# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...