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

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29
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5answers
4k views

Help understanding Algebraic Geometry

I while ago I started reading Hartshorne's Algebraic Geometry and it almost immediately felt like I hit a brick wall. I have some experience with category theory and abstract algebra but not with ...
27
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4answers
1k views

Geometry or topology behind the “impossible staircase”

This question on the topology of Escher games reminded me of a question I've had in my head for a little while now. Is there anything interesting geometric or topological that can be said about the ...
19
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4answers
8k views

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 ...
17
votes
2answers
970 views

How should I think about very ample sheaves?

Definition. [Hartshorne] If $X$ is any scheme over $Y$, an invertible sheaf $\mathcal{L}$ is very ample relative to $Y$, if there is an imersion $i:X \to \mathbb{P}_Y^r$ for some $r$ such that ...
14
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1answer
2k views

Plücker Relations

Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Plücker relations are quadratic forms on $\wedge^d V$ whose zero set is exactly the set of decomposable ...
14
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1answer
213 views

Can a continuous map $S^2 \rightarrow S^2$ preserve orthogonality without being an isometry?

Suppose I have a map $\phi: S^2 \rightarrow S^2$ and I know that a) $\phi$ is continuous and bijective b) If $a$ and $b$ subtend an angle of $\pi / 2$ at the center of the sphere, then so do ...
14
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1answer
435 views

topologies of spaces in escher games

There have been a couple of games released (or in development) in the past couple of years which do some weird topological tricks: Echochrome (video), Crush (video), and Fez (video). Do the spaces ...
13
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1answer
274 views

Quaternionic veronese Embedding

I know that the complex projective line $\mathbb{C}P^1$ can be embedded in the complex projective space $\mathbb{C}P^n$ (Veronese embedding). For example, $\mathbb{C}P^1\rightarrow\mathbb{C}P^3$ is ...
11
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2answers
166 views

Computing cohomology of hypersurface

I'm taking a course on differential geometry now, and we got the following exercise from the lecturer: compute the (de Rham) cohomology groups $H_{dR}^i(M)$ of your favourite space. In all the ...
11
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3answers
1k views

What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed ...
10
votes
2answers
324 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 $V$. ...
10
votes
1answer
130 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 ...
10
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4answers
1k views

Textbook for Projective Geometry

So while not actually a specific problem I'm struggling with, I was hoping for some of your insight! For a course, I'm currently reading Stillwell's Four Pillars of Geometry. While it does a nice ...
10
votes
2answers
1k views

Augmented Reality Transformation Matrix Optimization

i am a software developer, i'm working on an Augmented Reality system. I'd like to receive some advice in order to optimize my math model. My program has to be slim and fast. Here's the situation: ...
10
votes
1answer
413 views

A modern textbook on affine and projective spaces

Requirements: Scalar fields other than $\mathbb{R}$ and $\mathbb{C}$. Precise. Visual explanations are good, but they must complement definitions and proofs, not replace them. No repetition of text. ...
10
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1answer
135 views

History of the point at infinity?

I'm curious to learn more about the history of the introduction of the concept of the point at infinity into mathematics. The sum of my knowledge of the historical aspect is from this paragraph (which ...
10
votes
1answer
151 views

Packing infinitely many ellipses into a circle

Given a circle $C$, and an infinite set $S$ of mutually disjoint ellipses which are inside and tangent to $C$, prove that there must exist a disk $D$ which lies inside $C$ but outside every ellipse. ...
10
votes
0answers
508 views

Prerequisite of Projective Geometry for Algebraic Geometry

I studied Euclidean Geometry in high-school, and I have not studied anything relates to geometry since I started studying in university. I am now intending to study Algebraic Geometry, however, I ...
9
votes
1answer
446 views

Learning projective geometry

My ultimate goal is to learn some algebraic geometry with the more concrete immediate goal of understanding things like how $\mathbb R\mathrm P^2$ is embedded in $\mathbb C\mathrm P^2$ or how $\mathbb ...
9
votes
1answer
104 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): ...
8
votes
1answer
299 views

Is the graph of morphism of projective varieties $X \rightarrow Y$ closed in $X \times Y$?

