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

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Background for reading Atiyah's first paper on the twisted cubic

What should an undergraduate student need to 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 ...
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Formula for curvature of a hyperbolic plane

I would like to understand the curvature of a hyperbolic plane better in relation to the underlying Euclidean model and intrinsically without a model. I only consider the Beltrami-Klein model and the ...
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23 views

Curvature of a hyperbolic plane

Consider a projective plane and a real quadric. According to the Klein-Beltrami-model the inside of the quadric is a hyperbolic plane. Klein proved that this plane has a constant negative curvature. ...
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Are there any mathematical equations or a math procedure by which you can define a perspective drawing?

I have seen so many mathematical procedures including matrix. But all don't seem to plot a perspective view. Why is it hard to establish simple vector equations by which you can input the true x,y,z ...
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How to estimate the maximum projection area of a set of spheres?

I have a set of spheres P. The spheres have a known, finite range of radii. It seems that there must be at least one 2 dimensional plane such that the bounding circle around the projection of P onto ...
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How to prove the parallel projection of an ellipsoid is an ellipse?

Take the following ellipsoid in implicit form as an example: $$x^2 + 2 y^2 + 3 z^2 + x y + y z - 2 xz = 5$$ which shows: The parallel projection of the ellipsoid onto $xoy$ coordinate plane can ...
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+200

How to complete Vakil's proof that the composition of projective morphisms are projective when the target is quasicompact?

For this question, a morphism $\pi : X \rightarrow Y$ is projective iff there exists a finite type quasicoherent sheaf $\mathcal{E}$ on $Y$ such that $X$ is isomorphic (as a $Y-$scheme) to a closed ...
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21 views

Orthogonal lines on Mercator projection?

I am currently struggling with the following task: We have two pairs of latitude/longitude which determine a small line segment It is needed to get two pairs of latitude/longitude for a small line ...
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21 views

Question about preserving cross ratios in projective geometry

Let $ABCDEF$ be a cyclic hexagon, such that $AF,BE,CD$ concur. Prove that $(F,D;E,C)=(A,C;B,D)$. I'm relatively new to projective geometry. This problem would be solved by perspectivity through the ...
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427 views

Extension of regular function

This is an exercise in Hartshorne's book. For a quasi projective variety $Y$ with dimension $\geq 2$ and $p \in Y$ a normal point, if $f$ is regular on $Y-\{p\}$ then $f$ can be extended to a ...
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51 views

Finding ratio of cevian lines

I am preparing for an exam and doing some pratice problems. So I'm having a difficult time with this problem. At first I thought the ratio was 2:1 and then I also thought I would be able to use the ...
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1answer
39 views

Proximal operators on Balls (Projection)

I was following this tutorial, In section 21 it is given Proximal operator over a ball $B_\epsilon$ of radius $\epsilon$ as $$\text{Proj}_{B_\epsilon(y)}(u) = y + (u-y) \max({1 , \frac{\epsilon}{||{u-...
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22 views

How are the following two projective maps equivalent?

Any element of $PGL(2,\Bbb{C})$ determines a map $$\Bbb{P}^1\to\Bbb{P}^1$$ $$[z:w]\to [az+bw:cz+dw]$$ in other words, a Mobius map $$[z:1]\to [\frac{az+b}{cz+d}:1]$$ How are the two maps equivalent? ...
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198 views

Is the projective line minus one point always isomorphic to the affine space?

I'm thinking about the following problem: If I take a general point $p \in \mathbb{P}^1$ out of the projective line, is $\mathbb{P}^1 - \{ p \}$ isomorphic to the affine space $\mathbb{A}^1$? I ask ...
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138 views

Good book introducing Inconics

General Requirements Book Conic Sections Triange Geometry Inconics of Triangle Ideal Topics Projective properties arising from inconics like collinearity/concurrency relationships Other basic ...
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getting camera vector from turntable world

so I see that there are a lot of answers for the problem of getting a camera projection given a camera coordinate and direction in the world space. But what if you do the opposite? I have a world ...
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1answer
32 views

Intuitive explanation of Pascal's Theorem

I am wondering why Pascal's Theorem, as well as other 'Euclidean' results in projective geometry like Brianchon's Theorem should be true for not only circles, but also conics in general. Is there ...
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28 views

Conic's connection in projective space

How to show that a conic/quadric in the projective space (real or complex, dimension $n>1$) is connected.
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Simplifying linear combination of matrices

Abstract: I've got a matrix described as a linear combination of matrices, with coefficients computed using scalar products and the likes. I'd like to obtain a simple formula for this matrix, by ...
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1answer
26 views

Finding mapping transform in homogenous coordinate system using vanishing points

This question has been bothering me for some time, help would be appreciated! Suppose we have an image of a building facade with vanishing points at Vx = (x,0) and Vy = (0,y) which are horizontal and ...
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1answer
45 views

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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1answer
16 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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34 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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23 views

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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39 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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28 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
47 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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817 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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95 views

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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52 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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1answer
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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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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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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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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58 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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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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1answer
41 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 $A=\begin{pmatrix} 1& 1& 4\\ 1& 2& 0\\ 1& 3& 2 \end{pmatrix}$ ...