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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What is a rotation number in topology in relation to complex numbers? [closed]

I believe this has something to do with algebraic cycles on complex projective planes?
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The direction of parallel projection of an affine camera?

In this book on page 173, it states Given an affine camera of form: $$P_A = \begin{bmatrix} m_{11}&m_{12}&m_{13}&t_1\\ m_{21}&m_{22}&m_{23}&t_2\\ 0&0&0&1\end{...
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Non singular degenerate conics in the projective plane

I was trying to count degenerate conics in the projective plane $\mathbb{P}^2(\mathbb{F}_q)$ and I discovered what ooks like a "paradox" (I know it's not but I can't solve it). Given a point ...
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What “concept” is invariant through a coordinate transformation here?

I am reading Kunz's "Introduction to Commutative Algebra and Algebraic Geometry" and I'm not understanding what is meant by the underlined sentence in the screenshot below. What exactly is ...
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Countable algebraically closed field inside an incountable algebraically closed of characteristic zero

I would like to know if the following claim is true. I found this claim in a paper without proof :(. Let $k$ an uncountable algebraically closed field of characteristic $0$. Let $S=\operatorname{Spec}...
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How to obtain the principal/optical axis vector of the projective camera model such that it points towards the front of camera

So the goal is to obtain the principal/optical axis vector such that it points in the direction towards the front of the camera (the positive direction) given the camera projection matrix, which can ...
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What are the possible shapes a square can create on a projection plane?

What are the possible shapes a square can create on a projection plane? I have, square kite right trapezoid isosceles trapezoid scalene trapezoid trapezium (U.S.) Are there any I have missed or ...
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Is there a name for this sort of concave region, or points in it?

I am wondering if there is a mathematical term for this, since I'm trying to search up algorithms that deal with these sort of points/regions, but I am having difficulty finding what I'm looking for. ...
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Is it true that the intersection of a plane $\mathbf{\pi}=(\mathbf{v}^T,k)^T$ with the plane at infinity is just a line represented by $\mathbf{v}$?

Is it true in $\mathbb{P}^3$, that the intersection of a plane $\mathbf{\pi}=(\mathbf{v}^T,k)^T$ with the plane at infinity $\mathbf{\pi_\infty}$ is just a line represented by $\mathbf{v}$? In the ...
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Will the perspective projections of an equally sized sphere and circle be congruent?

I am not a mathematician (yet), but rather more of an artist and have a question about perspective which can be described with projective geometry. The diameter of a sphere and a circle are equal. A ...
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Why is a line $\mathbf{l}$ in $\mathbb{P}^2$ defined as a linear combination of its 2D null-space?

Why is a line $\mathbf{l}$ in $\mathbb{P}^2$ defined as a linear combination of its 2D null-space as $\mathbf{x} = \mu \mathbf{a} + \lambda \mathbf{b}$ where $\mathbf{l}^T\mathbf{a} = \mathbf{l}^T\...
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Projecting $\mathbb R^3$ onto $\mathbb RP^2$

I am reading Introduction to Algebraic Geometry by Justin R. Smith, and Exercise 2 of section 1.6 (p.33) states the following: In computer graphics, after a scene in $\mathbb R P^3$ has been ...
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Finding Measuring Points in three-point perspective drawing

In his Complete Guide to Perspective Drawing (page 39), Craig Attebery's places Measuring Points of two-point perspective in a line perperdicular to that crossing the Station Point and the Center of ...
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How to formulate principle of duality in projective geometry in terms of category theory?

In projective geometry, the principle of duality states that any theorem that holds for an incidence structure $(P, L, I)$, where $P$ are the points, $L$ are the lines and $I \subseteq P \times L$ is ...
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Why do I need to normalize before applying SVD/Cholesky Decomposition in this situation?

So to give some context, I am trying to perform metric rectification (ie. remove projection and affine distortion) in an image. I am following the method from example 2.17 on pg 56 of Multiple View ...
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Varieties receiving a morphism from projective space.

