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

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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 ...
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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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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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101 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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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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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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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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1answer
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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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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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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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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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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1answer
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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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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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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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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166 views

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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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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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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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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1answer
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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1answer
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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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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73 views

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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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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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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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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1answer
14 views

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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36 views

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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1answer
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 $\Pi_1(\mathbb{P}^n(\mathbb{C}))$ ?
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Does the complete quadrangle transform to a rectangle or a parallelogram, when the vertices of the triangle is projected to infinity?

By pairing the vertices of a given quadrangle we draw lines joining them and the lines intersect forming triangles with the sides of the quadrangle. We now project the points of intersection, located ...
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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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48 views

Intersection of Segre variety with linear spaces

Consider the intersection of the Segre variety associated to product of $n$ copies of $\mathbb P^2$, with $k$ linearly independent hyperplanes. Is it possible to drop one of the hyperplanes and obtain ...
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Triangles form a harmonic set with their medians and altitudes

In a triangle $\triangle ABC$, let $AD,BE,CF$ be its altitudes and $AK,BL,CM$ their medians. Show that $D\{EF,AB\} = -1$ and $K\{LM,AB\} = -1$ I don't get any of the problems here. Not any of these ...
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32 views

If $K_X$ is not $\mathbb Q$-Cartier then it is not nef

Let $X$ be a projective variety. Is it true that if the canonical divisor $K_X$ is not $\mathbb Q$-Cartier then it is not nef?
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36 views

Find the N versors more 'spaced' [closed]

I have to deal with a concrete problem that is: Given a 3d object I want to select N directions with N integer and N>=3 for projection that would maximize the information I gain and thus my ability to ...
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Applying homography to ellipse derived from normal distribution

I need to apply a homography to an elliptic area. First question: Is the resulting also elliptic in every case? I think so, but actually i don't really know. Anyway, I assume it for this question. ...
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1answer
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A problem in projective geometry…

I have the following projectivity: $$ f[x_1,x_2,x_3]=[4x_1+2x_2-x_3,2x_2,x_3,-x_2-x_3]. $$ I have to find all the lines $L$ such that $f(L) \subset L$. I've found the eigenvalues of this matrix, ...
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Is the Projective Real Plane Compact?

I feel like $\Bbb P (\Bbb R^2)$ is compact, but I know that $\Bbb R^2$ is locally compact, therefore it has a one-point compactification. $\Bbb P (\Bbb R^2)$ adds more than one point to the real ...
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34 views

Surjectivity of a map $D^{2n} \to \mathbb{CP}^n$

I'm solving an exercise about the complex projective space, and during a step of the solution I'm asked to find a surjective map $D^{2n} \to \mathbb{CP}^n$. I defined the map in this way $$ ...
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Project triangle from $\mathbb{R}^3$ into $\mathbb{R}^2$ with to fixed vertex texture coordinates

I have a triangle made out of the three vertices p1, p2, p3. I know the positions of the vertices in $\mathbb{R}^3$, called $x_i, y_i, z_i$ for $i=1,2,3$. I now want to assign each vertex a texture ...