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

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Arc length and radius of a helix

I have a cylinder of diameter $7.5\operatorname{cm}$, I want to make a helix with angle $19^o$ from horizontal plane. What will be the profile of the helix on the helix plane? Will it be circular and ...
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12 views

Eliminate asymptote using projective transform

I have a well-behaved curve $f:\mathbb{R}\rightarrow \mathbb{R}^2$ which has exactly one linear asymptote passing through points $p$ and $q$ in $\mathbb{R}^2$. I would like to find a projective ...
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31 views

Inclusion Mapping?

Is the hooked arrow map notation not supposed to mean an inclusion mapping? His definition is clearly showing inputs from $\mathbb{R}^2$ who's images are elements in $\mathbb{R}^{3}$ with fixed $z$ ...
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How can I determine distance of an object over a surface from two images?

I have a picture of an object over a surface. The object is not affixed to anything. I know the dimensions of the ball. If needed, I can have multiple pictures of the same scene from different ...
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36 views

General equation of a cone

What is the general equation of a cone in $\mathbb{R}^3$ space? There should be no assumptions about the location of the vertex, direction of the axis or aperture angle, these should all be variable.
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Planar ternary ring point operations

I have the following topic in my exam questions' list: Prove that point operations in a planar ternary ring satisfy field axioms. I know Proposition 1 from this paper but this only says something ...
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25 views

Poncelet's closure theorem

Need some help understanding the proof made by Kneebone and Semple in "Algebraic Projective Geometry". I loose it in the sentence about the (2,2) correspondance. As I understand it, they setup an ...
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2answers
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Why the word “projective” for $PGL_n(\mathbb{F})$?

I wrote the title for this question exactly as I had it exactly in my mind. Let me denote by $G=GL_n(\mathbb{F})$ for simplicity; I was working throughout the previous years many times with the ...
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1answer
46 views

The associated Schubert variety of a flag of subspaces of a vector space.

Let $V$ be a vector space and $W_1 \subsetneq W_2 \subsetneq ... \subsetneq W_\ell \subsetneq V $ a flag of subspaces. The associated Schubert variety is defined as : $ \Omega ( W_{ \bullet } ) = \{ \...
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Triangle in perspective to a given triangle but similar to another

Is it always possible to construct a triangle that is in perspective to a given triangle and have it also be similar to a different given triangle? If you create a triangle in perspective to another, ...
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810 views

The genus of the Fermat curve $x^d+y^d+z^d=0$

I need to calculate the genus of the Fermat Curve, and I'd like to be reviewed on what I have done so far; I'm not secure of my argumentation. Such curve is defined as the zero locus $$X=\{[x:y:z]\...
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Hilbert polynomial for a dimension zero projective variety by taking an affine chart

I am looking at exercise 12.21 from Gathmann's notes on algebraic geometry. I am given a homogeneous ideal $$I \unlhd k[x, y, z] $$ with a dimension $0$ projective locus. WLOG, we assume that this ...
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Reference Request: Complete Proof of Braikenridge–Maclaurin Theorem

Where can I find a reference to a complete proof of the Braikenridge–Maclaurin theorem, which is stated as: If the three pairs of opposite sides of (an irregular) hexagon meet at three collinear ...
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Conics not contained in any plane in $\mathbb{P}^3$

The cubic twisted curve is the most common example of a curve in $\mathbb{P}^3$ which is not contained in any plane. I was wondering if it is possible to find a conic in $\mathbb{P}^3$ that is not ...
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51 views

Exercise about the fundamental 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 continuous closed path (let's say $\...
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Reference for complex curve theory

Recently, I started to study complex curve theory with textbook written by Clemens. The thing is, I think I need a little more references for this study. I think my background is not enough. What I ...
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113 views

Show that any smooth projective curve of genus zero over a field $K$ is isomorphic to a plane conic over $K$

I have the following question: Show that any smooth projective curve of genus zero over a field $K$ is isomorphic to a plane conic over $K$. Assuming that a plane conic is a conic cut by a plane,...
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How to think of 2 intersecting planes in $\mathbb{R}^3$ as a cone?

