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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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Bounded domains in the complex projective space.

Let $\Omega$ be a bounded convex domain of $\mathbb{C}^3$ and, $$\pi:\mathbb{C}^3\setminus\lbrace0\rbrace\rightarrow \mathbb{C} P^2$$ be the canonical projection into the complex projective plan $\...
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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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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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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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Object's plane on image [closed]

I have a segmented image. So, for example, I know 2D coordinates of pixels for road on image. Now I want to know a plane of this road. How can I do this? I assume that decision rests on Ransac, The ...
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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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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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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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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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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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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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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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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 = (\...
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Fundamental group of $Q:=\{[x,y,z,t,u,v] \in \mathbb{R}P^5 |x^2+y^2+z^2-t^2-u^2-v^2=0\}$ and a covering of a quadric

I was doing some exercises in order to prepare myself for the writing exam of General Topology. We didn't give enough attention to quadrics, so this exercise is giving me hard times, any tips about it?...
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Camera Location ( Projective Geometry - Computer Vision)

I want to find the location of a camera in 'world coordinates' based on projective geometry. My source of theory is the book Multiple View geometry in Computer Vision, as well as A Flexible New ...
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Relationship between Affine definition of singular point and projective definition

Let $C : F(X,Y,Z) = 0$ be a projective curve given by a homogeneous polynomial $F \in \mathbb{C}[X,Y,Z]$, and let $P \in \mathbb{P}^2$ be a point. Prove that $P$ is a singular point of $C$ if ...
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A question about morphism of projective spaces

Consider the morphism: $$ f: (\mathbb{P}^2 -\{(0:0:1),(0:1:0) \} )\to \mathbb{P}^3 $$ Given by $f((x:y:z))=(x^2:xy:xz:yz)$, my problem is to find the closure of the image of $f$, my argument was: ...
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How to complete a noisy perspective rectangle?

I have an image of a rectangle that is noisy (perspective image). I know the value of $x$, $y$, and angle $3$. Angles "$2$" and "$4$" are not accurate. Also, I know the real rectangle size. The angle ...
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Checking smoothness of a projective variety

Let $X$ be the proj variety defined by $X= Z(x_1^2-x_0x_2, x_3^2-x_2x_4)$ I want to check the singular points of that. The Jacobian is given by: $$ \begin{bmatrix} -x_2 & 2x_1 & -x_0 & 0 ...
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Geometric interpretation of the Logarithm (in $\mathbb{R}$)

(Note: limited to $\mathbb{R}$) (Note: Geometric here means with straightedge and compass) Standard approaches to introducing the concept of Logarithm rely on a previous exposition of the ...
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About the image of an affine variety into a projective space.

Assume to have a morphism $f: \mathbb{A}^n \to \mathbb{P}^m$, I want to compute the ideal of the the projective closure of $ f(\mathbb{A}^n)$. I think that in general the image of an affine variety ...
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Visualising dot product P.l in Projective geometry

I am trying to understand how the following identity in homogeneous coordinates comes about and how to visualise the two multiplying parts P and l in the 3d cartesian coordinates. Are they orthogonal ...
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Multiplicity and degree of irreducible projective subschemes.

Suppose $X \subset \mathbb{P}^n$ is an irreducible projective scheme. Then its multiplicity $\mu_X$ is defined as the length of the local ring $\mathcal{O}_{X,\eta}$ over itself, where $\eta$ is the ...
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FInding pre-image of the Veronese map of degree d

I'm having trouble understanding how to find the preimage of the degree $d$ Veronese map, following these steps: (the projective space are the projectivization of $k^{n+1}$ and $k^{N+1}$, where $k$ ...
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Affine rectification via vanishing line

I'm trying to understand how to rectify an image given some lines that should actually be parallel in the final image. For example: from the book Multiple View Geometry. I know that the idea is to ...
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Prove that a map is open in the Zariski topology

