Projective geometry is closely related to perspective geometry. These types of geometry originated with artists around the 14th century.
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Mapping an object's projected 3D path to a pre-defined top-down 2D path.
The title of the question may be misleading and the context simpler. Please suggest more appropriate tags for this question.
Consider looking at a plane from two different perspectives.
Perspective ...
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1answer
27 views
Finding the inverse of a map from $CP^1$ to $S^2$
Given the map:
$$f:CP^1 \to S^2\ ,\ f[z:w] = \left(\frac{2\mbox{Re}(w\bar{z})}{|w|^2+|z|^2},\frac{2\mbox{Im}(w\bar{z})}{|w|^2+|z|^2}, \frac{|w|^2-|z|^2}{|w|^2+|z|^2}\right)$$
How would I go about ...
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45 views
Projection property implies Desarguian.
Let $\mathbb{P}$ be a projective plane with the following property:
For every three distinct concurrent lines $K$, $L$ and $M$, along with two projections
$$\pi_a:\ K\ \longrightarrow\ L\qquad\text{ ...
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20 views
Mathematical Basis for Dimetric Projection
For a school project, I need to make a program that can plot $y = f(x,z)$ using a form of dimetric projection. I was given the projection formulae
$$\begin{align*}
x' &= x + sz\cos(\theta)\\
y' ...
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35 views
Desargues' theorem: If the corresponding sides intersect in collinear points, why do the lines joining corresponding vertices have to be concurrent?
What I understand from Desargues' theorem is:
Let $p_{1}$ and $p_{2}$ be two different planes which intersect in a line. Let $l_{1},l_{2},l_{3}\dots$ be lines on $p_{1}$, and ...
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37 views
viewing ray geometry - with multiple aerial photographs
I am working with multiple aerial images. My idea is to model 3d objects (only upper parts). I am having known orientation parameters. As I am new to this field so that, I want to clarify few general ...
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2answers
34 views
Definitions and questions related to projective space $\mathbb{R}P^3$
I have the following questions regarding the definition of a quadric in a real projective space.
What is the precise definition on a quadric of signature (1,1) in the projective space ...
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47 views
Sample Code to Generate Points on the Rim of a Randomly Rotated Cone : What's Going On Here?
Related to this question:
http://math.stackexchange.com/questions/407897/randomly-generate-point-on-shell-from-3-points-2-angles-with-uniform-angle-dis
I'm trying to reverse engineer the ...
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1answer
24 views
Conics in Complex Projective Spaces
I was reading classification of complex hyperquadrics, I am stuck in $\mathbb CP^2$ what is $X_0^2+X_1^2=0$ in $\mathbb CP^2$, ok in $\mathbb CP^1$ this represents just two points, my attempt if $X_1$ ...
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2answers
77 views
Construction of the projective plane over $\mathbb{F}_3$
I have a question about constructing projective plane over $\mathbb{F}_3$. We first establish seven equivalence classes $P= \{ [0,0,1], [0,1,0], [1,0,0], [0,1,1], [1,1,0], [1,0,1], [1,1,1] \}$.
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2answers
47 views
Proving the concurrence of three lines.
Let $p_{1}, p_{2} \text{ and } p_{3}$ be three planes which intersect in a straight line (and not a point, which is generally the case).
Let a fourth plane $p_{4}$ cut these planes (not at the line ...
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27 views
Configuration analogues of projective spaces?
In a configuration, each point is incident to the same number of lines and each line is incident to the same number of points.
The Fano plane is a configuration, with 3 points on each line, and 3 ...
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1answer
27 views
Problem on hyperbolic hyperboloid generated by a rotation
This is the problem:
In $\mathbb{E}^3$ we consider the conic $\gamma$ of equations $x=yz-2=0$ , the line $a$ of equations $x=y+z=0$ and the surface $Q$, that is generated by the rotation of $\gamma$ ...
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1answer
37 views
Why in the affine space can not we use Grassmann formula?
For example, in space three-dimensional affine space generated by two skew lines is all the space three-dimensional, since they are not coplanar.
For this reason it is not worth the Grassmann ...
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1answer
32 views
Possible description of closed subset of a projective variety
Let $k$ be an algebraically closed field. Let $P\subset \mathbb{P}^n(k)$ be a projective variety, and $X\subset P$ be a subset.
Suppose that $X_i = X\cap U_i$ is an affine closed subset for every ...
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1answer
51 views
Find the transform
I have the paper with 3 points on it. I have also a photo of this paper. How can I determine where is the paper on the photo, if I know just the positions of these points? And are 3 points enough?
