in projective and hyperbolic geometry

I'm studying Projective Geometry. The book I'm using begins with a little bit of history of Geometry, more precisely the history of the fifth postulate, the discovery of other geometries, etc. ...
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27 views

Show that an irregular 1-set in the plane is totally disconnected

A $1$-set is a Borel set such that $0 < \mathcal{H}^1(A) < \infty$, where $\mathcal{H}^s$ is the Hausdorff measure. Let $A$ be an irregular $1$-set in the plane. Deduce from the theorem below ...
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42 views

What is the center of the Valentiner group $\mathcal{V}=\langle I, Q \rangle$?

(Please refer to this question first: Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$? ) I want to understand the center of the Valentiner group: $$\mathcal{V}=\langle I, Q ...
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Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$?

My question is the one in the title. If you want to understand the context of the problem, please read further. I reduced a problem to proving the question. Background is: Valentiner group ...
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318 views

Is a divisor in the hyperplane class necessarily a hyperplane divisor?

Let $V$ be a smooth irreducible projective curve over an algebraically closed field $k$, embedded in some projective space $\mathbb{P}^n$, and let $[H]$ be the induced hyperplane divisor class on $V$. ...
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30 views

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?
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Can one prove Brianchon's theorem using Ceva's theorem?

Can I prove Brianchon's theorem using Ceva's? I am also wondering if parabola and hyperbola can be inscribed in a hexagon?
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1answer
100 views

A “trivial” implication I don't understand.

I'm reading the article "Belyi's theorem for complex surfaces - Gabino Gonzalez Diez" and there are few lines of a certain proof that I don't understand (the author claims that all is trivial): ...
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49 views

Geometric Interpretation of the Cross-Ratio

The cross ratio of 4 points $A,B,C,D$ in the plane is defined by $$(A,B,C,D) = \frac{AC}{AD} \frac{BD}{BC}$$ And it's a ratio which is preserved under projections, inversions and in general, by ...
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Finding a specific camera transformation matrix

I have the following situation: - two targets with known coordinates with respect to the "world". They are on a fixed xy plane on a height 0 in the z-direction. - Both targets have an angle associated ...
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256 views

3D reconstruction from 2 images with baseline and single camera calibration

first posted: http://stackoverflow.com/q/24852151/3858076 i was forwarded to here cause this is more a mathematical problem so if anyone here could help me i would be very thankful. plz ignore the ...
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topologies of real projective plane models

Consider a sphere upon a plane (say $\mathbb{R}^2$). Let $C$ be the center of the sphere. Consider the lower hemisphere plus the boundary (bowl) and project lines from $C$ across the surface until ...
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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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How to determine 3d measurements

I am trying to reproduce an artwork that is both a 2D drawing and 3D paper sculpture by Romanian artist Liviu Stoicoviu done in the 80s, The Triangle: I have tried to trace the 2D artwork which ...
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1answer
41 views

Looking for a supplement to my Projective Geometry course

this is my first time posting to Math Stack Exchange! So currently, I am taking a Projective Geometry course and I am struggling. I was wondering if anyone knew of any textbook I could read to help ...
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24 views

Camera Calibration

In a camera model, in order to find the camera calibration, how do we find the the parameters from the vector a in the equation $Ca=0$? I know that the camera matrix to convert a world point to image ...
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171 views

Characterizing $\text{PGL}_2(\mathbb F_p)$

Where can I find a description and proof of the basic structure of $\text{PGL}_2(\mathbb{F}_p)$ (number of elements with each order, conjugacy classes, etc.) which is understandable by an ...
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39 views

Fundamental group of $\mathbb{P}^n(\mathbb{C})$

We know that $$\Pi_1(\mathbb{P}^n(\mathbb{R})) \cong \mathbb{Z}_2 $$ for $n \geq 2 $. Is there a similar statement for $\Pi_1(\mathbb{P}^n(\mathbb{C}))$ ?