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
2answers
29 views

Area of piece of paper folded around straight line of orientation $\theta$

Imagine drawing a straight line $l$ through the center of a square piece of paper with area $1$. Now fold the paper along that line. Q: What is the function for the area covered by the folded ...
4
votes
2answers
82 views

Exact Expression for numerical Solution 0.9595767

I need you to do just what any math genuis in a shallow Hollywood movie does: looking at big tables of numbers and seeing exact structure! These $3 \times 3$ matrices are solutions to a well-posed ...
2
votes
1answer
15 views

Two new conics associated with two triangles inscribed a conic

1-Let $ABC$ and $A'B'C'$ be two triangle inscribed a conic. Let $P$ be a point on the conic. $PA$ meets $B'C'$ at $A_1$, define $B_1, C_1$ cyclically. $PA'$ meets $BC$ at $A'_1$ define $B'_1, C'_1$ ...
0
votes
1answer
43 views

The set of polynomials which “cut out” smooth subsets of projective space is open and dense

Let $k[x_0,x_1,...,x_n]_d$ be a space of all forms (in other words, homogenous polynomials) of degree $d$ of variables $x_0, x_1,...,x_n$ over algebraically closed field $k$. Let's think of ...
-3
votes
2answers
33 views

How to determine that the 3 points given in homogeneous coordinates are collinear? [on hold]

How do I prove that the 3 points given in homogeneous coordinates are collinear? $$A=(1,3,2)^T, B=(0,6,8)^T, C=(3,3,-2)^T$$
0
votes
0answers
19 views

Transform in eigenvetor space

Hello I have a squared matrix C C = 0 2.2361 63.7887 2.2361 0 61.6117 63.7887 61.6117 0 and I calculate its ...
1
vote
1answer
61 views

Image of a line or conic on Veronese surface.

This is part of Exercise 5.13 from Undergraduate Algebraic Geometry by Reid: Consider the Veronese surface $S$ defined by the map: $$\phi: \mathbb{P}^2\rightarrow \mathbb{P}^5$$ where ...
2
votes
1answer
53 views

Is the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ birational?

This is Exercise 5.3 (a) in Undergraduate Algebraic Geometry by Reid. Does the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ define a rational map? Determine ...
0
votes
1answer
26 views

Irreducibility of an affine variety in an affince space vs in a projective space.

Proposition 5.5 in Undergraduate Algebraic Geometry by Reid says (I only write down a brief idea since the proposition is long and involves some other notations to define): The affine variety $U$ ...
1
vote
0answers
20 views

Homography with line correspondences

When calculating a homography with line instead of point correspondences, what is the derivation of the formula: $$ l_i = H^T\cdot l^{'}_i $$ I know that: $$ l^T\cdot x = 0 \quad\text{and}\quad ...
1
vote
1answer
31 views

Differential geometry of projective bundles

Can someone give me a reference about projective bundles from a differential-geometric point of view? I am not very familiar with algebraic geometry. I would like, for example, some theory about when ...
0
votes
0answers
25 views

Projective Transformation best fit for multiple 4-point sets

I have a photo that shows some (for example 10) jumbled paper sheets, each with known size, situated on a plain large table. The distances between these sheets are unknown. So I would have 10 ...
1
vote
1answer
13 views

Circle equation in homogeneous coordinates

Can someone give me a derivation why the circle equation is expressed in homogeneous coordinates like this (as described in Hartley): $$ (x-a\cdot w)^2 + (y-b\cdot w)^2 = r^2\cdot w^2 $$
1
vote
1answer
28 views

Number of points on a line in a finite projective plane

I've been reading some proofs regarding finite projective planes of order n, and often they start out by assuming that each line contains n+1 points. Is this a fact that follows from the axioms for ...
0
votes
0answers
13 views

Project 4 cones onto a sphere

I have four cones. The angle of each cones is 140 degree. I need to project it onto a sphere(place it ) such that, the cones cover the maximum area with minimum overlap. I initially thought that ...
0
votes
0answers
20 views

$OABCD$ tetrahedron with $OA ⊥ OB ⊥ OC ⊥ OA$

I've got stuck at this problem: Let $OABCD$ be a tetrahedron with $OA ⊥ OB ⊥ OC ⊥ OA$. If $OH$ is the orthocentre of triangle $ABC$, show that $OH$ is perpendicular on plan $(ABC)$. Then ...
0
votes
1answer
24 views

Group law on elliptic curves.