It ...
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1answer
21 views
Left-ratio and right-ratio in (not necessarily commutative) field
I am reading Birkhoff–Neumann (1936) about (not necessarily commutative) field $F$.
The authors use terms right-ratio and left-ratio in section 13.
Right-ratio is denoted as $[x_1, x_2, ... ...
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12 views
Hilbert Functions of a projective variety
Let $X=\{P_1,P_2,P_3\}$ be the union of three points in $\mathbb{P}_K^2$. Find $h_X(d)$ (hilbert function of $X$ of degree $d$) in the following two cases:
1. $P_1,P_2,P_3$ lie on one line;
2. ...
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1answer
20 views
why the last rule of projective planes looking for 4 points instead of 3?
Looking at the rules of projective planes the rule indicates:
There exists a set of four points, no three of which belong to the same line.
But I'm wondering why should there be a set of 4 ...
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2answers
67 views
Permutations and Cross-ratios
Pick four distinct numbers, list all 24 permutations, and compute the cross-ratio of each permutation. Show that at most six numbers have occurred, given by the cross-ratio group
$y, \frac{1}{y}, 1-y, ...
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134 views
Is a divisor in the hyperplane class necessarily a hyperplane divisor?
Let $V$ be a smooth irreducible projective curve over an algebraically closed field $k$, embedded in some projective space $\mathbb{P}^n$, and let $[H]$ be the induced hyperplane divisor class on $V$.
...
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33 views
How to find a transformation a direct sum of two subspace to null space of one of them
I have a problem in algebra which need your help.
Assume that we have the fixed $N$-dimentional $W$ over the finite field $F_{q^N}$. $U$ and $V$ are the subspace of W, satified that:
...
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0answers
60 views
Irreducible closed subsets of projective varieties
I want to prove the following lemma:
Let $X \subset \mathbb{P}^n$ be a projective variety. Let $W \subset X$ be a closed irreducible set. Then $W$ is also a projective variety.
My idea is as ...
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2answers
26 views
Rational projective variety given by coprime homogeneous polynomials
Let $K$ be algebraically closed field.
Let $f_k, f_{k-1} \in K[x_1,...,x_n]$ coprime homogeneous polynomial of degree $k$ and $k-1$ respectively. I want to prove that:
The variety
$$ X = ...
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0answers
19 views
How to visulize surface link in four dimension?
I am now facing a problem with "surface link" in four dimension. I have heard that three 2-torus can be linked in four dimension. And I have created a movie by cutting four dimensional space with ...
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1answer
94 views
Meaning and types of geometry
I heard that there's several kind of geometries for instance projective geometry and non euclidean geometry besides the euclidean geometry. So the question is what do you mean by a geometry, do you ...
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0answers
31 views
How can I compute the multiplicities of the projection map over a smooth projective plane curve?
Suppose that $F$ is a non-singular homogeneous polynomial in $\mathbb{C}^3$ and let $X$ be its zero locus, which is well-defined in $\mathbb{P}^2$. Consider the function $\pi:X\rightarrow\mathbb{P}^1$ ...
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2answers
90 views
Intersection of smooth projective plane curves
I need to calculate the number of intersections of the smooth projective plane curves defined by the zero locus of the homogeneous polynomials
$$
F(x,y,z)=xy^3+yz^3+zx^3\text{ (its zero locus is ...
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1answer
73 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
...
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0answers
30 views
How can I define a chart in a smooth projective plane curve?
Consider $F(x,y,z)$ a non-singular homogeneous polynomial of degree $d$. If I consider the zero locus $X=\{[x:y:z]\in\mathbb{P}^2:F(x,y,z)=0\}$
How can I define a complex structure in $X$ in order ...
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2answers
40 views
to get a MDS code from a hyperoval in a projective plane
explain how we can get a MDS code of length q+2 and dimension q-1 from a hyperoval
in a projective plane PG2(q) with q a power of 2?
HINT:a hyperoval Q is a set of q+2 points such that no three ...
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97 views
Existence of a special set of q+2 points in the finite affine plane over $\mathbb F_q$
I am working in the finite affine plane over $\mathbb F_q$ with $q=2^n$.
Such a plane has $q^2$ points, $q^2+q$ lines, each line has $q$ points, and by a point is passing $q+1$ lines.
There are $q+1$ ...
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1answer
40 views
An interesting question in planar geometry
Several points in space are projected orthogonally on some three planes alpha, beta, and gamma. Could it happen that in the three projections, plane alpha contains 3 points, plane beta contains 7 ...
