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

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Clifford Algebra and Fano Plane

Thank you for reading, I'm a novice, not a mathematician by trade this question could seem very simple (or even perhaps obvious) to many of you here. I've not yet found examples of this on the web. ...
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Book suggestions on projective geometry

I want to be acquainted with projective geometry, so I'm asking for a reference. I need some words to explain my specific background and motivation. There are many things I learnt related to ...
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Property of Projection Operator

Let $p: \mathbb{R}^n \rightarrow X$ be the projection mapping onto $X \subset \mathbb{R}^n$ compact convex. I am wondering if $$ \left( p(x) - p(y) \right)^\top z \leq \left( x -y\right)^\top z $$ ...
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31 views

Calculating the projective closure with more than one generator

I am given a variety $X = Z(f_1,f_2)$ in affine 3-space (in $x,y,z$), and I would like to compute its projective closure $Y = Z(g_1,\dots,g_n)$ in projective 3-space (in $x,y,z,w$). I have seen this ...
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Dimension of embeddings of Segre variety (product of projective spaces)

The Segre map gives an embedding of the Segre variety $\Sigma_{n,m}$ (i.e. of the categorical product of two projective spaces of dimension $n$ and $m$) into a projective space of dimension $nm+n+m$. ...
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Zorn lemma and existence of a basis

THis is what I have : In projective geometry, the zorn lemma guarantees that any filtered family of independent subsets of U has a basis with maximal element. My question is, what if I don't have ...
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How to prove with sets that if a finite subset is dependent then the set is dependent

$X'$ is the finite subset and $X$ is the whole set. Independent means if you remove an element, then it no longer is a subspace $< x >$. Dependent means $< X/{b} >=< X >$(ie if we ...
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Generalization of a projective plane?

In the area of finite geometry, a projective plane is an incidence structure of points and lines with the following properties: Every two points are incident with a unique line Every two lines are ...
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2answers
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Is the categorical product for projective spaces essentially the tensor product?

I wonder whether the categorical product of two projective spaces is essentially given by the tensor product of the underlying vector spaces. Is this at least true for projective Hilbert spaces? One ...
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24 views

arithmetic and geometric genus for a reducible plane curve

If $C$ is an irreducible plane curve we have the well known formula relating the airthmetic (obtained via the degree-genus formula) and the geometric genus $$\frac{(d-1)(d-2)}{2} - \sum ...
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Transformation matrix from quadrilateral to rectangle

There exists a rectangle somewhere in space with some orientation. A camera from the coordinate center point is looking along the z axis and is seeing the rectangle as a quadrilateral (due to ...
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Creating a Cube-based 3-Dimensional Game [migrated]

I am trying to create a 3-dimensional game that is based entirely off of cubes of the exact same size. I wanted to learn how to make my own 3-dimensional game using only 2-dimensional game libraries. ...
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1answer
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Why does it take 5 points to construct a projective frame in $\mathbb{R}^4$?

I am reading this answer http://math.stackexchange.com/a/186254/130408. The original question in that post is about deducing the projection matrix. But I have difficulties in understanding that ...
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1answer
29 views

Calculate angles of a projection of a tetrahedron

ABCD is a regular tetrahedron. It is projected on a plane in such a way that the projection forms an isosceles triangle ABC (AB = BC ≠ AC) with D lying in the middle of AC (right image): Problem: ...
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36 views

projective plane over algebraic structure

I know that it is possible to build a projective plane over any field. I also know that it is possible to build a projective plane over Hamilton's quaternions. My question is: Is it possible to build ...
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10 views

From 3d to 2d and back

I have a the internal parameters: A = [1.1349e+04 0 imWidth/2 0 11341.35490 imHeight/2 0 0 1] and the camera position placed at the word coordinates 0 0 0. ...
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Translation and Rotation for projections.

Lets say I place a camera in the air pointing downwatch to the surface and taking pictures. For all experiments right now I use this estimate of my internal camera paramters: ...
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Projection from a point to a plane - confused about terminology.

