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

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About an example of normal bundle of a curve over a surface

I know the definition of the normal bundle $N_{C/S}$ of a curve $C$ over a surface $S$ as the cokernel of the injection $T_C \subset T_S|_C$ where $T$ is the tangent bundle. I would like to exhibit an ...
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1answer
55 views
+200

3D projection coordinates onto 2D plane to determine transformation matrix?

I'm not sure if there is an actual solution to this problem or not, but thought I would give it a shot here to see if anyone has any ideas. So here goes: I basically have three vertices of a rigid ...
2
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1answer
36 views

What is the intersection of the Segre variety in $\mathbb{P}^5$ and the Veronese surface in $\mathbb{P}^5$?

This is an exercise from Chapter 8 of Ideals, Varieties and Algorithms by Cox et al. The projective Veronese surface in $\mathbb{P}^5$ is defined as the projective closure of the surface $S$ which ...
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10 views

Closure of Schubert cell is the Schubert variety

My question concerns Proposition 1.4.6 in the following article: http://www.mi.uni-koeln.de/~littelma/SMTkurz.pdf . There's just one, apparently straightforward detail of the argument which I can't ...
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19 views
+50

Invariant points and lines under homography

Given a matrix representation of an homography in a real projective space $P(\mathbb{R^3})$, what is the general procedure to calcule the invariant subspaces? A brief description would be enough.
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27 views

Inverting an isometric projection?

I'm trying to invert a function that takes points on a 2-d plane to an isometric projection of that plane. This function is encoded as follows (as part of the Isomer library): ...
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1answer
32 views

question about the dimension of the global section space of a vector bundle

Suppose that $L,L^{'}$ are a line bundle over a compact riemann surface $C$. Take $H^0(C,L\otimes L^{'})$. Is it true that $h^0(C,L\otimes L^{'})=h^0(C,L)+h^0(C,L^{'})$ where $h^0(V)$ ,means the ...
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7 views

Cross-verifying a homography on known correspondences

Context I have two sets of known 2D correspondences $S_1$ and $S_2$, from which I have constructed homographies $h_1$ and $h_2$. This was achieved using the homogeneous estimation method, ie. by ...
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21 views

Help to understand the proof of the Riemann Munford relation

Here i post a file where from page 617 to 618 there is the proof of the Riemann mumford relation that is the theorem 1.13. My problem is to understand the beginning of that proof. In particular ...
0
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1answer
26 views

constructing segments with equal cross ratio

I was puzzeling again and had the following problem: Given: two segments $AD$ and $PS$ on $AD$ there are points $B$ and $C$ so that $AD \gt AC \gt AB$ (so they are in order A, B , C, D ) on $PS$ ...
0
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1answer
39 views

Question about the degree of a morphism

Suppose that $\phi$ is a morphism between compleax algebraic varieties named $X$ and $Y$. I know that the degree of the morphism $\phi= [Rat(X):Rat(Y)]$. Suppose that $\phi$ is a one degree morphism. ...
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46 views

Morphism between surfaces

Suppose that $S$ is a surface of general type. Let $K_S$ the canonical bundle of $S$ and $\phi=\phi_{K_S}$ the canonical map. Suppose that the canonical map is a morphism from $S$ to ...
2
votes
2answers
28 views

Alternative definition of projective space

I just found this definition of the projective space over a vector space: "Given a vector space V of dimension $n+1$, we will denote by $\mathbb{P}^n= \mathbb{P}(V)$ the projective space of all ...
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0answers
18 views

For polytopes, does union and linear transformation commute?

Given two (convex) polytopes $P_1$ and $P_2$ and a linear transformation $T$, is it true that: $$T(P_1 \cup P_2) = TP_1 \cup TP_2$$ What if $P_1$ and $P_2$ are not convex?
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61 views

Optimizing over intersection of polytopes inside polytope

I have the following situation, as shown in the following diagram: $W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in blue) located within a regular simplex and having coordinates $\in ...
7
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2answers
111 views

How to find the vertices of a particular ellipse with straightedge and compass?

In order to provide and alternative solution to a well-known problem $^{(*)}$ I would like to solve the following sub-problem in the most effective way (i.e. in the least number of steps). ...
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0answers
12 views

Cross Ratio of two rays through origin

There are two fixed and two variable concurrent rays of unit length in 3 space through the origin. How should the spherical coordinates of the two variable points be related to result in a constant ...
2
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0answers
41 views

3D projection onto 2D plane to determine transformation matrix?