The graph of a morphism $X \rightarrow Y$ is closed in $X \times Y$ if $X$ and $Y$ are affine varieties. What if $X$ and $Y$ are projective varieties? I am still not quite familiar with projective ...
8
votes
1answer
176 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 ...
8
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1answer
94 views
+100

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 ...
7
votes
3answers
10k views

The intersection of two parallel lines

My teacher told me that two parallel lines have a point of intersection, it is called Point at infinity. But I I can't understand how it can be true or how was it proved, can someone explain this to ...
7
votes
2answers
353 views

Is there a slick way to show that finite projective planes of $7$ points are unique up to isomorphism?

I was reading about the Fano plane, the smallest possible projective plane. After playing around with it, it seems that any projective plane of 7 points will be isomorphic to the Fano plane. ...
7
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2answers
415 views

How should I think of lines and planes in projective space?

I have been learning about projective varieties recently and I realised that I have some trouble trying to grasp what lines and planes are even in say $\Bbb{P}^3$. For one, how should I think about a ...
7
votes
1answer
675 views

Rectify image from congruent planar shape objects

I am implementing an algorithm to remove projective distortions on the following image. I understand this is possible by applying the following transformation: $$ \begin{matrix} 1 ...
7
votes
2answers
350 views

Perspective problem - trapezium turned square

True or false: If you draw a trapezium on the ground, there always exists a point above (but not necessarily directly above) the trapezium such that the trapezium looks like a square from that point. ...
7
votes
1answer
439 views

The genus of the Fermat curve $x^d+y^d+z^d=0$

I need to calculate the genus of the Fermat Curve, and I'd like to be reviewed on what I have done so far; I'm not secure of my argumentation. Such curve is defined as the zero locus ...
6
votes
2answers
147 views

Construction of the projective plane over $\mathbb{F}_3$

I have a question about constructing projective plane over $\mathbb{F}_3$. We first establish seven equivalence classes $P= \{ [0,0,1], [0,1,0], [1,0,0], [0,1,1], [1,1,0], [1,0,1], [1,1,1] \}$. ...
6
votes
3answers
297 views

Understanding cross ratio and harmonic conjugates

I'm studying projective geometry and I'm really having trouble with ''grokking'' what's it all about. Is there an easy/intuitive/visual way to understand cross ratio? I understand that it's ...
6
votes
1answer
78 views

Intersection of two quadrics

How to understand (maybe, informally) why the intersection of two quadrics in general position in $\mathbb{CP}^3$ is an elliptic curve? It is obvious that it is a compact 2-manifold, i.e. a sphere ...
6
votes
1answer
142 views

Why is $\mathbb{R} P^n$ called projective space?

I know that: If one defines an equivalence relation on $\mathbb{R}^{n+1}-\{0\}$ by $$x\sim y \iff y=tx$$ for some nonzero real number $t$, where $x,y\in\mathbb{R}^{n+1}-\{0\}$, Then The real ...
6
votes
1answer
151 views

Existence of a special set of q+2 points in the finite affine plane over $\mathbb F_q$

I am working in the finite affine plane over $\mathbb F_q$ with $q=2^n$. Such a plane has $q^2$ points, $q^2+q$ lines, each line has $q$ points, and by a point is passing $q+1$ lines. There are $q+1$ ...
5
votes
2answers
249 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 ...
5
votes
1answer
229 views

If a line bundle and its dual both have a section (on a projective variety) does this imply that the bundle is trivial?

