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

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What does this Perspective-projection matrix in 2D do?

Given a projection axis $X$, camera positioned in the origin and $d$ the distance to the projection plane, this is the perspective projection matrix: $$ P = \left[ \begin{array}{@{}ccc@{}} 1 & 0 ...
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Parallel Lines Intersecting in the Projective Plane

My question is about visualizing projective space, in particular the real projective plane $\mathbb{P}^2(\mathbb{R})$. I know there are different ways to define this space, but in each we can say that ...
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How can I convert four dimensions into two?

I am trying to generalize the following problem (or at least extend it to 4 dimensions). If I have a 3 dimensional vector with the coordinates A,B,C and the constraints that A+B+C=1 and ...
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Lipschitz constants of projections

Consider two compact sets $A, B \subset \mathbb{R}^n$. Assume that the projection mappings $P_A: \mathbb{R}^n \rightarrow A$, $P_B : \mathbb{R}^n \rightarrow B$ have Lipschitz constant $1$ and $L$, ...
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3-Point Shoot using Quadratic Equation [closed]

This is my assignment. The question is "In what part of the three-point line can a player do best the three-point shoot to gain 3 point but using quadratic equation." There are no data given but we ...
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Ellipse as projection of a disk - function describing ellipse diameter with disk rotation?

Say I have got a disk of radius $r$ and a plane $p$ in $3D$ space. The disk is "aligned" to $p$ and lies at an arbitrary distance, so that its orthogonal projection on $p$ is an identical disk of ...
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Topological properties of the so-called “plane at infinity”.

When 3D-Euclidean geometry is extended with ideal points at infinity, a whole "plane at infinity" is added to the geometry. Apart from metric properties it has become a 3D projective space and the ...
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Line varieties in Projective Geometry!

I'm an Engineering student. All of the sudden I need to know about "Family of Lines" which is a topic in "Projective Geometry". I've found the old book of Veblen & Young (and two other books) but ...
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Why are projective transformations $3$-transitive on points?

I can see why Mobius transformations can take $3$ points to any $3$ points, but I can't see geometrically why a projective transformation can do that. If you want the points $A, B, C$ to be send to ...
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Equations of a projective variety from parametric ones

How does one find equations of a variety given parametric equations (i.e. a regular map) in projective space? For example, I got stuck in finding the equations of the curve in $\Bbb{P}^2$ described by ...
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Computing the tangential and cross components of one quantity using gnomonic projection

I have a spin-2 field given called shape distortion of galaxies as $$\gamma=\gamma_1+i\gamma_2=|\gamma|e^{-2i\phi}$$ where $\phi$ is the orientation angle. If this quantity has been measured on ...
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29 views

A question about lines in the projective space.

Let $ax+by+cz=0$ be a line in projective space. Let the line be satisfied by two points $(a_1,a_2,a_3)$ and $(b_1,b_2,b_3)$. Then we have $$a_1x+a_2y+a_3z=0$$ $$b_1x+b_2y+b_3z=0$$ This implies that ...
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762 views

Lat/Long grid points covered by projecting rectangle onto sphere

Before my question proper, a little background: I'm wanting to optimise some computer rendering by eliminating the drawing of things that aren't visible given the current view. Suppose we have a ...
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1answer
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Projective Spaces which are not Vector Spaces

I'm studying Projective Spaces, I've collected a few books and most of them define Projective Spaces in terms of Vector Spaces, that is, they define a 'projective space structure" in the vector space ...
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prove that the vanishing line can be determined given three coplanar equally spaced parallel lines

This is an exercise of the book " multiple view geometry in computer vision" (eqn.8.15, p218), ( not homework). It says that under projective geometry, a set of equally spaced parallel lines, which ...
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238 views

fixed point projective geometry

I am thinking about the following: Let $\sigma:\mathbb C P^n\rightarrow\mathbb C P^n$ be a projectivity with $\sigma\circ\sigma=id_{\mathbb C P^n}$. I define the set of all fix points by ...
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Blow Up: Resolution of Singularity