I'm not sure if there is an actual solution to this problem or not, but thought I would give it a shot here to see if anyone has any ideas. So here goes: I basically have three vertices of a rigid ...
1
vote
1answer
54 views

Projective bundles

I am studying about projective bundles now. And I have the following doubts. 1) If we have an exact sequence of vector bundles over a scheme $X$, $0\longrightarrow E'\longrightarrow E\longrightarrow ...
0
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0answers
9 views

Cross ratio for two variable rays

Given two fixed and two variable rays in the plane with ends of rotating vectors between origin O (0,0) and the unit circle centered at O: $\alpha $ constant, $t_1,t_2$ variable. $$ A (1, 0), B ( ...
0
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0answers
25 views

How would one define polynomials over the projective line $P_K^1$

May $K$ be a field. If I set $\varXi=(X:Y)$ as a "projective variable" and "projective coefficients" $a_k=(x_k:y_k)\in P_K^1$ - may I then write a polynomial map $P_K^1\longrightarrow P_K^1$ in a form ...
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0answers
34 views

Open subscheme in special fiber

Let $X$ be a projective scheme over $R$ a discrete valuation ring with generic fiber irreducible. Can an open subscheme of $X$ be contained in the special fiber of $X$ ? Or is it true that every open ...
2
votes
1answer
29 views

Three points are collinear iff the determinant of the matrix of their barycentric coordinates vanishes

Let $A,B,C\in \mathbb{R}^2$ be noncollinear points. Then we have that for every point $P\in\mathbb{R}^2$ there exist $\alpha_1,\alpha_2,\alpha_3\in\mathbb{R}$ such that $P=\alpha_1A+\alpha_2 ...
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0answers
27 views

Intuitive argument for the transitivity of $PGL(n+1)$ acting on $\mathbb{P}^n$

In spherical geometry we can consider the action of $\operatorname{O}(n+1)$ on the unit sphere $\mathbb{S}^n$. It's easy to see that this action is transitive, because for any two points $x,y \in ...
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0answers
58 views

What is the degree of the pull back of a line bundle?

Let $X=\mathbb{P}^2$, and let $(y_1,y_2,y_3)$ be homogeneous coordinates on $X$. Consider a map $\phi:\mathbb{P}^1\longrightarrow X$, given by $\phi(x_1,x_2)=(x_1^2,x_1x_2,x_2^2)$, where $(x_1,x_2)$ ...
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1answer
43 views

Complete intersection curve

I have two very basic questions/clarifications. Let $X=\mathbb{P}^n_k$, and let $Y$ be a subvariety of $\mathbb{P}^n_k$ of dimension $m$. Then we say that $V$ is a complete intersection variety if ...
2
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1answer
20 views

Perspective projection of a sphere on a plane

I know the perspective projection of a sphere on a plane is an ellipse. How would I find the parametric equation for this ellipse? Say I have a camera at $(0, 0, z_2)$, a plane at $z=z_1$, and a ...
3
votes
1answer
84 views

Affine varieties over finite fields

I read in this paper (http://www.math.iitb.ac.in/~srg/preprints/Chandigarh.pdf) that the following set is an affine variety: $V_f=\{(t_0,...,t_N)\in \mathbb{F}_p^{N+1} : f(t_0,...,t_N)=0 \}$ where ...
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1answer
15 views

The significance of the 3-dimensional plot from homogeneous coordinates of a 2-dimensional function

If $(x,y)$ are your standard Cartesian coordinates and $(X,Y,Z)$ are homogeneous coordinates, then $x=X/Z$ and $y=Y/Z$. So if we have a function $f(x,y)$ we can convert it to a function $F(X,Y,Z)$ by ...
2
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2answers
24 views

The simplest way to find a parametrization of the plane projective curve $XZ-Y^2=0$.

I have to explain to some first year math students that the projective algebraic set $\textbf{Z}(XZ-Y^2)\subset\mathbb P^2_k$ is $$V=\{(a^2_0:a_0a_1:a^2_1)\subset\mathbb P^2_k \,:\, \textrm{for}\; ...
3
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0answers
43 views

Weighted blow-ups

I would like to understand what's a weighted blow-up in a very simple case: $\mathbb{C}^2$ blown-up in the origin with weights $(a,b)$. In found some notes online saying that this is the surface $X$ ...
5
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2answers
148 views

Characterizations of the cross-ratio

$$ (z_1,z_2;z_3,z_4) = \frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)} $$ What are the most prominent or most interesting theorems of the following form? Theorem: The cross-ratio is the only function ...
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0answers
11 views

A couple of doubts regarding the proof of the Desargues theorem

I've just seen the Desargues' theorem and its proof. Now I have a couple of doubts. In the 3D space when the triangles are not coplanar, it's obvious that if the lines intersect at a finite point, ...
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1answer
30 views

The projective space of all lines through the origin

I have a question to the following example: Assume that $\mathbb{A}_2$ is an affine plane over a field $\mathbb{K}$, and we have fixed affine coordinates $x, y$ on $\mathbb{A}_2$. Let $\mathbb{P}$ be ...
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13 views

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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60 views

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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2answers
69 views

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 ...
5
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0answers
91 views

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. ...
2
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1answer
52 views

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 ...
2
votes
1answer
59 views

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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1answer
33 views

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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0answers
13 views

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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1answer
18 views

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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1answer
27 views

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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1answer
49 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 ...
4
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1answer
52 views

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 ...
2
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42 views

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 ...
0
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1answer
8 views

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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0answers
22 views

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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14 views

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 ...