Say $k$ is an algebraically closed field and define the equivalence relation on $k^{n+1}$ given by $x \sim y \iff x=\lambda y $ for some $\lambda \in \mathbb{k}^{\times}$. Clearly $\mathbb{P}^{n} = k^{...
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Perspective projection plane, calculating squares on the plane

Let's say I have a road I'm looking at from the top, have a square on it. Then I have a different location from which I look at the road, the square now is a convex quadrilateral. https://i.imgur.com/...
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Show that these 3 projective lines intersect on the same point

I'm stuck with the following problem of projective geometry from an assignment: Let $P_1$, $P_2$, $P_3$ and $Q$ be four points in the projective plane (over an algebraically closed field) such that ...
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Classification of Complex Surfaces. Illustrations of the failure of genus to provide a good classification.

Suppose that $X$ is a complex projective curve, i.e., a compact Riemann surface. In this case, a very useful invariant of such objects is the genus, a topological property of the manifold which ...
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Unique projective maps preserving lines

Let $\mathbb{P}(V)$ be a projective space of dimension $3$ and let $L_i$, $i=1,2,3$ be pairwise non-intersecting lines in $\mathbb{P}(V)$. If $\phi: L_1 \to L_1$ is a projective transformation, prove ...
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Intriguing geometry problem regarding isogonal lines

A line $r$ contains the points $A,B,C,D$ in this order. Let $P\notin r$ such that $$\angle APB=\angle CPD$$ Denote furthermore by $G$ the intersection of the angle bisector of $\angle APD$ and $r$. ...
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How to find the equation of an image under a central projection

Let $\pi:\mathbb{P}^3 \to V(x_2) \cong \mathbb{P}^2$ the linear projection with center $P =(0:1:0:0)$. Find the equation for the image of $C=\{(s^3:s^2t:st^2:t^3)|~(s:t) \in \mathbb{P}^1 \}$ under $\...
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Image Analysis using cross ratios

I'm stuck trying to solve an exercise regarding an image analysis. Consider a book that measures 16 cm $\times$ 24 cm lying on a table. Let the vertices of the book be denoted by A,B,C,D and the ...
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Why does the real projective plane / Boy surface look like this?

In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901 My question is, you can see that the Boy surface is made up of three ...
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Determining the matrix of a projection

I have a question about the correctness of my ideas regarding the following exercise. Define $A_0=[(1,0,0,0)]$ $B_0=[(0,1,0,0)]$ $A_1=[(0,0,1,0)]$ $B_1=[(0,0,0,1)] \in \...
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Equivalent Definitions of Lines in Projective Space

I’ve been working with two definitions of lines in $\mathbb{P}_\mathbb{R}^2$, and tried to show their equivalence. The first is that, given two points $a=(a_0:a_1:a_2)$ and $b=(b_0:b_1:b_2)$, the ...
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Project point onto surface using yaw, pitch, roll rotation and standard trig formulas

Given the known coordinates of location A (0,0,50), at which we have a laser pointer aimed downward, I need a series of equations that calculate the projected point onto the flat surface below (x,y,0) ...
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Find a projective change of coordinates

Find a projective change of coordinates that takes the projective completion of the circumference C: $x^2 + y^2 = 1$ to the projective completion of the parabola P: $y^2=2px$, $p \geq 0$ (i.e. $...
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Projective-invariant differential operator

This question has been cross-posted to MathOverflow. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = 0 \...
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Desingularisation of curves (Lorenzini-Invitiation to Arithmetic Geometry, chap 6,ex 7)

Given a nonsingular complete curve over algebraically closed $\bar{k}$, which is interpreted as a field $\bar{k}(X)$ of transcendence degree 1 and its set of valuations trivial on $\bar{k}$, we may ...
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Do all projections matrices take this form?

Do all projection matrices take the form $P = A{(A^TA)}^{-1}A^T$? If so, can you help me derive it and explain it intuitively?
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What transformations can be set by projecting a straight line onto a straight line

What transformations can be set by projecting a straight line onto a straight line (without adding an infinitely distant point)? I said that the homothety with coefficient $k \neq 1$ and the ...