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

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How would one define polynomials over the projective line $P_K^1$

May $K$ be a field. If I set $\varXi=(X:Y)$ as a "projective variable" and "projective coefficients" $a_k=(x_k:y_k)\in P_K^1$ - may I then write a polynomial map $P_K^1\longrightarrow P_K^1$ in a form ...
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32 views

Open subscheme in special fiber

Let $X$ be a projective scheme over $R$ a discrete valuation ring with generic fiber irreducible. Can an open subscheme of $X$ be contained in the special fiber of $X$ ? Or is it true that every open ...
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25 views

Three points are collinear iff the determinant of the matrix of their barycentric coordinates vanishes

Let $A,B,C\in \mathbb{R}^2$ be noncollinear points. Then we have that for every point $P\in\mathbb{R}^2$ there exist $\alpha_1,\alpha_2,\alpha_3\in\mathbb{R}$ such that $P=\alpha_1A+\alpha_2 ...
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Intuitive argument for the transitivity of $PGL(n+1)$ acting on $\mathbb{P}^n$

In spherical geometry we can consider the action of $\operatorname{O}(n+1)$ on the unit sphere $\mathbb{S}^n$. It's easy to see that this action is transitive, because for any two points $x,y \in ...
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55 views

What is the degree of the pull back of a line bundle?

Let $X=\mathbb{P}^2$, and let $(y_1,y_2,y_3)$ be homogeneous coordinates on $X$. Consider a map $\phi:\mathbb{P}^1\longrightarrow X$, given by $\phi(x_1,x_2)=(x_1^2,x_1x_2,x_2^2)$, where $(x_1,x_2)$ ...
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1answer
30 views

Complete intersection curve

I have two very basic questions/clarifications. Let $X=\mathbb{P}^n_k$, and let $Y$ be a subvariety of $\mathbb{P}^n_k$ of dimension $m$. Then we say that $V$ is a complete intersection variety if ...
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1answer
15 views

Perspective projection of a sphere on a plane

I know the perspective projection of a sphere on a plane is an ellipse. How would I find the parametric equation for this ellipse? Say I have a camera at $(0, 0, z_2)$, a plane at $z=z_1$, and a ...
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83 views

Affine varieties over finite fields

I read in this paper (http://www.math.iitb.ac.in/~srg/preprints/Chandigarh.pdf) that the following set is an affine variety: $V_f=\{(t_0,...,t_N)\in \mathbb{F}_p^{N+1} : f(t_0,...,t_N)=0 \}$ where ...
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1answer
14 views

The significance of the 3-dimensional plot from homogeneous coordinates of a 2-dimensional function

If $(x,y)$ are your standard Cartesian coordinates and $(X,Y,Z)$ are homogeneous coordinates, then $x=X/Z$ and $y=Y/Z$. So if we have a function $f(x,y)$ we can convert it to a function $F(X,Y,Z)$ by ...
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The simplest way to find a parametrization of the plane projective curve $XZ-Y^2=0$.

I have to explain to some first year math students that the projective algebraic set $\textbf{Z}(XZ-Y^2)\subset\mathbb P^2_k$ is $$V=\{(a^2_0:a_0a_1:a^2_1)\subset\mathbb P^2_k \,:\, \textrm{for}\; ...
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Weighted blow-ups

I would like to understand what's a weighted blow-up in a very simple case: $\mathbb{C}^2$ blown-up in the origin with weights $(a,b)$. In found some notes online saying that this is the surface $X$ ...
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36 views

Proof of the Inscribed Angle Theorem

I want to give a proof of the Inscribed Angle Theorem by using the Laguerre formula. Let $C$ denote the circle. Take three different points $A,B$ and $P$ on $C$. Write $a := \overline{AP}$ and $b:= ...
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145 views

Characterizations of the cross-ratio

$$ (z_1,z_2;z_3,z_4) = \frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)} $$ What are the most prominent or most interesting theorems of the following form? Theorem: The cross-ratio is the only function ...
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11 views

A couple of doubts regarding the proof of the Desargues theorem

I've just seen the Desargues' theorem and its proof. Now I have a couple of doubts. In the 3D space when the triangles are not coplanar, it's obvious that if the lines intersect at a finite point, ...
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1answer
27 views

The projective space of all lines through the origin

I have a question to the following example: Assume that $\mathbb{A}_2$ is an affine plane over a field $\mathbb{K}$, and we have fixed affine coordinates $x, y$ on $\mathbb{A}_2$. Let $\mathbb{P}$ be ...
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Application of projective duality

What are the applications of the projective duality principle in another scientific areas such as Physics and Chemistry?
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34 views

Camera calibration: how does checkerboard size/numbers/placement affect accuracy

I am trying to calibrate a camera using a checkerboard by the well known Zhang's method followed by bundle adjustment, which is available in both Matlab and OpenCV. There are a lot of empirical ...
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2answers
63 views

What kind of geometry is useful to study for mathematical competitions?

