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

learn more… | top users | synonyms

26
votes
5answers
3k 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 ...
25
votes
4answers
938 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 ...
14
votes
3answers
5k 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 ...
14
votes
2answers
678 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
votes
1answer
207 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
votes
1answer
406 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
votes
1answer
266 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 ...
12
votes
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 ...
11
votes
2answers
112 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 ...
10
votes
2answers
264 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
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
3answers
680 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
902 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
342 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
votes
1answer
96 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 ...
9
votes
1answer
80 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. ...
8
votes
1answer
269 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
357 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 ...
7
votes
2answers
328 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
votes
2answers
332 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
526 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
321 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
318 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 ...
7
votes
0answers
417 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 ...
6
votes
3answers
6k 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 ...
6
votes
2answers
141 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
1answer
130 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 ...
5
votes
1answer
168 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
88 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
165 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 ...
5
votes
3answers
327 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
1answer
108 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
58 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 ...
5
votes
1answer
50 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
202 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
388 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
57 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
265 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
274 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
90 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
216 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 ...
5
votes
0answers
130 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 $q+1$ ...
4
votes
3answers
205 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 ...
4
votes
1answer
114 views

Difference between the real projective plane and the complex projective plane

Well the title says it all. If we consider the $P^2(\Bbb R)$ and the $P^2(\Bbb C)$, and we compare them, my guess is that it will be like a round $\Bbb R^2$ versus a sphere. I don't have very good ...
4
votes
1answer
77 views

Description of varieties in $\mathbb{P}^2\times \mathbb{P}^1$

If $[x:y]$ are coordinates of $\mathbb{P}^1$ and $[X:Y:Z]$ are coordinates of $\mathbb{P}^2$, what do the following varieties look like? $V(x^2X+y^2Y+xyZ)\subset \mathbb{P}^2\times \mathbb{P}^1$ ...
4
votes
1answer
185 views

How to show $\mu(\overline{\mathbb{R}}) = \overline{\mathbb{R}}$ iff $a, b, c, d \in \mathbb{R}$ for $\mu(z) = \frac{az+b}{cz+d}$?

Let $\mu(z) = \frac{az+b}{cz+d}$ be a Möbius transformation. I want to show that $$\mu(\overline{\mathbb{R}}) = \overline{\mathbb{R}} \iff a, b, c, d \in \mathbb{R}.$$ What would be an elegant, and ...
4
votes
1answer
179 views

Proof of Moisezon Theorem

We call a compact complex manifold Moisezon manifold, if its dimension coincides with the algebraic dimension, i.e. it has as many algebraically independent meromorphical functions as its complex ...
4
votes
1answer
38 views

Closed subschemes of projective space

We work over the complex numbers. If $X$ is an integral closed subscheme of $\mathbb P^n$, then there exists a homogeneous ideal $I$ of $A=\mathbb C[x_0,\ldots,x_n]$ such that $X = V_+(I) =$ ...
4
votes
1answer
78 views

Isomorphisms of $\mathbb P^1$

Prove that every isomorphism of $\mathbb P^1$ (over an algebrically closed field $\mathbb K$) is of the form $$ \phi(x_0: x_1) = (ax_0+bx_1 : cx_0 + dx_1) $$ where $\begin{pmatrix} a & b ...
4
votes
1answer
118 views

Dual projective surface

Let $X \subset \mathbb{P}^3$ be a smooth surface of degree $d>1$ and consider the Gauss map $X \to \mathbb{P}^{3*}$, which sends a point of $X$ to its tangent plane. To see that the image $X^*$ of ...