Is there any reason that, on a projective variety X, if a line bundle L has a (non-zero) section and also its dual has a section then this implies that L is the trivial line bundle?
5
votes
1answer
114 views

Distinguished open sets in $\mathbb{P}^n$

I have given a homogenous polynomial $F\in\mathbb{C}[x_0,\ldots,x_n]$ of degree $d>0$ and consider the open set $U_F=\{p\in\mathbb{P}^n; F(p)\neq 0\}$. I'd like to prove that $U_F$ is isomorphic to ...
5
votes
3answers
356 views

Projective spaces with Zariski topology

Why $\mathbb{P}^1\times\mathbb{P}^1\not\cong\mathbb{P}^2$ where the projective spaces have the Zariski topology?
5
votes
3answers
723 views

How do you create projective plane out of a finite field?

I have heard and read unclear mentions of links between projective planes and finite fields. Is it possible to construct a projective plane (or a Steiner system) starting out with a field? Could you, ...
5
votes
1answer
111 views

projective cubic

I have some difficulties to prove that the image of the function $f:\,\mathbb P^1\longrightarrow\mathbb P^3$ such that $$(u,v)\longmapsto (u^3,\,u^2v,\,uv^2,\,v^3)$$ is the algebraic projective set ...
5
votes
1answer
88 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 $n+1$ ...
5
votes
1answer
53 views

Finding the inverse of a map from $CP^1$ to $S^2$

Given the map: $$f:CP^1 \to S^2\ ,\ f[z:w] = \left(\frac{2\mbox{Re}(w\bar{z})}{|w|^2+|z|^2},\frac{2\mbox{Im}(w\bar{z})}{|w|^2+|z|^2}, \frac{|w|^2-|z|^2}{|w|^2+|z|^2}\right)$$ How would I go about ...
5
votes
2answers
208 views

Implicitization of a cubic

Let $\nu$ be the map $\mathbb{P}^1 \to \mathbb{P}^2$ defined by $$\nu([X_0,X_1])=[A_0(X_0,X_1),A_1(X_0,X_1),A_2(X_0,X_1)]$$ where the $(A_i)$ are homogenous polynomials of degree 3, without common ...
5
votes
2answers
447 views

Analytically flavored book in projective geometry

I am looking for a book in projective geometry, using the apparatus of linear algebra, complex analysis, and, perhaps, modern algebra, in full. The counterexample is the Hartshorne's book on ...
5
votes
1answer
62 views

Can you define a vector space in terms of a pre-existing projective space?

Projective spaces are usually defined as the quotient of a vector space (by the equivalence relation that identifies collinear vectors). However, in my opinion, projective spaces seem intuitively less ...
5
votes
1answer
399 views

Problem in Deducing Perspective Projection Matrix

I understand the traditional way(use similar triangle and make depth value linear) to deduce the perspective projection matrix. But I want to try another approach after I read this text: Fundamentals ...
5
votes
1answer
314 views

Parametrizing a conic in projective space

I am just beginning to learn algebraic geometry. An exercise in Reid, p. 24 is to prove that if $Q(x,y,z)$ is a quadratic form over a field $k$ with at least 4 elements, and $Q$ vanishes on the zero ...
5
votes
0answers
102 views

How to calculate the dimension of the intersection of projection of varieties?

In $\mathbb P_{n+1}$ we consider $d$ varieties $V_1,\ldots V_d$, each of them is defined by $d-1$ equations, as follows: we have $d-1$ polynomials $f_2(X_0,\ldots,X_n),\ldots,f_d(X_0,\ldots,X_n)$ and ...
5
votes
1answer
231 views

Möbius transforms on the projective line

Let's suppose that I have a mapping $$ \phi:\mathbb{P}^1\rightarrow\mathbb{P}^1 $$ of degree two. How can I use Möbius transforms to write this map as $$ \phi([x:y])=\frac{y^2}{x^2}? $$ I already ...
4
votes
3answers
218 views

what is the definition of a line in $\mathbb{P}^n(k)$ + how to compute the hilbert polynomial of two intersecting lines?

(1) I have never studied any projective/affine geometry or algebraic curves. I'd like to see a clear definition of a line in the projective space $\mathbb{P}^n(k)$, since I need it for my algebraic ...