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

learn more… | top users | synonyms

1
vote
0answers
19 views

Relativistic Projective Geometry

If we assume that space-time has an extra two dimensions so that there is more symmetry between space (with 3) and time (now with 3). What would the corresponding cross ratio equation look like if we ...
1
vote
0answers
20 views

Relativity and Projective Geometry [migrated]

How do you identify the cross ratio equation of projective geometry from the hyperbolic geometry of relativity? Specifically, what relativistic variables would correspond to A,B,C,D in the standard ...
1
vote
1answer
17 views

determine cube orientation given one side in a perspective projection

Suppose that we are given an arbitrary quadrilateral T that does not have any parallel edges. I want to draw a cube in a three-point perspective projection such that T is one of its sides. The ...
2
votes
1answer
15 views

Lines and projective isomorphism

Let $\rho : P(\mathbb{R^3}) \to P(\mathbb{R^3})$ be an homography, this is, a projective isomorphism induced from the isomorphism of vector spaces. I'm trying to understand what information about ...
3
votes
0answers
34 views

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 ...
5
votes
2answers
98 views

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
votes
1answer
37 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 ...
0
votes
0answers
11 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 ...
2
votes
0answers
34 views

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.
2
votes
0answers
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): ...
1
vote
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 ...
0
votes
0answers
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 ...
0
votes
0answers
22 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
votes
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
votes
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. ...
4
votes
0answers
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
29 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 ...
1
vote
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?
1
vote
0answers
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
votes
2answers
114 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). ...
0
votes
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
votes
0answers
49 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
55 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
votes
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
votes
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 ...
0
votes
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 ...
1
vote
0answers
29 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 ...
1
vote
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 $(x_1,x_2)$ ...
1
vote
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
votes
1answer
21 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
85 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 ...
1
vote
1answer
17 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
votes
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
votes
0answers
44 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
votes
2answers
149 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 ...
0
votes
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, ...
1
vote
1answer
31 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 ...
0
votes
0answers
13 views

Application of projective duality

What are the applications of the projective duality principle in another scientific areas such as Physics and Chemistry?
1
vote
0answers
64 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 ...
1
vote
2answers
71 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
votes
0answers
92 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
votes
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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
1answer
28 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 ...
0
votes
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
votes
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 ...