# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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: $$(x(t),y(t)) = \bigg(\frac{1+t^2}{2+t^3}, \frac{t}{2+t^3}\bigg)$$ 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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### 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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### 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: ...
This is from my textbook Why the column of the new design matrix are orthogonal? for example, let say $A=\begin{pmatrix} 1& 1& 4\\ 1& 2& 0\\ 1& 3& 2 \end{pmatrix}$ ...