Are any interesting family of smooth projective varieties that receive a surjective regular morphism (and maybe finite) from projective space? Now what we replace projective space by varieties ...
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Determining depth in perspective drawing

In his Complete Guide to Perspective Drawing (page 27), Craig Attebery proposes a method for determining depth in a line going towards its vanishing point (VP): VP = Vanishing Point HL = Horizon Line ...
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Meaning and determination of Stand Point in Perspective Drawing

In his Complete Guide to Perspective Drawing, Craig Attebery defines the concept of Station Point as the following: Later on, the author tries to represent this Station Point in a picture plane, ...
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Geometric construction of vanishing points from a length ratio.

Can someone please explain why the vanishing point can be constructed in such a way? The textbook I am studying shows this example without much explanation. From my studies, I know using the cross ...
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Counting dimension of hypersurfaces in $\mathbb{P}^3$ containing three disjoint lines

Assume $k$ is algebraically closed field of char zero. Let $l_1,l_2,l_3$ be three disjoint lines in $\mathbb{P}^3_k $. The dimension of the $k$-vector $\operatorname{Homog}_2 k[x_0, x_1, x_2, x_3]$ ...
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Recovery of affine properties from images

In the textbook I am studying Multiple View Geometry in Computer Vision, Second Edition. Richard Hartley, Andrew Zisserman. pg 49, it is stated that one can recover affine properties from images by ...
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Degenerate conics and its matrix form

I am confused about parts of the following excerpt from a textbook I am studying: Degenerate conics. If the matrix $C$ is not of full rank, then the conic is termed degenerate. Degenerate point ...
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Veronese embedding

I am reading the text "An invitation to Algebraic geometry" by Karen Smith et. al. In the text there is a proposition on Veronese mappings: Prop: The Veronese mapping $v_d$ defines an ...
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Are global sections of varieties determined by their local rings?

Let $(X,O_X)$ be the ringed space associated to a "classical projective" variety over an algebraically closed field $k$. (i.e. $O_X$ is sheaf of regular functions, in the sense of Chapter I. ...
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Proof that the line $\textbf{l}$ tangent to a conic $C$ is $\textbf{l}=C\textbf{x}$

Prove that the line $\textbf{l}$ tangent to a conic $C$ at a point $\textbf{x}$ on C is given by $\textbf{l}=C\textbf{x}$. The equation of a conic in matrix form is defined as $\textbf{x}^TC\textbf{x}=...
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What is the point of representing a conic in homogeneous form?

In this textbook I am studying, they state the following: The equation of a conic in inhomogeneous coordinates is $$ax^2+bxy+cy^2+dx+ey+f=0$$ ie. a polynomial of degree 2. "Homogenizing" ...
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How to get the angle between the inner common tangents to two general ellipse

I have been trying to find the angle made by inner tangents to two general ellipse by following the method described by Futurologist to find four homogeneous equations of tangents, this method is ...
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Solutions to a linear equation over a subfield of a finite field

I know that if we have a line defined over a finite field $\mathbb{F}_p=K$ then this line has p+1 solutions in $P^2(K)$, $p^2+1$ in $P^2(\mathbb{F}_{p^2})$ and so on. But what about the opposite? If ...
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Is every projective variety, zero set of system equations? [closed]

Is every projective variety, zero set of system equations with homogeneous polynomials with same degree ?
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is this theorem true for case quasi-projective?

Definition. If $X, Y$ are two varieties, a morphism $\varphi: X \rightarrow Y$ is a continuous map such that for every open set $V \subseteq Y,$ and for every regular function $f: V \rightarrow k$, ...
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Projective plane and concurrency of three lines formula

I was reading AMD's answer here , and he considers three plane which represent three lines, now for three planes to intersect other than origin it must be that the all the normals together are not ...
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Expressing a projective conic in standard form

I was given the task to find a projective transformation $\phi: \mathbb{P}_\mathbb{C} \rightarrow \mathbb{P}_\mathbb{C}$ that transforms the following conic: $$2\,{x}^{2}+2\,xy-3\,xz+4\,yz+{z}^{2}=0$$ ...
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How to express by a single equation a region delimited by confocal ellipse and hyperbola

I am trying to find the equation of a shape formed by a conic (ellipse) that excludes its intersection with another conic (hyperbola). The hyperbola and ellipse are assumed to share common foci. But ...
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Short Exact Sequence of a Hyperplane in the Projective Space

Let $H$ be a hyperplane in $\mathbb P^n$ where $f:H\rightarrow \mathbb P^n$ is the closed immersion. Let $\mathcal F$ be a coherent subsheaf of $\mathcal E = \oplus\mathcal O_{\mathbb P^n}(l)$. There ...
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using diagonals of the a trapezoid inside a triangle you can find the midpoint of the bases

In the following triangle, AB is parallel to DE. F is an intersection point of the segments BD, AE and CG. Is it true that AG = GB?
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Calculating the distance between two objects with known heights in an image in which forced perspective appears to make them the same height?