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2answers
66 views
Number of lines passing through a fixed point in a quadric surface
Suppose that $S \subset \mathbb{P}^3$ is a smooth quadric surface. Let $p \in S$. Can more than two lines $l \subset S$ pass through $p$? If not, why?
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1answer
35 views
Reference request: Projective space $\mathbb{RP}^3$
I have to write seminar paper in Projective geometry and the name of the seminar is (I would try to translate):
Representation of the lines in the projective space $\mathbb{RP}^3-$Kernel, Lineal, ...
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0answers
29 views
formula for inverse multidimensional stereographic projection
i'm in need of formula for inverse multidimensional stereographic projection with variant radius of the sphere. Sadly the only ones i'm able to find have either fixed number of dimensions or don't ...
7
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2answers
140 views
How should I think of lines and planes in projective space?
I have been learning about projective varieties recently and I realised that I have some trouble trying to grasp what lines and planes are even in say $\Bbb{P}^3$. For one, how should I think about a ...
2
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1answer
48 views
A linear subspace of projective space
If we have a linear subspace of projective $P^n_k$ where $k=\bar k$ that contains all the points $[0:\ldots:0:x:0:\ldots:0]$ with the nonzero $x$ in the $i$th slot for all $i$, how can we see that ...
6
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1answer
121 views
Learning projective geometry
My ultimate goal is to learn some algebraic geometry with the more concrete immediate goal of understanding things like how $\mathbb R\mathrm P^2$ is embedded in $\mathbb C\mathrm P^2$ or how $\mathbb ...
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1answer
38 views
Projecting two vectors to have constant element-wise euclidean distances
Let $x = (x_{0}, \ldots, x_{n})$ and $y = (y_{0}, \ldots, y_{n})$ be two vectors, and $d = (d_{0} = x_{0} - y_{0}, \ldots, d_{n} = x_{n} - y_{n})$ their element-wise euclidean distances.
My question ...
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1answer
87 views
A controlled trapezoid transformation with perspective projecton
I'm trying to implement a controlled trapezoid transformation in Adobe Flash's ActionScript using the built-in perspective projection facility. To give you an idea of how the effect looks like:
...
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1answer
42 views
Proving that the angle between the main diagonal of a cube and a skew diagonal of a face of the cube is 90 degrees
I need to prove that the angle between the main diagonal of a cube and a skew diagonal of the face of the cube is 90 degrees. I can do this with vectors, but I have to use applications from projective ...
2
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2answers
48 views
Prove that the intersection of a sphere and plane is a circle - Projective Geometry
Prove that if a plane has two distinct common points with a sphere centered at $O$, then the intersection is a circle with some center $O_{1}$, where segment $OO_{1}$ is orthogonal to the plane.
...
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0answers
22 views
Structure of $\text{P}\Gamma\text{L}(d,V)$
Is the following true regardless of the vector space $V$:$$\text{P}\Gamma\text{L}(V)\cong\text{PGL}(V)\rtimes\langle\sigma\rangle~?$$
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1answer
47 views
Ways to project arbitrary Fractals on 2D objects and 3D objects w different dimensions?
I am trying to create a house/texture in 3D and in 2D with fractals, perhaps related. My friend said that fractals can have different dimensions such as 1.74, 1, 4.71111... and pretty much anything. ...
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1answer
44 views
Finding the orthogonal projection
The angle between a line and a plane is thirty degrees. Segment $MN$ on the line has length $10$. What is the length of $MN's$ orthogonal projection on the plane?
I got 5 as an answer. Is that ...
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1answer
37 views
Solving for vector from equations. Can Dot product product be bought down as the denominator
I am new to vector geometry. I have a set of two equations:-
$p_2 \cdot x = (p_3 - \bar{p}) \cdot b_1$
$p_2 \cdot y = (p_3 - \bar{p}) \cdot b_2$
where, $p_2(x,y)$ is a 2D vector, $p_3(x,y,z)$ is a ...
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1answer
118 views
Examples of isometries of $S^2$ [duplicate]
Hey guys I am trying to do some practice problems for my course and I came across this problem, like I know what isometries are and how they work. But I am getting confused on how I would apply the ...
2
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0answers
34 views
Do we have homogeneous coordinates for probabilities?
As a roboticist, implementing visual odometry on a robot, homogeneous coordinates are convenient for projections of a non-moving object on an image sensor at $t$ and $t+1$ to estimate its position, ...
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1answer
105 views
Projecting a point on a plane through a matrix
I need to render some shadows in opengl, one way to do this is to render your object twice, a first time multiplying it by a special "shadow matrix" that flat your object on a plane generating the ...
