Projective geometry is closely related to perspective geometry. These types of geometry originated with artists around the 14th century.
23
votes
3answers
658 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 ...
23
votes
5answers
1k 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 ...
14
votes
1answer
193 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
330 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 ...
12
votes
2answers
151 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 ...
11
votes
1answer
222 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 ...
9
votes
2answers
122 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 $V$.
...
9
votes
2answers
521 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:
...
8
votes
1answer
682 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 ...
7
votes
4answers
491 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 ...
7
votes
2answers
234 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
133 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
215 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:
$$
\begin{matrix}
1 ...
7
votes
1answer
209 views
A modern textbook on affine and projective spaces
Requirements:
Scalar fields other than $\mathbb{R}$ and $\mathbb{C}$.
Precise. Visual explanations are good, but they must complement definitions and proofs, not replace them.
No repetition of text. ...
6
votes
3answers
514 views
Finding the Transform matrix from 4 projected points (with Javascript)
I'm working on a project using Chrome - JS and Webkit 3D CSS3 transform matrix.
The final goal is to create a tool for artistic projects using projectors and animation - somewhat far away from using ...
6
votes
1answer
115 views
Learning projective geometry
My ultimate goal is to learn some algebraic geometry with the more concrete immediate goal of understanding things like how $\mathbb R\mathrm P^2$ is embedded in $\mathbb C\mathrm P^2$ or how $\mathbb ...
6
votes
2answers
237 views
Perspective problem - trapezium turned square
True or false: If you draw a trapezium on the ground, there always exists a point above (but not necessarily directly above) the trapezium such that the trapezium looks like a square from that point.
...
5
votes
1answer
211 views
Is the graph of morphism of projective varieties $X \rightarrow Y$ closed in $X \times Y$?
The graph of a morphism $X \rightarrow Y$ is closed in $X \times Y$ if $X$ and $Y$ are affine varieties. What if $X$ and $Y$ are projective varieties?
I am still not quite familiar with projective ...
5
votes
3answers
257 views
Projective spaces with Zariski topology
Why $\mathbb{P}^1\times\mathbb{P}^1\not\cong\mathbb{P}^2$ where the projective spaces have the Zariski topology?
5
votes
1answer
75 views
projective cubic
I have some difficulties to prove that the image of the function $f:\,\mathbb P^1\longrightarrow\mathbb P^3$ such that
$$(u,v)\longmapsto (u^3,\,u^2v,\,uv^2,\,v^3)$$
is the algebraic projective set
...
5
votes
2answers
173 views
Implicitization of a cubic
Let $\nu$ be the map $\mathbb{P}^1 \to \mathbb{P}^2$ defined by $$\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
1answer
61 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
...
5
votes
1answer
140 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
0answers
104 views
How should I think about homogeneous coordinates?
It's quite easy to make sense of ordinary coordinates on a space $X$: it is just an open embedding of an open subset of $X$ in some vector space. Along the same lines, one could say that homogeneous ...
4
votes
3answers
172 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
92 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 ...
4
votes
1answer
89 views
Pullback of very ample sheaf again very ample? And other questions.
Let $S \subseteq \mathbb{P}^n$ be a smooth projective surface with given embedding in projective space. Moreover, let $X$ be another smooth surface and let there be a map $\pi: X \rightarrow S$ that ...
4
votes
1answer
79 views
intrinsic and geometric definition of blow-up
Suppose I am given an algebraic variety $X$ and a (closed) point $x \in X$. I know of two descriptions of the blow-up of $X$ at $x$. One is intrinsic but not geometric: if $\mathcal{I}_x$ denotes the ...
4
votes
1answer
134 views
How to show that the image of a certain projective embedding is an algebraic curve?
I found the following claim in a paper by Griffiths and Harris :
Start with a complex torus $\mathbb{C}/\Lambda$. The vector space of meromorphic functions having period lattice $\Lambda$ and a pole ...
4
votes
1answer
42 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 ...
4
votes
2answers
222 views
How to calculate x,y position of 3D points?
I have points in 3D system like this
$$p1=(2,3,4)$$
$$p2=(3,5,5)$$
Here I would like draw point $p1$ and $p2$ in $2D$ view.
Project type = orthographic.
Coordinate system = Cartesian
X- axis, ...
4
votes
1answer
274 views
A basic question on the terminology used in Desargues' Theorem
This is a pretty elementary concerning the terminology commonly used in Desargues' Theorem
from plane geometry (or really, projective geometry).