I have a photograph containing two objects with known dimensions. Object A, in the foreground, is 23 inches tall. Object B, in the background, is 30 inches tall. In the photograph, however, both ...
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Polytope of cone over a rational normal curve

Consider a rational normal curve $C\subset\mathbb{P}^d$ of degree $d$, and let $W\subset\mathbb{P}^{d+1}$ be a cone over $C$. Since $C$ is a toric variety $W$ is toric as well. I would like to ask ...
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If a line and its points are removed from a projective geometry, is the resulting is affine geometry?

I saw that it's written in a couple of places, but couldn't find proper proof for this. I saw it for example in: here and here I would like to find a reference to a proof of this (as I'm sure some ...
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Different representations of the perspective projection matrix?

So in the textbook, we want to map $(x,y,z,1)$ to $\big(\frac{x}{1-\frac{z}{d}}, \frac{y}{1-\frac{z}{d}}, 0, 1\big)$. Since we are working with homogenous coordinates, this is the same thing as ...
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How to make a cusp catastrophes' graph model in 3D?

I have a problem of finding a way or at least an example how to make a catastrophe model in 3D. I have googled and searched in books, but all I have is a description with no examples, I have only a ...
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Prove the locus of $M$ is a conic passing though $I$ and $J$.

Problem: In the perspective plane $\mathbb{P}^2$, given a conic $(S)$ and a line $d$ having no intersection with $(S)$. Fixed two points $I$ and $J$ on $d$. Take a variety point $M$ such that the ...
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How to count constraints in algebraic geometry?

I am interested in where to look to find techniques to approach the following problem: Let $M$ be a smooth projective manifold, and let $S \subset M$ be some submanifold. Consider on $M$ a complete ...
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dual of an ovoid in $PG(3,q)$

In $PG(3,q)$, a $(q^2+1)$-cap is an ovoid. A $(q^2+1)$-cap is a set of $q^2+1$ points, no three of which are collinear. What is the dual of an ovoid? I know how to get the dual of a statement ...
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Charts for the complex projective line .

Let $\mathbb{C}P^1$ be the complex projective line quotient of $\mathbb C^2 \setminus \{0\}$ by the equivalence relation : $$(z_1,z_2) \sim \;(z_1',z_2') \iff \exists \rho \in \mathbb C^*, \quad (z_1'...
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Are these two curves birationally equivalent?

I'm trying to work out if the following two curves are birationally equivalent: $$Y^2 = 2X^4 + 17X^2 + 12$$ $$2Y^2 = X^4 - 17$$ (I'm considering the above as the affine shorthands for the projective ...
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Projective line $P_{1}$ as bivariate family

Is a projective line $P_{1}$ (brought about by 2 projective points in projective space $P_{n}$ ) a bivariate parametric family? The aim of the question is to get the concept of a basis in a n-...
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Properties of Measuring Points in perspective drawing

In his Complete Guide to Perspective Drawing, Craig Attebery defines a central concept in two-point perspective: Measuring Points, which can be used to determine depth in lines going away from the ...
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How to portray a dome as a projection

I'm currently working on an analysis of a special dome in a church in Rome: The St. Ignazio dome. In this picture I have already provided my analysis as for the vanishing point and vanishing line. ...
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Consider the set of points in $\operatorname{PG}(3,q)$ given by $X_0^2+X_1^2+X_2^2+X_3^2=0$. Show that no plane is disjoint to this set.

My attempt: call this set $\Omega$. I feel like we will have to look at $q$ even and $q$ odd separately. Suppose that $q$ is even, then (since the characteristic of the underlying field is 2), we can ...
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What is the meaning of a homography matrix determinant?

I am trying to find one image (needle) within another (haystack). The method is to find some important points in the needle and try to match them in the haystack. Then to use homography to map the ...

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