For blow ups, I have worked only in $\mathbb{CP}^2$. Once I locate the base-point, say $[x,y,z]=[0,1,0]$, I go back to $\mathbb{C}^2$ by considering the chart $y=1$. I then proceed to blow up ...
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Rotation plane on the sphere (quarternion)

I asked similar question on stackoverflow but still no answers.http://stackoverflow.com/questions/25185329/image-rotation-with-the-gyro-data-math I assume it is more math than programming problem. ...
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Saari's homographic conjecture and the actual definition of homography

By the wikipedia definitions found here and here, and especially by the definition implicit in this MSE post, it seems two images are homographic if they are renderings of the same set of points in ...
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1answer
113 views

Space of inscribed n-gons modulo projective transformations

Say $P$ ~ $Q (P$ and $Q$ are «projectively equivalent») iff there is a projective transformation $f$ such that $f(P) = Q$. Then ~ is an equivalence relation. I read that the space of inscribed ...
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26 views

Projective and affine conic classification

I have a doubt on the classification of non-degenerate conics (parabola, ellipse, hyperbola) in projective geometry (my textbook is "Multiple View Geometry in Computer Vision", which, as the title ...
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If $A$ and $B$ are lines in $\mathbf{CP}^3$ and $P$ is a point in $\mathbf{P}^3$, is there a line incident to $A,B,P$?

I want to figure out the following: If $A$ and $B$ are lines in $\mathbf{CP}^3$ and $P$ is a point in $\mathbf{P}^3$ then is it always possible to find a line in $\mathbf{CP}^3$ which meets all of ...
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Gaussian curvature of a complex projective curve

Let $X \subset \mathbb CP^2$ be a complex curve inheriting metric from $\mathbb CP^2$. Suppose that locally $X$ is given by a holomorphic map $z \to [h_1(z) \colon h_2(z) \colon h_3(z)]$. What is the ...
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Can one prove Brianchon's theorem using Ceva's theorem?

Can I prove Brianchon's theorem using Ceva's? I am also wondering if parabola and hyperbola can be inscribed in a hexagon?
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Injective and continuous function that is an embedding

Consider $n,d\in \mathbb N$ and $N= {n+d\choose d}-1$, then the well known $d$-uple embedding: $$\rho_d: \mathbb P^n(\mathbb C)\longrightarrow\mathbb P^N(\mathbb C)$$ is a continuous (respect to ...
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Finding the Transform matrix from 4 projected points (with Javascript)

I'm working on a project using Chrome - JS and Webkit 3D CSS3 transform matrix. The final goal is to create a tool for artistic projects using projectors and animation - somewhat far away from using ...
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1answer
51 views

Blow-up in Projective Space: Choosing the Appropriate Chart

Consider $x^2+y+\alpha=0$ (made-up example), where $(x,y)\in\mathbb{C}^2$ and $\alpha\in\mathbb C$ is a free parameter. The equation defines a family of curves. The curves have no common points in ...
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46 views

Geometries (Euclidean and Projective)

We can think of Euclidean Geometry and Cartesian (Coordinate) Geometry as equivalent, in the sense that some proposition is true in Euclidean Geometry iff it's true in Coordinate Geometry. It makes ...
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59 views

Area of projection of cube in $\mathbb{Z}^3$ onto a hyperplane

A cube with vertices $(\pm 1, \pm 1, \pm 1)$ gets projected into the plane perpendicular to vector $\mathbf{n}\in S^2$. The projection is a hexagon, how do I find the area? I think I can just ...
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48 views

Proving a theorem using Pappus' theorem

I need some help. I want to prove Desargues' theorem via using Pappus' theorem. And I don't know how. Please, help me!
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102 views

Packing infinitely many ellipses into a circle

Given a circle $C$, and an infinite set $S$ of mutually disjoint ellipses which are inside and tangent to $C$, prove that there must exist a disk $D$ which lies inside $C$ but outside every ellipse. ...
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40 views