Let $k$ perfect field. If we have a cubic non-singular projective curve $C(k)$ (over a field $k$), take two diferent points $P_1,P_2 \in C(k)$ and consider the line through the points, by Bezout ...
0
votes
1answer
38 views

exercize about the foundamental group of $\mathbb{P}^n(\mathbb{R})$

Let $p$ be a point in $\mathbb{P}^n(\mathbb{R})$ and $\Sigma$ the set containing all the projective lines passing through $p$. Given $s\in \Sigma$ we can define a continous closed path (let's say ...
0
votes
1answer
22 views

Finite geometry - how to determine parallel classes

I try to learn a little about finite geometry and I have now encountered the following exercise: Exercise: Construct the affine plane $\mathrm{AP}(\mathbb{Z}_3)$. Determine it's parallel classes ...
1
vote
1answer
18 views

Projection along an axis

I have problem with understanding what projection along an axis means in practise. For example I have two paraboloids $H=\{(x,y,z)\in \Bbb{R}^3:z=2xy \} $ and $E=\{(x,y,z)\in \Bbb{R}^3:z=x^2 +y^2 \} $ ...
0
votes
0answers
18 views

Isomorphism between semi-orthogonal group $O(2,2)$ and direct product of projective general linear group $PGL(2,\mathbb{R})$ with itself

Is there any natural isomorphism between semi-orthogonal group $O(2,2)$ and direct product of projective general linear group $PGL(2,\mathbb{R})=GL(2,\mathbb{R})/R^+$ with itself?, where $O(2,2)$ is ...
3
votes
0answers
50 views

What was the genesis of Hua's identity?

Many resources I have read prove Hua's identity more-or-less mechanically. I have seen there is more than one raison d'être for Hua's identity: e.g. its connection to the fundamental theorem of ...
0
votes
0answers
24 views

$V\subset \mathbb{P}^n(k)$ is irreducible, prove that $\dim(V)$ is the transcendence degree of $k(V)$ over $k$.

This is a problem from Ideals, Varieties, and Algorithms by Cox et. al. $V\subset \mathbb{P}^n(k)$ is irreducible, prove that $\dim(V)$ is the transcendence degree of $k(V)$ over $k$, where $k(V)$ ...
2
votes
1answer
48 views

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 ...
1
vote
1answer
22 views

Can 3 transformations (V, Σ, U) of SVD to describe a perspective transformation?

As known SVD (Singular value decomposition) is a factorization of the form M = UΣV∗. https://en.wikipedia.org/wiki/Singular_value_decomposition SVD of the linear map T can be easily analysed as a ...
2
votes
0answers
15 views

Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
0
votes
2answers
64 views

Why are orthogonal projection matrices not … orthogonal?

I know that given an orthogonal matrix U, then orthogonal projection onto the column space of U is represented by the matrix $UU^t$, which is again orthogonal. I've computed these types of matrices ...
0
votes
0answers
15 views

Projection of an oblique circle on XZ plane

While going through an exercise of surface integration, I got confused in this problem.The surface is the intersection of sphere $S:x^2+y^2+z^2-1=0$ and the plane $P:y-x=0$. Clearly, the curve of ...
0
votes
0answers
36 views

Why is cot(a) function used in perspective projection?

I'm working with the different space projections. But I wonder about the perspective projection. Let me remind you one template, which may be used in 3D rendering software: ...
2
votes
1answer
24 views

Projecting a sphere from inside

I am trying to make a renderer for a programming project, and yet I am having trouble projecting the points onto the screen (the way it works so far, the camera can't look down on a face because the ...
4
votes
0answers
123 views

Two twisted cubic curves in $\mathbb P^3$ intersect iff they lie in a common cubic surface

Let $C_1$ and $C_2$ be twisted cubic curves in $\mathbb P^3$. I want to prove that they intersect if and only if they lie in common cubic surface, perhaps singular. The second condition can be ...
0
votes
0answers
14 views

Why do we have to normalize the eigen-vectors before orthographic projection?

Given such a matrix about the grades of 6 students in Maths, Computer Sciences and French: \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 2 \\ 2 & 2 & 1 ...
1
vote
0answers
24 views

“General position” in a poset

Does anyone have a reference for this notion of "general position" in a poset: A set $S$ in a poset $(X,\le)$ is said to be in general position if for any $A,B\subseteq S$, $\{x:\forall y\in ...
7
votes
1answer
67 views

How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of tangent $2$-planes? [duplicate]

A manifold $M$ is said to admit a field of tangent $k$-planes if its tangent bundle admits a subbundle of dimension $k$. How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of ...
8
votes
2answers
67 views

Diffeomorphism between $\mathbb{P}^n$ and the submanifold of $\mathbb{R}^{(n+1)^2}$ consisting of certain matrices?

