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Questions tagged [projective-geometry]

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

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Distance between homogeneous transforms

I am working in homogeneous coordinates to represent affine transforms in 3-d. As such, my points in $\mathbb{R^3}$ get a fourth coordinate, and my transforms are 4x4 matrices. To be clear, I ...
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Transvection theorem 4.9 of Artin Geometric Algebra $\sigma C = C, \sigma D = -D$

I can't understand quite well a passage from Artin Geometric Algebra (fantastic book btw). He's considering the case of $F_5$, $\beta = \pm 1$, $\gamma=0$. He say: [...] If $\beta=1$, them $\sigma ...
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smooth projective quadric $Q_{n,m}$ in $\mathbb{P}_n(\mathbb{R})$ with planarity $m$

How to show that smooth projective quadric $Q_{n,m}$ in $\mathbb{P}_n(\mathbb{R})$ with planarity $m$, dimension $n$ and $i=p-q$ (where $(p,q)$ is signature of the quadratic form) are related by $2m+i=...
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Find the equation of line d such that H(a,b;c,d)

In the Extended Projective Plane, let a be the line with equation x=0, b the line with equation y=x-4, and c with equation y=-x+4. Find the equation of the line d such that H(a,b;c,d). I ...
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Lines in projective and affine space

I'm trying to understand lines in affine and projective space in order to solve problems 2.15 and 4.13 in Algebraic Curves by William Fulton: https://www.google.com/url?sa=t&source=web&rct=j&...
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Universal mapping property for projective schemes and globalising it to projective bundles

Let $Y = \text{Spec}A$ be a noetherian affine scheme. Let $S$ be a graded $A$-algebra which is finitely generated by $S_{1}$ as an $S_{0}$ algebra. In other words, $S$ looks like $$ S = A[x_{0}, x_{1},...
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An attempt to improve Plücker coordinates

Consider a 2-dimensional subspace of 4-dimensional vector space (of quaternions). It is a line in projective 3-space $P^3$. Let $u,v$ be the following 3-vectors (or vector quaternions) $u=d+m$ $v=d−...
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What knowledge should I have before learning projective geometry? (I'm an 11th grade student)

I wanted to learn projective geometry. I don't know much about it. I came across projective geometry in a book called 'Euclidean geometry in mathematical olympiads' and I was very interested in it. So ...
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38 views

On a sheet of paper it is drawn a point $A$ and 2 lines $L_1$ and $L_2$ which intersect outside the sheet in a point B. [duplicate]

On a sheet of paper it is drawn a point $A$ and 2 lines $L_1$ and $L_2$ which intersect outside the sheet in a point B. How one can draw a line $AB$ using only a ruler?
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On double points of a Projective space

$\underline {Background}$: Let, $ p\in Proj(K[x_0,....,x_n])$ . A double point at $ p\in Proj(K[x_0,....,x_n])$ is the scheme given by the square of the ideal (sheaf) of p. i.e we can consider it ...
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$W=\mathbb{P}(S)$ for some (k+1)-dimensional subspace $S$ of $\mathbb{k}^{n+1}$

Let $W$ be a subset of $\mathbb{P}_n(\mathbb{k})$. It is known that for every affine map $A$ (with which $W$ intersects) $A \cap W$ is a k-dimensional affine space. Is it true that $W=\mathbb{P}(S)$ ...
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Finite projective plane order 11

Consider a finite projective plane of order 11, which is there more of: a) Unordered 4-tuples of lines with a non-empty intersection? b) Unordered 7-tuples of points which belong to the same line? ...
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Projective Transformation for two distinct vertices and a linear objective function

I am trying to understand how to prove the following, which seems to be a quite useful insight in terms of linear optimization. Unfortunately, I have a hard time with projective geometry. I'd greatly ...
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Cross-ratio of points in the real projective plane

I would like to compute the cross-ratio of the points $A,B,C,D \in \mathbb{RP}^2$, in the projective plane, given by: $$ A=(0:1:2) \quad B=(1:2:3) \quad C=(2:3:4) \quad D=(3:4:5) $$ First I want to ...
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Proving that X,Y,Z in Quadrangle is collinear using axiom P5

Let P be a projective plane satisfying P5 and P6. Let ABCD be a complete quadrangle with diagonal points P,Q, and R. So, P=AB (int)CD, Q=AC(int)BD, and R=AD(int)BC. Let X,Y,Z be the points given by H(...
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There cannot be a concept of parallelism in a homogeneous projective space?

Page 4 of my computer vision textbook, Multiple View Geometry in Computer Vision (Second Edition), by Hartley and Zisserman, states the following: Affine geometry. We will take the point of view ...
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Computing the structure sheaf $\mathcal{O}_X(U)$ for $U=X\setminus \mathbb{V}(x_0^2+x_1^2+x_2^2)$ and $X=\mathbb{P}^2$.

