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

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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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15 views

What is the meaning of projectivized quadratic forms?

I have a quadratic form $$C_1=z_1^2 + z_2^2 +z_3^2.$$ What does it mean to projectivize $C_1$? I am guessing that it is $$[C_1]=\{C : C=\lambda C_1, \ \lambda \in \mathbb{C} \}.$$ Is this correct? ...
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16 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
42 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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1answer
34 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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1answer
70 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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1answer
24 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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1answer
28 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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1answer
98 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): ...
6
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1answer
43 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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15 views

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
12 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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30 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
36 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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21 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
37 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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163 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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13 views

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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1answer
31 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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1answer
35 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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13 views

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
13 views

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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84 views

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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1answer
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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21 views

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 ...
2
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1answer
32 views

Conic through 4 points

Let $p_1,\ p_2,\ p_3,\ p_4$ be any 4 different points on $\mathbb{CP}^1$ and $x_1,\ x_2,\ x_3,\ x_4$ are 4 different points on $\mathbb{CP}^2$. How can I show that there is unique conic $Q$ passing ...
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1answer
22 views

Can projective spaces be given structure of a linear space.

Let $\mathbb{RP^{n-1}}=\mathbb{R^n}/ \sim $ where x ~ y iff $\exists \ \lambda \in \mathbb{R} \ s.t \ \lambda x=y$ Can $\mathbb{RP^{n}}$ be given the structure of an $\mathbb{R}$-module . Moreover, ...
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86 views

Bounding the cohomology of a smooth projective variety

Let $X/\mathbb C$ be a smooth projective variety. Suppose it is smoothly embedded in $\mathbf P^n$ as the zero locus of an ideal generated by homogeneous polynomials $f_1, f_2, \dots, f_r$ in $n+1$ ...
4
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1answer
79 views

CP(2) = SU(3)/U(2)?

In my understanding the complex projective line $CP^1 = \mathbb{C}^2/\mathbb{C^*}$ where $\mathbb{C^*}$ is $\mathbb{C}$ without $0$, i.e. just 2 complex coordinates and a homogeneous factor. And the ...
3
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1answer
60 views

Why is $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$ constant if dim $\phi(\mathbb{P}^n)<n$?

Let $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$, $n\leq m$. I want to demonstrate that if dim $\phi(\mathbb{P}^n)<n$ then $\phi(\mathbb{P}^n)=pt$ (ex. 7.3(a), ch.II from Hartshorne). It's well ...
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56 views

Hyperplanes as dual projective spaces

I was reading through Harris's Algebraic Geometry book, and was slightly perplexed by the following paragraph: "Note that the set of hyperplanes in a projective space $\mathbb{P}^{n}$ is again a ...
3
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1answer
45 views

Module over an associative ring with unity and axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
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2answers
81 views

Module over a ring which satisfies Whitehead's axioms of projective geometry

I asked this question(Characterization of a vector space over an associative division ring). It was pointed out that the answer was negative. So I reconsidered the problem and have come up with the ...
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1answer
33 views

Whitehead's axioms of projective geometry and a vector space over a field

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
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1answer
28 views

Whitehead's axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
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1answer
24 views

Application of Desargues' theorem for constructions

I found this interesting document (german) on the internet. On page 8 it says: "Draw a line segment between two given points only using compass and ruler, while the distance between the two points is ...
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1answer
14 views

What is the plane gradient?

My professor recently used the following phrase "the unknown 3D point is in a plane whose gradient is $(a,b,c)^T$". I can't seem to place his terminology anywhere on the internet. What does he mean by ...
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0answers
26 views

Determining 3D position of point from 2D projection.

Say I have a 3D point $p$ and I project this point (using perspective projection) onto the image plane at 2D point $u$. Knowing that $p$ is on a plane with gradient $(a,b,c)^T$, how can I express the ...
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1answer
16 views

Showing angles are preserved by isometry.

Im trying to show that a rigid transformation (isometry) preserves angles. Here is my approach so far. Let $x,y \in \mathbb{R}^n$ and $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a rigid motion ...
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2answers
33 views

Description of real projective spaces in various contexts

What I want to know is : What is the description of real projective spaces (specially $RP^0$, $RP^1$, $RP^2$) respectively in context of topology, geometry and algebra? I'm searching for simple ...
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1answer
29 views

Polar correlation and conics in $\Bbb RP^2$

I'm stuck on a small detail in Proposition 1.2.8 in Geiges' Introduction to Contact Topology. Let $C$ be a conic in $\mathbb{R}P^2$ given by $q^tAq=0$, where $A$ is a nonsingular, symmetric 3x3 ...
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1answer
16 views

“projective maassbestimmung” in Automorphic Functions by Fricke + Klein

I was reading a copy of Fricke and Klein's Theory of Automorphic Forms, and I came across the phrase projective maassbestimmung in the first chapter. Google translate returns: maßbestimmung as ...
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1answer
45 views

Is it true that ${\mathbb P}^1_{(1,2)} \cong {\mathbb P}^1$?

In a class on Algebraic geometry, we learnt the following - ${\mathbb P}^1_{(1,2)} \cong {\mathbb P}^1$ over the field ${\mathbb C}$. I'm not sure I followed the entire argument exactly. I'll ...
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10 views

If all I have is this image of a projection, how can I get screen coordinates for a object of a certain size?

All I have is this image and I want to figure out the coordinates relative to the 0,0 in the depicted coordinate system for an planar object of any size. Eg. (1m x 1m or 2.31m x 3.23m) Think a rug ...
0
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1answer
13 views

How to multiply a vector and matrix when the matrix includes a translation?

What is the proper way to right multiply an $N$ x $N$ matrix $H$ by an $N$ x $1$ vector $\mathbf{v}$, if $H$ includes a translation vector? For example, say $$H=R-\mathbf{tn}^T$$ where $R$ is a ...
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1answer
33 views

When are two 3D Lines parallel in Plücker matrix form?

When are two lines in 3 dimensional space parallel, when the lines are both represented by Plücker matrices $L$ and $L'$. I'm trying to prove the solution to this question: ...
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1answer
17 views

Is a Spread Unique?

Let $V$ be a vector space of dimension $n$. It is well known that when $r | n$, there is a set of disjoint $r$-dimensional subspaces of $V$, which covers $V$, called Spread. My question is that is a ...
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33 views

3D Vector projection on a Plane

I want to Project a Vector on to a Plane. Assume, you have a Central Point (1,1,1) and you want to move (0,0,3) in z-direction. How can I project the end of this movement (point) on a plane with ...