Let $\mathbb{P}^n$ denote the set of all lines through the origin in the coordinates space $\mathbb{R}^{n+1}$. Define a function$$q: \mathbb{R}^{n+1} - \{0\} \to \mathbb{P}^n$$ by $q(x) = \mathbb{R}x ...
4
votes
1answer
45 views

Tangent Space to Grassmannian

I have a second question today. In Harris' "Algebraic Geometry: A First course" he constructs (on page 200) an isomorphism between the tangent space of the Grassmannians and some homomorphisms: He ...
0
votes
0answers
21 views

Metric Image Rectification using Camera Angle and Focal Length

I'm trying to measure the size of an object in millimetres from an close-range image of the object captured with an angled camera. The application is intended to be from a smartphone, so we can't ...
0
votes
0answers
12 views

Transformation Matrix for particular problem

I have a question regarding transformation matrices. I have two images both showing a table. I have coordinates of the corners of the tables, and now I want to apply a transform to 1 of the images so ...
0
votes
3answers
97 views

Basic conceptual questions about orthogonality in Linear Algebra,

I have the Gram-Schmidt algorithm memorized, so that I can always compute an orthonormal basis, when I need it (on pen and paper, I don't studying mathematical / scientific computing ... yet). Could ...
0
votes
0answers
29 views

Poncelet's Porism

Let $E_0$ and $E_1$ be two ellipses contained in the projective plane $\textbf{P}$. Each ellipse bound a disk on one side and a Möbius band on the other. We assume that the disk bounded by $E_1$ ...
0
votes
1answer
39 views

How to compute an orthogonal projection of a vector that is not in the vector space?

I am working on a problem where I found an orthonormal basis of polynomials that span the vector space V of polynomials $\vec p$ of degree at most 2. But part(b) of the problem asks to use part(a) to ...
0
votes
0answers
12 views

Last Row of the perspective projection matrix

Could you explain to me what is the purpose of -1 in the last row of the projection matrix? And how it affects the perspective division step ?
0
votes
0answers
11 views

Construction of perspective matrix

Is it possible to construct a perspective matrix by multiplying a perspective transformation matrix with a parallel projection matrix ?
2
votes
1answer
32 views

Projectivization and stereographic projection, or: Why nonlinear + nonlinear = linear?

This previous question had me thinking about something I've taken for granted. Consider $\mathrm{SL}_2(\mathbb{C})$ acting on $\mathbb{C}^2$, which descends to an action on the complex projective line ...
0
votes
0answers
22 views

Coordinate-free definition of the twistor correspondence

Given a point $x^\mu \in \mathbb{C}^4$, define the $2\times 2$ matrix $x^{AA'}$ by $x^\mu=\sigma^\mu_{AA'}x^{AA'}$. Here, the $\sigma^\mu$ are the van der Waerden symbols, where $\sigma^0=I_2$ and ...
0
votes
0answers
19 views

Homography between the projective plane and the affine plane

I have encountered a simple projective geometry problem studying computer vision that I do not understand completely. Let $P$ be the projective plane of which we have taken an image. Let ...
0
votes
1answer
51 views

$Tor^*_{\mathcal O_{\mathbb P^3}}(\mathcal O_{L_1}, \mathcal O_{L_2})$ for two lines $L_i$ in the projective space

I need to calculate $Tor^*_{\mathcal O_{\mathbb P^3}}(\mathcal O_{L_1}, \mathcal O_{L_2})$, where $L_1$ and $L_2$ are lines on $\mathbb P^3$. If they are intersecting at a point, I believe that they ...
4
votes
3answers
122 views

Is it possible to project orthogonally an ellipse with major and minor axes $2a$,$2b$ so that its image is a circle with diameter $2b$?

Problem: Prove that the area of an ellipse with major axis and minor axis of lengths $2a$ and $2b$,respectively, is $ab \pi$ . Proof: We do this by projecting the ellipse into a figure whose ...
4
votes
1answer
42 views

Convex hull in projective space

Let $S \subseteq \mathbb R\mathbb P^n$ be a closed connected set that does not intersect every hyperplane. If I choose any affine chart containing $S$, I can consider its convex hull, and it seems ...
0
votes
3answers
26 views

What is the projection of the vector $v = [-3, 2, 3]^T$ on the plane with equation $1x + 1y - 2z = 0$?

What is the projection of the vector $v = [-3, 2, 3]^T$ on the plane with equation $1x + 1y - 2z = 0$? So my solution is first i use the component vector formula of ${a \cdot b}\over\bigl|a\bigr| $ ...