It is well known that any (possibly degenerated) conic section in $\mathbb{R}P^2$ is given by, up to a projective transformation, a point, a line, two lines or a circle (given by the equation $x_0^2+...
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137 views

Photo image to find the screen orientation

I am trying to find the angle of tilts of a screen using projection of a circle from a source $S$. The light beam falls on the photo screen to expose it and what we get is an ellipse with major axis $...
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31 views

Intersection point of lines in projective geometry

Let $K$ be a field and let $g_1,g_2,g_3,h_1,h_2,h_3,$ be different lines in the projective plane $\mathcal{P^2}$$(K)$, so that $g_1,g_2,g_3$ have one intersection point $A$ and $h_1,h_2,h_3$ have ...
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32 views

How to project a 3d-line (represented in Plücker coordinates) into 2d image

I have a fully calibrated camera setup (that means $K$ and $P = [R|t]$ are known) and want to project a 3d line into the camera image. The 3d line is defined via world coordinates $A$ and $B$ (as $...
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Asymptotes of $(x(t),y(t)) = \bigg(\frac{1+t^2}{2+t^3}, \frac{t}{2+t^3}\bigg)$, collinear points, …

Consider the curve: \begin{equation} (x(t),y(t)) = \bigg(\frac{1+t^2}{2+t^3}, \frac{t}{2+t^3}\bigg) \end{equation} Question 1: What are his asymptotes? Answer: In projective space: $[(2+t^3,1+t^2,t)]$...
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Vanishing polynomial in complex projective space

Assume we are working in $n$-dimensional complex projective space. Why does a (homogeneous) polynomial of degree less than or equal to $d - 1$ which equals $0$ on $d$ points on a line $L$ in ...
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Difference between Projective Geometry and Affine Geometry

I recently started reading the book Multiple View Geometry by Hartley and Zisserman. In the first chapter I came across the following concepts.. Projective geometry is an extension of Euclidean ...
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36 views

Alignment of one 3D Coordinate system to another 3D Coordinate system

I'm working on a project depicted by this picture(taken from internet) where there are different coordinate system involved which corresponds to camera coordinate system and local 3D coordinate system ...
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1answer
57 views

How many Fano Planes Can We Build with the Numbers from $1$ to $35$

The Fano plane is the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point. Assume that ...
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34 views

Triangulation with two camera setup - Result in world or camera coordinate system?

I have some problems to understand how I can triangulate a 3D-point using a two camera setup. Let's assume I'm using a right handed coordinate system and the camera is looking in positive z-...
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37 views

Relationship of camera matrices and real world units

I have problem to understand the relationship between a camera matrix and real world coordinates. Let's say I have a camera, with the following (calibration) parameters: ...
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1answer
46 views

Explain why none of the lines through a point inside a hyperbola is a tangent line to that hyperbola?

Explain why none of the lines through a point inside a hyperbola is a tangent line to that hyperbola? I'm thinking since points must be on the hyperbola in order to be tangent, then they can't be ...
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mean-deviation form, why orthogonal?

This is from my textbook Why the column of the new design matrix are orthogonal? for example, let say $A=\begin{pmatrix} 1& 1& 4\\ 1& 2& 0\\ 1& 3& 2 \end{pmatrix}$ ...
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Non-tangent lines to lines in $\mathbb{P}^3(\mathbb{C})$.

Let $\pi:\mathbb{C}^4\setminus\{0\}\to\mathbb{P}^3(\mathbb{C})$ be the quotient map. Let $Q\subset\mathbb{P}^3(\mathbb{C})$ be a smooth quadric, let $q$ be the quadratic form such that $Q=\pi[\{v\in\...
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cubic surface equation

If $[1,0,0,],[0,1,0],[0,0,1],[1,1,1],[1,3,2],[1,4,3]$ are six points on $P^2$ in general position and $f_0, f_1, f_2, f_3$ are the generators of the four dimensional vector space generated by cubics ...
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Equation for a circle in homogeneous coordinates

The equation for a circle in homogeneous coordinates is given by $(x - aw)^2 + (y - bw)^2 = r^2w^2$. I understand that the center of the circle, given by (a, b) in euclidian space is given by (a, b, ...
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Projection from high dimension to lower, for visualization

I want to project high dimensional data points onto 2D screen coordinates, for visualization purposes. I want to be able to control the angles of projection manually (eg, with the mouse). I have ...
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1answer
26 views

Geometric Significance that 2D Points Form a Line

I'm reading through Multiple View Geometry in Computer Vision, by Hartley and Zisserman, and on page 2 it is stated that points at infinity in the two-dimensional projective space form a line, and in ...
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How large can a set of pairwise disjoint 2-(7,3,1) designs (Fano planes) be?