At least in some representative cases, I totally buy ...
4
votes
0answers
96 views
Existence of a special set of q+2 points in the finite affine plane over $\mathbb F_q$
I am working in the finite affine plane over $\mathbb F_q$ with $q=2^n$.
Such a plane has $q^2$ points, $q^2+q$ lines, each line has $q$ points, and by a point is passing $q+1$ lines.
There are $q+1$ ...
3
votes
2answers
122 views
Explicitly writing out all elements of $\mathbb{P}^{1}(\mathbb{F}_{n})$
Explicitly the elements of $\mathbb{P}^{1}(\mathbb{F}_{3})$ are $[1:0], [0:1], [1:1]$, and $[1:-1]$. Why is this so? How would I do this for $\mathbb{P}^{1}(\mathbb{F}_{4})$? What about general ...
3
votes
2answers
84 views
Intersection of smooth projective plane curves
I need to calculate the number of intersections of the smooth projective plane curves defined by the zero locus of the homogeneous polynomials
$$
F(x,y,z)=xy^3+yz^3+zx^3\text{ (its zero locus is ...
3
votes
1answer
100 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?
3
votes
1answer
192 views
What is a very simple example of the way projective geometry is used in quantum mechanics?
In The Road to Reality by Roger Penrose, projective geometry as developed during the Renaissance is framed as (eventually) playing a pivotal role in quantum mechanics. (In fact, Penrose seems enamored ...
3
votes
1answer
66 views
Explicit bundle projection
Consider the principal bundle $S^7$ with base space $\mathbb{C}P^3$ (3-dimensional complex projective space) and fiber $S^1\cong U(1)$. Can someone write to me the bundle projection ...
3
votes
1answer
143 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 ...
3
votes
1answer
70 views
Projective lines which can be viewed in some sense as surfaces
The complex projective line can be viewed as the $2$-sphere.
I'd appreciate some examples of other projective lines (over any field or even ring) that can be viewed in some sense as a surface. I am ...
3
votes
1answer
101 views
Prove that $f$ is not a projectivity
Definition (projectivity). Let $V,W$ vector spaces over some field $k$ and $V\neq\{0\}$. Let $F\::V\to W$ an injective linear map and let $v \in V\ \setminus\{0\}$. A projective map $\mathbb{P}(F\ ...
3
votes
1answer
111 views
Projective Plane vs. Reference Plane
I was told that the Projective Plane was also known as the Reference Plane in Projective geometry, but when I told my professor this, he freaked and told me I was completely wrong. He said that the ...
3
votes
1answer
59 views
Cross-ratio relations
The way I define the cross-ratio in projectve geometry:
Let $P_0,P_1,P_2,P_3$ being four points on a projective line G, such that $P_0,P_1,P_2$ are pairwise distinct. Let $\pi:\mathbb KP^1\rightarrow ...
3
votes
1answer
71 views
A finite subgroup of $\mathrm{PGL}(2,k)$
Let $G$ be a finite subgroup of $\mathrm{Aut}(\mathbb P^{1}(k))=\mathrm{PGL}(2,k)$, where $k$ is an algebrically closed field of characteristic $0$.
Suppose that there is a common fixed point for all ...
3
votes
1answer
170 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 ...
3
votes
0answers
52 views
Irreducible closed subsets of projective varieties
I want to prove the following lemma:
Let $X \subset \mathbb{P}^n$ be a projective variety. Let $W \subset X$ be a closed irreducible set. Then $W$ is also a projective variety.
My idea is as ...
3
votes
0answers
219 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 ...
3
votes
2answers
392 views
Lat/Long grid points covered by projecting rectangle onto sphere
Before my question proper, a little background: I'm wanting to optimise some computer rendering by eliminating the drawing of things that aren't visible given the current view.
Suppose we have a ...
3
votes
0answers
109 views
Projecting Projective Curves
I've been stuck for quite a while on what is probably a trivial problem. Let $X\subset\mathbb{P}^n$ be a smooth projective curve, and let
$$\mathcal{I}=\{(p,q,r):p,q\in X,p\neq ...
3
votes
0answers
104 views
The geometry of $\operatorname{PSO}(4)$ and the quaternions
Question: Given a twist of the projective space, how do I find unit quaternions that represent it?
Backgroud and what do I mean: Following Conway & Smith's On Quaternions and Octonions, every ...