5d Basis Vectors of Penrose's Tilings

I have been writing some software to display/render Penrose tilings. I was hoping to use the approach of projecting a 5-dimensional lattice into 2d and apply some coloring based on regions etc. I ...
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1answer
21 views

Projective roots of a homogeneous polynomial

Suppose that $f(X,Y)\in\mathbb C[X,Y]$ is a homogeneous polynomial of degree $n$, then we can consider it as a function on $\mathbb P^1_\mathbb C$. It has at most $n+1$ projective roots (points of ...
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Preserving incidence relation proof

How can one prove via analytic method that projective map preserves incidence relation?
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The cross-ratio of four points is a real number exactly when the four points lye on a line or a circle

Let $z,z_1,z_2,z_3$ be four points on the extended plane. Their cross-ratio $(z,z_2,z_3,z_4)$ by definition is the image $Tz$ of $z$ under the Möbius transformation $T$ that sends $z_1,z_2,z_3$ to ...
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While extending the $\pi_0$ configuration, shouldn't it end at $\pi_3$?

Kindly refer to pg. 9 of Foundations of Projective Geometry by Hartshorne. $\pi_0$ contains points such that there can only be $1$ line between two points. There many not be lines joining all pairs ...
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A doubt in the proof of Desargues' Theorem.

I have a question regarding the proof of Desargues' Theorem. When the traingles $ABC$ and $A'B'C'$ are assumed to be lying on the same plane. A point $X$ is taken outside that plane, and the lines ...
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Area of an ellipse proportional to integral of cross-ellipse distances?

I am curious if the area of an ellipse can be shown to be proportional to the integral of all cross-ellipse distances. Before I define cross-ellipse distance, I will give a motivating example from a ...
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A little problem about $4$ points in $\mathbb P^1(\mathbb C)$

I have to solve the following problem: Consider a set $D\subseteq\mathbb P^1(\mathbb Q)\subset \mathbb P^1(\mathbb C)$ such that $|D|>3$. Then there exists a Moebius transformation $M$, and ...
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A question from “Foundations of Projective Geometry” by Hartshorne.

"Foundations of Projective Geometry" by Hartshorne says the following: The completion of the affine plane of four points is a projective plane with 7 points. The affine plane of $4$ points is ...
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Enumerative projective geometry

I am wondering whether for any two lines $\mathfrak{L}, \mathfrak{L'}$ and any point $\mathfrak{P}$ in $\mathbf{P}^3$ there is a line having nonempty intersection with all of $\mathfrak{L}, ...
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Vertices and indices of a tesseract before and after projection

how to define the vertices and indices of a tesseract before and after projection into 3D ,is the way in which vertices are connected to form lines " Wireframe " in 4D remains the same after ...
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120 views

Computing cohomology of hypersurface

I'm taking a course on differential geometry now, and we got the following exercise from the lecturer: compute the (de Rham) cohomology groups $H_{dR}^i(M)$ of your favourite space. In all the ...
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Origin in homogeneous coordinates [closed]

In homogeneous coordinates in projective geometry, is the origin given by: (0,0,w) for all w?
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126 views

Blow-ups in Projective Space

This is in regards to a question (no solutions or comments thus far :-() I asked earlier in regards to the blow-up of an elliptic curve: Question Let $f(x,y)=y^2-4x^3+ax+b$, where $(x,y)\in\mathbb ...
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187 views

Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
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A question about three collinear points

This video (at 44:00) says that in a projective space if three points are collinear, and two of those points lie at infinity, then the third point will also have to lie at infinity. I wonder why ...
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Projection matrix to project a point in a plane

How to determinate the 4x4 S matrix so that the P gets projected into Q, on the XZ (Y=0) plane? Q = S P
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A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
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1answer
75 views

Chirality of a Möbius band without boundary?

In this answer it is remarked that the real projective plane minus one point is homeomorphic to the Möbius strip without boundary. A normal Möbius strip is topologically equivalent to a real ...