I was wondering how one computes $\mathcal{O}_X(U)$ for $U=X\setminus \mathbb{V}(x_0^2+x_1^2+x_2^2)$ and $X=\mathbb{P}^2$ by considering the usual cover of $\mathbb{P}^n$ by affine charts $U_i$. ...
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Vertical lines in the projective plane $P^2$

How do you get that two vertical lines in $P^2$ intersect at $(0 : 1 : 0)$ or how do you calculate it? If we look at two parallel lines, their point of intersection is at $(1 : s : 0)$ with s as the ...
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Eccentricity going to zero — Geometric definition conic

Given a straight line $D$, a point $F\notin D$ and a positive real number $e$, a conic is a subset of ${\cal P}_2$ defined as: $$ \mathcal{C}(e,F,D) = \{M\in {\cal P}_2,\, d(F,M)=e\,d(M,D) \} $$ where ...
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A demonstration with Cross Ratio

In the Extended Euclidean Plane, H, let A=(0,0), B=(5,0), and C=(4,0). Show that there exists a point D=(x,0) for some real number x, such that Rx(A,B;C,D)=pi. I have the formulas for these things. ...
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Image of a line in $\mathbb{P}^2$ under Veronese embedding; a conicc [duplicate]

Consider the Veronese surface $S$ defined by the map: $$\phi: \mathbb{P}^2\rightarrow \mathbb{P}^5$$ where $\phi(x_0,x_1,x_2)=(x_0^2,x_0x_1,x_0x_2,x_1^2,x_1x_2,x_2^2)$. The problem asks to show ...
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The map of graded rings $k[w, x, y, z] \rightarrow k[s, t]$ induces a closed embedding $\mathbb{P}_k^1 \rightarrow \mathbb{P}_k^3$

Show that the map of graded rings $k[w, x, y, z] \rightarrow k[s, t]$ given by $(w, x, y, z) \mapsto (s^3, s^2t, st^2, t^3)$ induces a closed embedding $\mathbb{P}_k^1 \rightarrow \mathbb{P}_k^3$, ...
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Unique line in $\mathbb{P}^4$ intersecting three pairwise non-intersecting lines not in a hyperplane.

I need to show that there is a unique line (in $\mathbb{P}^4$ I assume, or could they also mean in $\mathbb{R}^5$?) that intersects three lines $L,M,N$ which are pairwise non-intersecting and not in ...
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1answer
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Dimension in $\mathbb{P}^4$ of $\langle L,M \rangle \cap N$ with $L,M,N$ pairwise non-intersecting and not in one hyperplane

Given three lines, $L, M, N \in\mathbb{P}^4$, not in one hyperplane and not pairwise intersecting, I need to calculate $$\dim(\langle L,M\rangle\cap N).$$ By the dimension of intersection theorem ...
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What are pairwise non-intersecting lines in $\mathbb{P}^4$

Given three lines, $L, M, N \in\mathbb{P}^4$, not in one hyperplane and not pairwise intersecting, I need to calculate $$\dim(\langle L,M\rangle\cap N)$$. I can however not find a definition for ...
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Line intersecting three lines in $\mathbb{P}^4$ that are not in one hyperplane

Given three lines $L,M,N$ in $\mathbb{P}^4$, not all in one hyperplane, I want to show by example that it is possible that there are multiples lines intersecting $L,M$ and $N$. What I know: A ...
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Is it possible to arrange the routes and the bus stops so that if one route is closed, it is still possible to get from any one stop to any other [closed]

A certain City has 10bus routes. Is it possible to arrange the routes and the bus stops so that if one route is closed, it is still possible to get from any one stop to any other, but if any two ...
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Barycentric and projective coordinates

I have a question: what is the relation between the barycentric and the projective coordinates? Are the first one a particular case of the second? Thank! Edit: the setting is the plane, and in ...
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Dualize this Projective Geometry Statement

In the Extended Euclidean Plane, let l and m be two lines that intersect at the point ). Let A, B, C be three points on l other than o and let A', B', C' be three points on m other than o. Assume that ...
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Finding correct projective change of coordinates $M$.

Find a $3 \times 3$ matrix $M$ such that, under the change of varibles $$\begin{bmatrix} x \\ y \\ z \end{bmatrix} = M^{-1}\begin{bmatrix} x_1 \\ y_1 \\ z_1 \end{bmatrix},$...
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29 views

Projective Plane of order n

Let $P=(S,L)$ be a projective plane of order $n$. Let $K$ be a nonempty subset of $S$ with the property that no three points belonging to $K$ are collinear. Prove that $|K|$ is less than or equal to $...
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Kernel of ${\rm GL}(n,F)$ on ${\rm PG}(n-1,F)$ over a division ring $F$

I am reading Peter Cameron's note on Classical Groups and I got confused with Proposition 2.1 on page 14. I have no problem in proving that the elements in kernel are scalars. However, I don't ...
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Given the projectivity $\textbf x'=H\textbf x$, why is $\textbf x' \times H\textbf x = 0$?