I'm bad in geometry but I would like to be better. What kind of geometry is useful to learn olympiad level geometry? I mean, can projective geometry solve more problems than geometry with complex ...
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91 views

Find the intersection of two lines entirely outside the given sheet of paper by straightedge alone

This is a problem from Courant:"Two straight lines entirely outside the given sheet of paper are each given by two pairs of straight lines intersecting at points of the lines outside the paper. ...
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1answer
44 views

How does the multiplicative group of a finite field, considered as a vector space, act on subspaces?

Given that a finite vector space $V = \operatorname{GF}(p)^n$ corresponds to the finite field $F = \operatorname{GF}(p^n)$, I'm wondering about how the multiplicative subgroup of $F$, $F^*$, acts on ...
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1answer
57 views

How can I get a cohomology of hypersurfaces by using their equation?

While studying about complex projective hypersurfaces, I attempts to find a cohomology of this hypersurface : $$X_n=\{(x_0:x_1:x_2:x_3) \in \mathbb{C}\mathbb{P}^3~|~x_0^n+x_1^n+x_2^n+x_3^n=0\}$$ I ...
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32 views

Given a point $P$ and a hyperplane $H$ in $\mathbb{P}^n$ such that $P \in H$, there is $T$ linear such that $T(P)=(0:\cdots:0:1)$ and $H:X_0=0$

Show that given a point $P$ and a hyperplane $H \subseteq \mathbb{P}^n$ such that $P \in H$, there is a linear transformation $T$ such that $T(P)=(0:\cdots:0:1)$ and $H$ is given by the equation ...
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Projective conic generated by a set of tangent triangles.

I need to proof the following result: Let C be a real projective conic and P, Q two points interiors to C then there is another real projective conic such that every triangle inscribed on that conic ...
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1answer
18 views

Projective space of a module

I'm studying projective geometry in a basic course of geometry. My question is: Is there an equivalent definition of projective space not of a vector space but of a module? I think the basic ...
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1answer
26 views

Are projective transformation linear and why?

As the title, I would like to know: Are projective transformation linear and why? I'm talking about projective transformation, linear transformation in terms of homogeneous coordinates. I can prove ...
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1answer
48 views

Standard Cremona Involution

Let $\varphi$ be the standard Cremona involution on $\mathbb{P}^r$, which is defined as $[x_0,\dots,x_r]\mapsto [\frac{x_0\dots x_r}{x_0},\dots, \frac{x_0\dots x_r}{x_r} ]$. I came across to the ...
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51 views

Show that any quadric in $\mathbb{P}^3$ is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$

Show that any non-singular irreducible quadric in $\mathbb{P}^3$ is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$ I know that every non-singular and irreducible quadric in $\mathbb{P}^3$ can ...
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Show that there exists a coordinate system in $\mathbb{P}^n$ such that $P_0=(1:0:\cdots:0),\ldots,P_n=(0:0:\cdots:1),P_{n+1}=(1:\cdots:1)$

Proposition: Let $P_0,\ldots,P_{n+1}$ be $n+2$ points in $\mathbb{P}^n$ such that every $n+1$ are in general position. There exists a coordinate system in $\mathbb{P}^n$ such that ...
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1answer
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Preserving shapes in perspective projection

Perspective projection is very simple to perform, but when I tried to prove that certain geometric elements preserve their identity when projected I faced many difficulties though intuitively it seems ...
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22 views

Generalization of a Result Concerning Projective Planes

Let $\mathcal P$ denote the set of all possible orders of projective planes. For $q\in\mathcal P$, let $PG_2(q)$ denote the projective plane of order $q$. There is a theorem due to James Singler ...
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Method for Visualizing Projective Space

I've been trying to understand projective space as follows: Consider the plane at z=1 as the 2D affine plane, and for any curve in this affine plane, let the inclusion in projective space be the set ...
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1answer
43 views

How is it that dot product does projection but so does division?

Can anyone explain how it is that dot products can perform projection but so can division, and how the processes are related (or not)? With dot product, you can project one vector onto another ...
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47 views

Metric in $\mathbb{P}_2$

I have to prove that $\mathbb{P}_2$ with the function $\delta(P,Q)$ defined by "Sine of the angle between two vector in $\mathbb{R}^3$ such that they correspond respectively to P and Q" is effectively ...
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Visualizing projective closures - is it okay to just think of the affine case?

