# 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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### Rigorous intro to projective transformations?

I'm a math Ph.D. having worked mostly in analysis, so I'm not too familiar with projective geometry. My job has recently got me into pretty hardcore multicamera computer vision stuff and I was ...
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### Show that the set of lines in $\mathbb{R}^n$ is a (smooth) manifold of dimension $2(n-1)$

I was recently made aware of the result in the title. It's easy to show for $\mathbb{R}^2$, but I'm having trouble coming up with a generalization for $\mathbb{R}^n$. There are a couple of ways to ...
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### Number of Rational Points on $C : ax^2 + bxy + cy^2 = dz^2$ over finite field

Let $p \neq 2$ be a prime, let $a,b,c,d \in \mathbb{F}_p$ satisfy $acd \neq 0$, and let $C$ be the conic given by the homogeneous equation $$C : ax^2 + bxy + cy^2 = dz^2.$$ a) If $b^2 \neq 4ac$...
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### Object's plane on image [closed]

I have a segmented image. So, for example, I know 2D coordinates of pixels for road on image. Now I want to know a plane of this road. How can I do this? I assume that decision rests on Ransac, The ...
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### Triangle Inequality for Angles in Projective Space

I want to show that the angle between two lines through the origin in a (complex or real) inner product vector space $(V,\langle \cdot,\cdot\rangle)$ is a distance function which turns $\mathbb{P}V$, ...
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### Are superelliptic curves singular?

It is an easy corollary of the Riemann-Hurwitz formula that smooth double covers of $\mathbb{P}^1$ can only be branched over an even number of points. Let $F(x,z) \in \mathbb{C}[x,z]$ be a homogeneous ...
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### What would be a geometric Euclidean interpretation of the homogeneous plane (0,0,0,8)?

Homogeneous coordinates have one dimension more than the corresponding Euclidean coordinates. The Euclidean origin can be described with projective coordinates as (0,0,0,1). So, geometrically, what ...
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### Examples of finite projective planes with combinatorial properties

I just learned of the Fano plane which looks like it has some interesting combinatorial properties. I am not too versed in this area so wanted to ask if there are some good resources out there on ...
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### Inclusion–exclusion principle in projective geometry

In the problems that I have to apply Grassmann in projective geometry, can I use the inclusion-exclusion principle? Consider the following problem: We consider three linear varieties of dimension ...
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### Projective Geometry: Prove that the mapping in P2R is not well-defined.

Prove that the mapping F: P2(R) to P2(R) given by F(x1,x2,x3) = (x1x2, x2, x3) is not well-defined. I know that to determine whether a mapping is well-defined, you should pick two points that are ...
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### Mathematics and the art of linearizing the circle

[I edited the question and put stronger emphasis on "constant curvature" than on "naturalness".] One of the most prominent problems of ancient mathematics was the squaring of the circle: to construct ...
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### Would finding perspective distance using infinite subdivisions be impossible due to self-recursion?

I am currently trying to solve a perspective problem. Say you had a projected 3D rectangle (not neccesarily square) face defined by the four points $P_0,P_1,P_2,P_3$ as follows: My goal is to obtain ...
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### Image Analysis using cross ratios

I'm stuck trying to solve an exercise regarding an image analysis. Consider a book that measures 16 cm $\times$ 24 cm lying on a table. Let the vertices of the book be denoted by A,B,C,D and the ...
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### Why does the real projective plane / Boy surface look like this?

In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901 My question is, you can see that the Boy surface is made up of three ...
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### Projective-invariant differential operator

This question has been cross-posted to MathOverflow. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = 0 \...
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### Desingularisation of curves (Lorenzini-Invitiation to Arithmetic Geometry, chap 6,ex 7)

Given a nonsingular complete curve over algebraically closed $\bar{k}$, which is interpreted as a field $\bar{k}(X)$ of transcendence degree 1 and its set of valuations trivial on $\bar{k}$, we may ...
Do all projection matrices take the form $P = A{(A^TA)}^{-1}A^T$? If so, can you help me derive it and explain it intuitively?
What transformations can be set by projecting a straight line onto a straight line (without adding an infinitely distant point)? I said that the homothety with coefficient $k \neq 1$ and the ...