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

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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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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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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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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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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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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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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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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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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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38 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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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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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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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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42 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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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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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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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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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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$3\mathrm D$ 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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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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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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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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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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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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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 $z_2$, ...
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Theorem** on page 288 from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris

1Let $V$ a $2n$ complex vector space and take on $V$ a quadratic form. Now define $$ \Sigma=\{\Lambda:Q(\Lambda,\Lambda)\equiv0 \} \subset Gr(n,2n)$$ where $\Lambda$ is a maximal subspace i.e it is ...
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An interesting point of a triangle. (Help needed to prove a statement.)

Consider a triangle whose sides are segments of $\color{red}{\text{line}}$, $\color{blue}{\text{line}}$, $\color{green}{\text{line}}$ falling in the circum-circle $c$. Let ...
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Atlas on the Grassmannian Variety

Let $G(k,n)$ the set of all $k$-dimensional sub-spaces of a vector complex space $V$ of dimension $n$. I know that it is possible to define the grassmannian as the quotient of $\chi(n,k)$ by $GL(k)$ ...
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Affine open sets of projective space and equations for lines

I am reading Introduction to Algebraic Geometry by Smith et al. and I have some questions about some vocabulary that they use but that is not explicitly defined (I guess it is probably obvious and I ...
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$\text{SO}(2,1)$ is isomorphic to the group of automorphism of $x^2+y^2-z^2$

I am not able to prove the following. Let $\mathbb{K}$ be a field with more than four elements and with characteristic different from $2$. Let $\mathcal{C} \subset \mathbb{P}_2(\mathbb{K})$ be the ...
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coordinates depending on perspection

The diagram below shows a perspective drawing of two squares, with coordinates given-on the drawing-for some of the corners of the squares(the line a the top is the horizon) The diagram below shows ...
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Projective transformation a parabola to a circle

Take the parabola $x^2 - y = 0$ in the cartesian plane. I'm not entirely sure about this, but we can express this using homogenous coordinates as $X^2 - Y = 0$ (the $W$ coefficient is $0$?) With the ...
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Cohomology of conic bundle 3-folds

It is known that for a smooth cubic 3fold $X\subset \mathbb{P}^4$ we have $H^3(X,\mathcal{O}_X)$ (or if you prefer $H^{0,3}(X)=0$). Moreover, if I project off a line $l\subset X$ I can resolve the map ...
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Computing local coordinates

Let $p=[x_0,y_0,1]\in \mathbb P_2(k)$ (projective space). Determine a projective transformation $\phi\in GL(3,k)$ such that $\phi(p)=[0,0,1]$ and name the coordinates explicit. Its easy to see that ...
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Variety of maximal isotropic subspaces

Suppose that $V$ is a complex vector space of even dimension $2n$. Let $Q:V \times V \rightarrow \mathbb{C}$ a bilinear, non degenerate, simmetric bilinear form over the field of complex number. Set ...
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Why are degenerate conics not projectively equivalent to nondegenerate conics?

This is what I understand about conics being projectively equivalent. Two conics $C1=V(F)$ and $C2=V(G)$ are projectively equivalent if there is an invertible matrix $A$ such that $F(X,Y,Z)=0$ iff ...
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Is the action of the matrix UU^(t) always a projection? What can I say about I - UU^t?

The way I understand it is that: if U is orthogonal, i.e., its columns (or rows) form an orthonormal basis for the n-dimensional Euclidean (coordinate) space, then the matrix $UU^t$ is an orthogonal ...
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What is the relationship between a complex manifold being Kähler, projective, nonprojective, and nonKähler?

I was wondering if this implication is true. I read a few places that $$\text{nonprojective} \Longrightarrow \text{nonKähler}$$ but I think I maybe have misunderstood. Equivalently, this is of course ...
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Constructing a conic from two point-polar pairs

Suppose I have two points $A$ and $B$ and two lines $a$ and $b$ in the (projective) plane. Can I construct a conic section for which $a$ is the polar of $A$ and $b$ is the polar of $B$? How unique ...
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Find a projective transformation that sends the locus $x^2-y^2=z^2 $ to $yz=x^2$

So I'm trying to find a function that sends $x^2-y^2=z^2$ to $yz=x^2$. I know it can be done because all conics are projectively equivalent. I think I have to use a matrix of some kind but I don't ...