This question is quite general and has been discussed on MSE before, however my case is a little bit different and I'm wondering about the geometric interpretation of a specific example. I think that ...
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1answer
23 views

Computing the matrix that represents orthogonal projection,

There is a theorem that says if $U$ is an orthogonal matrix, i.e., its columns (or rows) form an orthonormal basis, then the action of $UU^T$ represents orthogonal projection of the vector space onto ...
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34 views

Number of parameter of a quadric

Suppose for example that $S$ is an algebraic complex surface contained in $\mathbb{P}^6$. $S$ is the complete intersection of four quadrics in the six dimensional projective space. If i take a quadric ...
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41 views

affine vs projective tranformation

I'm trying to grasp the difference between the affine and projective transformations...I got the point of the line at infinity but their matrix representation is not yet clear enough: ...
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77 views

$3\mathrm D$ Projection Of $4\mathrm D$ Polyhedron

Can someone identify this shape? I think it is a $3\mathrm D$ projection of $4\mathrm D$ polyhedron. The body in the center seems to be a truncated octahedron, so as the body in the middle. The ...
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115 views

Moduli space of algebraic surfaces Vs moduli space of curves

Define the surface $S$ as the complete intersection of four quadrics $Q_i$ with $i=1,2,3,4$ in $\mathbb{P}^6$ (complex six dimensional projective space) i.e. $$S=Q_1 \cap Q_2 \cap Q_3 \cap Q_4$$. Put ...
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Computing projective coordinates and equation of a plane.

The question We are given a vector space with basis $\{e_1,e_2,e_3\}$ and a projective frame, $\{ [e_1],[2e_1 + e_2],[e_2 - e_3] \}$ and a fixed point $[3e_1 + e_3]$. What is the coordinates for the ...
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69 views

Intersection counting without Bézout

I am trying to solve the following problem: Let $C$ be a non-degenerate line (resp. conic) in $\mathbb{C}\mathbb{P}^2$ and $D$ a projective curve in $\mathbb{C}\mathbb{P}^2$ of degree $d$ such ...
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Finding the transformation matrix of a projective transformation in RP^2

So I want to understand how to find the matrix that represents the projective transformation that sends 4 given points to 4 given images, I know that 4 points are necessary to determine it but I can't ...
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36 views

A $d$-form on ${\mathbb R}^n$ that vanishes on $\binom{d+n-1}{n-1}$ general points, vanishes identically.

I'm looking for a reference for the fact that a $d$-form on ${\mathbb R}^n$ that vanishes on $p_1,..,p_{\binom{d+n-1}{n-1}}$ general points, vanishes identically. A specific construction of a set of ...
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1answer
28 views

Reducible cubic surface are always singular.

I want to prove that Any reducible cubic surface are always singular. A possible way may be to take a look at the intersection of the irreducible components. But I don't know how. Thanks for any ...
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1answer
33 views

Lines on a quadric surface

I want to show that: Let $Q$ be a quadric surface of rank 3 in $\mathbf{P}^3$(over $\mathbf{C}$), then any line in $Q$ must pass through the only singular point of $Q$. I know that up to a transform ...
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Question regarding Calibration while using Phase Measuring Profilometry (PMP)

We are using PMP to create the 3d model of a real world object in a summer project. However, to actually use PMP we need to relate the camera and the projector parameters and coordinates. To ...
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Weighted Projective Spaces and varieties

Consider the weighted projective space $\mathbb{P}(1,1,2)$ with variables $x_0,x_1,x_2$ of degrees 1,1,2 respectively. Consider the map $$(a_0,a_1,a_2) \to (a_0^2,a_0 a_1,a_1^2,a_2)$$ with ...
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119 views

Exercise about an algebraic surface

Let $\mathbb{P}^6$ the six-dimensional complex projective space. Suppose that $Q_{i}$ is a smooth quadric in $\mathbb{P}^6$ for $i=1,...,4$. Define $$S=Q_1 \cap Q_2 \cap Q_3 \cap Q_4 $$ as complete ...
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1answer
29 views

Is this a metric on $\mathbf P\mathcal H$?

Let $\mathcal H$ be a real or complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. On the projective space $\mathbf P\mathcal H = \left(\mathcal H\setminus\{0\}\right)\big/{\sim}$ ...
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27 views

Constructing a Mobius transformation that acts on any two points of the upper half complex plane:

I would like to construct a Mobius transformation that sends any two points $z_1$ and $z_2$ from the upper half of the complex plane to i and to $iR^+$, i.e., given any two points $z_1$ and $z_2$, ...