As wikipedia defines well, the Fano plane is a small symmetric block design, specifically a 2-(7,3,1)-design. The points of the design are the points of the plane, and the blocks of the design are the ...
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Definition of Finite Projective Plane clarification

I do not understand part iii. Why can't there be four collinear points? The Fano plane is an example of a $3$-uniform configuration. What about configurations that are $4$-uniform? You must have $4$ ...
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Reflection is not a collineation

Could you give me a collineation that proves that you can't construct the reflection of a line $e$ across a parallel line $f$ with a straightedge only? (That is, a collineation that maps $e$ and $f$ ...
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2answers
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Construction of a finite projective plane of order $p$, for any prime $p$

I have this construction of a finite projective plane (FPP) of prime order $p$, but I am not sure what's going on. We have already proved that FPPs of order $q$ have $q^2+q+1$ lines and points (if ...
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Projective Geometry: Combinatorially, but not projectively equivalent polytopes

I have a hard time understanding Projective Geometry. My task is to Find two polytopes, that are combinatorially, but not projectively equivalent. What combinatorially equivalent means is ...
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35 views

Degree of maps $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$

In the book I am reading right now, it is defined that for a map $\mu \colon \mathbb{P}^1\rightarrow \mathbb{P}^r$ the degree is the degree of the direct image cycle $\mu_{*}[\mathbb{P}^1]$. We are ...
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Why are degenerate conics not projectively equivalent to nondegenerate conics?

This is what I understand about conics being projectively equivalent. Two conics $C1=V(F)$ and $C2=V(G)$ are projectively equivalent if there is an invertible matrix $A$ such that $F(X,Y,Z)=0$ iff $G(...
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Drawing the reciprocal of a circle through the circle of inversion.

I have a general question about drawing the reciprocals of circles through the circle of reciprocation. I understand inversion and reciprocation are two entirely different things yet somehow ...
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The subgroup of $PGL(V)$ stabilizing a projective configuration

Let $P(V)$ be a projective space and consider the natural action of $G=PGL(V)$ on it. Let $S=\{p_1,\dots, p_k\}$ be a finite set of points in $V$ where $k\geq 2$. Is there any reference about the ...
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An example calculated in Principles of Algebraic Geometry of Griffiths and Harris'.

On page 413 they write: Example. We can now make a second computation for Chern classes of projective space. Let $X_0, \ldots , X_n$ be linear coordinates on $\mathbb{C}^{n+1}$, and let $\mathfrak{...
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Can the equation $x+y+z=1$ describe a sphere?

I know that in a three-dimensional Euclidean space, with the Euclidean distance, $x+y+z=1$ describes a plane. In the same conditions, $x^2+y^2+z^2=1$ would be a sphere (a 2-sphere to be exact). ...
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Background for reading Atiyah's first paper on the twisted cubic

What should an undergraduate know before being able to read Michael Atiyah's A Note on the Tangents of a Twisted Cubic ? Most of the words in the paper look foreign to me, but I'm very intrigued by ...
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18 views

Point, Line Duality

I am currently studying Projective Geometry and have trouble understanding the point-line duality concept. Why is the cross product of two points a line and the cross product of two lines a point?
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Projectivity maps that fix three points

Please check this definition: A projectivity is a bijection $PV\to PW$ induced by an isomophism $\phi: V\to W$ given by $\phi(kv)=k\phi (v)$. Now, i have seen here Old Question that an answer says ...
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Projectivised tangent bundle of 2 sphere

I'm trying to understand how rotations act on the "projectivised" tangent bundle of the sphere. Let $S^2$ be the two sphere and denote by $P(TS^2)$ the tangent bundle where each tangent space $T_xS^...