Articles of projective geometry

Area of projection of cube in $\mathbb{Z}^3$ onto a hyperplane

A cube with vertices $(\pm 1, \pm 1, \pm 1)$ gets projected into the plane perpendicular to vector $\mathbf{n}\in S^2$. The projection is a hexagon, how do I find the area? I think I can just compute $\mathbf{n}\cdot e_{\mathbf{x}},\mathbf{n}\cdot e_{\mathbf{y}}$ and $\mathbf{n}\cdot e_{\mathbf{z}}$ and the area would just be the some of the 3 […]

How to obtain the equation of the projection/shadow of an ellipsoid into 2D plane?

Given an ellipsoid equation of the form \begin{equation}\label{eq_1}x’Ax=1\end{equation} where $A\in\mathbb{R}^{n\times n}$ is positive definite and non-diagonal and $x\in\mathbb{R}^n$. So, how can I obtain the projection or shadow of the ellipsoid into a 2D plane? In these links: How to prove the parallel projection of an ellipsoid is an ellipse? and Projection of ellipsoid part of […]

Geodesics (2): Is the real projective plane intended to make shortest paths unique?

From the Wikipedia article on geodesics: In Riemannian geometry geodesics are not the same as “shortest curves” between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points […]. Going the “long way round” on a great circle between two points on a […]

Manifold over a Finite Field

I’m trying to either associate a manifold with a finite field, or, ideally find a way of considering finite fields as manifolds, in a non-trivial manner. I hope to be able to use this to extend topological methods to finite projective planes.

Hypersurfaces meet everything of dimension at least 1 in projective space

The following exercise is taken from ravi vakil’s notes on algebraic geometry. Suppose $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension at least $1$, and $H$ is a nonempty hypersurface in $\mathbb{P}^n_k$. Show that $H\cap X \ne \emptyset$. The clue suggests to consider the cone over $X$. I’m stuck on this and I realized […]

Projective and affine conic classification

I have a doubt on the classification of non-degenerate conics (parabola, ellipse, hyperbola) in projective geometry (my textbook is “Multiple View Geometry in Computer Vision”, which, as the title implies, is not specifically targeted at projective geometry). As far as I understand, we can classify conics from a projective and from an affine point of […]

Natural way of looking at projective transformations.

Let $k$ be a field and let $V$ and $W$ be finite-dimensional $k$-vector spaces, where $\dim(V)\ge1$ and $\dim(W)\ge1$. Let $q:V\to\mathbb{P}(V)$, $u\to[u]$ be the quotient map. By my teacher, a map $f:\mathbb{P}(V)\to\mathbb{P}(W)$ is defined to be a projective map if there exists an injective linear map $L:V\to W$ such that for all $[p]\in\mathbb{P}(V)$ we have that […]

Why is $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$ constant if dim $\phi(\mathbb{P}^n)<n$?

Let $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$, $n\leq m$. I want to demonstrate that if dim $\phi(\mathbb{P}^n)<n$ then $\phi(\mathbb{P}^n)=pt$ (ex. 7.3(a), ch.II from Hartshorne). It’s well known that $Pic(\mathbb{P}^n)\simeq\mathbb{Z}$, with generator $O_{\mathbb{P}^n}(1)$. First question : it is right that if I show that $\phi^*O_{\mathbb{P}^m}(1)$ is generated by less than n+1 global sections, then must be $\phi^*O_{\mathbb{P}^m}(1)\simeq O_{\mathbb{P}^n}$ and so […]

line at infinity

I tried solving the following question, could you have a look at my answer and tell me whether it’s right or wrong? All input is appreciated. Question: Let $ABCD$ be the vertexs of a parallelogram in the affine plane immersed in its projective closure. Find the equation of the line at infinity in the projective […]

What is the difference between the sphere and projective space?

I know about the antipodal mapping. What I want to know is what the most significant differences between the sphere and projective space are, and how to think of each of them and their relationship to one another. I come at this from a coding theory/vector quantization perspective; I’m trying to understand the difference between […]