Edit: It seems rude to delete the question, but I have my answer now thanks to rghthndsd. I'm a bit unsure about the terminology in a homework question I'm doing, and I can't find any clear answers ...
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Tangent lines of conics

Let $k$ be algebraically closed. Let $P\in k[x,y,z]$ be a homogeneous quadratic polynomial. Let $C$ be the zero locus of $(P)$ in $\mathbb{P}^2$. Let $Q \in \mathbb{P}^2$. Is there a tangent line at ...
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31 views

Tesseract projection into $3D$

I found this: The tesseract is a four dimensional cube. It has 16 edge points $v=(a,b,c,d)$, with $a,b,c,d$ either equal to $+1$ or $-1$. Two points are connected, if their distance is $2$. ...
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1answer
25 views

What does the closure of a set mean?

In my book for projective geometry, this symbol: < x > means a subspace containing points x. But my teacher calls it "the closure of x". Does this mean the same thing. He also described "closure ...
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Why is it that quadratic forms seem fundamental for reciprocal or dual mappings?

In projective geometry mappings between points and hyperplanes (a "reciprocal" or "dual" mapping) often involve quadratic forms, e.g. in 3d projective space the polarity against an ellipsoid. The ...
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Is it possible to transform projections?

From linear algebra I know how to transform a point or a direction from one space to another using transformation matrices (and I think it's pretty awesome by the way). But what about projections? Is ...
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31 views

How do I know if a configuration contains inscribed 4-gons

For example, the Mobius Kantor configuration. I'm supposed to prove that n3 configurations don't contain 2 inscribed 4-gons. I don't know where to start from this(Do I just stare at the 2-D ...
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1answer
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Relative camera matrix (pose) from global camera matrixes

I have a list of camera poses from a given ground truth. Each pose is given in the form of a quaternion and a translation, from some arbitrary world origin. Each pose can be assembled into a 4x4 ...
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59 views

Finding distance from point to line determined by vector

We are given a point, b. We are given a vector, v, which determines a line. We are given a point c such that there exists a line through c which has the same direction as the line determined by v. ...
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<M>:= the intersection of subspace U such that U is a subspace of L containing M

L is a linear space and set M is the set of points of L. The definition I put above is "the smallest subspace of L generated by M". The thing I don't understand in this definition is why do we need ...
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1answer
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cross ratio - is it ordering invariant

I want to know if the cross Ratio depends upon the ordering of the points around a particular point . I am calculating the cross ratio as : CR = A(1,2,3)*A(1,4,5)/(A(1,2,4)*A(1,3,5) where A is the ...
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Show that $C\in\mathbb{P}^2(k)$ is a rational curve

Let $k$ be an algebraically closed field of characteristic $p>0$. We consider the curve $$C = V(X^pZ^{p-1}-Y^{2p-1})\subset\mathbb{P}^2(k)$$ Show that $C$ is a rational cuve. We did ...
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44 views

Projection Matrix between two Vectors

Given a two normal vectors v1 = [a1;b1;c1] and v2 = [a2;b2;c2] as given in Fig1. How I can derive the projection matrix that ...
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Distinguished open sets in $\mathbb{P}^n$

I have given a homogenous polynomial $F\in\mathbb{C}[x_0,\ldots,x_n]$ of degree $d>0$ and consider the open set $U_F=\{p\in\mathbb{P}^n; F(p)\neq 0\}$. I'd like to prove that $U_F$ is isomorphic to ...
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Imagining the projective Space

I am trying to get used to work in the projective space. Therefore I wanted to know which tactics there are to imagine the projective space. $$\mathbb{P^n}(k):= (k^{n+1}\backslash \{0\})/k^{*}$$ I ...
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Projectivizing a group: how to go from AGL(n,K) to PGL(n+1,K)

$ \newcommand{\GL}{\operatorname{GL}} \newcommand{\AGL}{\operatorname{AGL}} \newcommand{\PGL}{\operatorname{PGL}} $Given an irreducible matrix group $G_{\infty,0} \leq \GL(n,K)$, I form the group ...
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Desargues Configuration