I'm referring to section 4.1 in Multiple View Geometry by Hartley, where the Direct Linear Transformation (DLT) algorithm is explained. I have the intuition that since the points $\textbf x_i'$ and $\...
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Rigorous intro to projective transformations?

I'm a math Ph.D. having worked mostly in analysis, so I'm not too familiar with projective geometry. My job has recently got me into pretty hardcore multicamera computer vision stuff and I was ...
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1answer
28 views

Show that the set of lines in $\mathbb{R}^n$ is a (smooth) manifold of dimension $2(n-1)$

I was recently made aware of the result in the title. It's easy to show for $\mathbb{R}^2$, but I'm having trouble coming up with a generalization for $\mathbb{R}^n$. There are a couple of ways to ...
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1answer
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Number of Rational Points on $C : ax^2 + bxy + cy^2 = dz^2$ over finite field

Let $p \neq 2$ be a prime, let $a,b,c,d \in \mathbb{F}_p$ satisfy $acd \neq 0$, and let $C$ be the conic given by the homogeneous equation $$ C : ax^2 + bxy + cy^2 = dz^2. $$ a) If $b^2 \neq 4ac$...
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Triangle Inequality for Angles in Projective Space

I want to show that the angle between two lines through the origin in a (complex or real) inner product vector space $(V,\langle \cdot,\cdot\rangle)$ is a distance function which turns $\mathbb{P}V$, ...
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Disjoint Lines in Projective Space [duplicate]

I’m trying to prove the following result: Given $3$ pairwise disjoint lines $L_1$,$L_2$ and $L_3$ in $\mathbb{P}^3(k)$, we can find a change of coordinates such that $L_1=V(Z,T)$, $L_2=V(X,Y)$ and $...
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1answer
33 views

Explicit formula for the projection from the line to an arbitrary circle

Overview: Given an arbitrary point $t$ on the horizontal axis of a Cartesian plane and a point $\textbf{p}$ on a circle, I would like to find the point $\textbf{t}'$ located at the intersection of the ...
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1answer
47 views

Transforming Quadrics in Characteristic 2

I’m trying to solve the following problem given in a textbook: Let $k$ be an algebraically closed field and $Q=V(F)$ a quadric in $\mathbb{P}^3(k)$, where $F$ is an irreducible polynomial in $X,Y,Z,...
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1answer
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Are superelliptic curves singular?

It is an easy corollary of the Riemann-Hurwitz formula that smooth double covers of $\mathbb{P}^1$ can only be branched over an even number of points. Let $F(x,z) \in \mathbb{C}[x,z]$ be a homogeneous ...
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What would be a geometric Euclidean interpretation of the homogeneous plane (0,0,0,8)?

Homogeneous coordinates have one dimension more than the corresponding Euclidean coordinates. The Euclidean origin can be described with projective coordinates as (0,0,0,1). So, geometrically, what ...
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1answer
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Find the dimension of the sum of three planes

I have the following problem: Let $\pi_1$, $\pi_2$, $\pi_3$ be three planes in a projective space such that $\pi_1 \cap \pi_2 \cap \pi_3$ = $\emptyset$, $\dim\,(\pi_1 \cap \pi_2) = 1$, $\dim\,(\...
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Projective geometry and planes equations

Let $d_1, d_2$ and $d_3$ be three non coplanar concurrent lines and $O=d_1\cap d_2\cap d_3$. In each line we place three points (2 by 2 distinct) $A_i,B_i$ and $C_i$, $i\in \{1,2,3\}$such that $A_i\in ...
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Examples of finite projective planes with combinatorial properties

I just learned of the Fano plane which looks like it has some interesting combinatorial properties. I am not too versed in this area so wanted to ask if there are some good resources out there on ...
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Inclusion–exclusion principle in projective geometry

In the problems that I have to apply Grassmann in projective geometry, can I use the inclusion-exclusion principle? Consider the following problem: We consider three linear varieties of dimension ...
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1answer
31 views

Projective Geometry: Prove that the mapping in P2R is not well-defined.

Prove that the mapping F: P2(R) to P2(R) given by F(x1,x2,x3) = (x1x2, x2, x3) is not well-defined. I know that to determine whether a mapping is well-defined, you should pick two points that are ...
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2answers
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Mathematics and the art of linearizing the circle

[I edited the question and put stronger emphasis on "constant curvature" than on "naturalness".] One of the most prominent problems of ancient mathematics was the squaring of the circle: to construct ...
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1answer
31 views

Would finding perspective distance using infinite subdivisions be impossible due to self-recursion?

I am currently trying to solve a perspective problem. Say you had a projected 3D rectangle (not neccesarily square) face defined by the four points $P_0,P_1,P_2,P_3$ as follows: My goal is to obtain ...
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Complex Projective Line is homeomorphic to the 2-sphere

I was wondering if someone could help me out with proving the fact that the complex projective line is homeomorphic to the 2-sphere. I've defined the complex projective line $\mathbb{CP}^1 = (\...