Section 3.2 of Coxeter's Projective Geometery discusses the self-dual $10_3$ configuration, ten points and ten lines, with three pointson each line and three lines through each point. I am looking ...
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Grassmannian, Plucker coordinates

In which books can I find something about the grassmannian and the plucker coordinates ?
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Specify any perspective transformation

I want to specify a perspective transformation using rotation angles. This should transform a homogeneous co-ordinate $q$ to another. Basically, a square would look like a trapezoid after the ...
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Plücker coordinates of the Clifford parallels

Let $$q=\cos\theta+(x_q\textbf{i}+y_q\textbf{j}+z_q\textbf{k})\sin\theta$$ be a unit quaternion parameterised by $\theta\in\mathbb{R}$, where $(x_q,y_q,z_q)$ is fixed and $x_q^2+y_q^2+z_q^2=1$, and ...
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Projective Co-ordinate Geometry

I am learning projective geometry in my computer vision course. So, we represent a co-ordinate point in an image as a homogeneous co-ordinate as $(x,y,1)$. My professor says that if we are given two ...
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Quotient of a proj variety by an involution

Usually, if you have an affine variety defined by some equations and have an involution on it, it's quite easy to immediately see what the equations of the quotient of the variety by the involution ...
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some fun with holomorphic line bundles

These are probably trivial questions... (for the experts) I'd like to get convinced (perhaps an intuitive/geometric explanation will be more effective than a formal one) of the following facts: i. ...
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53 views

Transforming a circle to get a parabola

On http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html I am unable to understand the following point Obviously, this transformation sends (x,y,w)=(1,0,1) to (x',y',w') = ...
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Is the universal hyperplane section the blowup of the baselocus?

I think I've heard this statement before but I'd like to make sure it's true. Let $X$ be a variety and $L$ a line bundle on it. Take $S < P\left(H^0(X,L)\right)$ to be a linear subspace of the ...
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Visualizing homology and elation in projective geometry

I'm reading through Coxeter's Projective Geometry and in the chapter where he talks about perspective collineations he divides them in two classes: elations - perspective collineations whose axis and ...
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230 views

Two circles intersect in two points

Take for example two circles $$\begin{cases}x^2+y^2=1\\x^2+y^2-x-y=0\end{cases}$$ These two circles intersect in two points namely $(0,1)$ and $(1,0)$. But by Bezout's theorem they must intersect four ...
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Give 2x2 matrices such that for $x \in \Bbb R$

$Ax$ is the vector obtained projecting $x$ onto the line $x_1=x_2$ I can not understand what the exercise asks, or how to start to solve it. The funny thing is that this exercise is within the ...
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37 views

Which 6x6 line-matrix corresponds to a 4x4 point/plane-matrix

In 3-dimensional projective geometry I have a point-point map (collineation) $c$ with matrix $A$. Then $A^{-1t}$ is the matrix for the plane-plane map for the same $c$. These matrices are 4x4 and ...
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Prove that a curve in P^n of degree n not contained in a hyperplane is rational

The set up is as stated above. We have a projective curve $X$ of degree n embedded in $\mathbb{P^n}$, which is not contained in any hyperplane. We claim that it is therefore rational. The way I have ...
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Why is $(0,0,0)$ not acceptable as a co-ordinate on the projective plane?

Reading "Elliptic Tales: Curves, Counting, and Number Theory" which states $(0,0,0)$ cannot be a co-ordinate on the projective plane but I find the argument advanced for this in the book to be ...
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Lost on projection. Projective Geometrey.

I was given this image, seen below, as a study guide,I color coded it for ease of reading, for the homework (which is to setup 3 perspectives that map $(a,b,c,d) \to (c,d,a,b)$). I am having trouble ...
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Incidence geometry: Prove that there is only 1 unique configuration 7_3_

$7_3$ also know as Fano plane. How can I prove that only 1 configuration exists for 7 points and 7 lines with 3 points on every line and 3 lines at every point. I would think an incidence matrix would ...