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

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Triangles form a harmonic set with their medians and altitudes

In a triangle $\triangle ABC$, let $AD,BE,CF$ be its altitudes and $AK,BL,CM$ their medians. Show that $D\{EF,AB\} = -1$ and $K\{LM,AB\} = -1$ I don't get any of the problems here. Not any of these ...
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25 views

If $K_X$ is not $\mathbb Q$-Cartier then it is not nef

Let $X$ be a projective variety. Is it true that if the canonical divisor $K_X$ is not $\mathbb Q$-Cartier then it is not nef?
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27 views

Find the N versors more 'spaced' [on hold]

I have to deal with a concrete problem that is: Given a 3d object I want to select N directions with N integer and N>=3 for projection that would maximize the information I gain and thus my ability to ...
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Applying homography to ellipse derived from normal distribution

I need to apply a homography to an elliptic area. First question: Is the resulting also elliptic in every case? I think so, but actually i don't really know. Anyway, I assume it for this question. ...
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A problem in projective geometry…

I have the following projectivity: $$ f[x_1,x_2,x_3]=[4x_1+2x_2-x_3,2x_2,x_3,-x_2-x_3]. $$ I have to find all the lines $L$ such that $f(L) \subset L$. I've found the eigenvalues of this matrix, ...
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semi-positiveness of canonical line bundle under the condition Kodaira dimension be positive.

Let $M$ be a projective variety with positive Kodaira dimension, then why the canonical line bundle is semi-positive?. Is there any reference?
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Is the Projective Real Plane Compact?

I feel like $\Bbb P (\Bbb R^2)$ is compact, but I know that $\Bbb R^2$ is locally compact, therefore it has a one-point compactification. $\Bbb P (\Bbb R^2)$ adds more than one point to the real ...
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30 views

Surjectivity of a map $D^{2n} \to \mathbb{CP}^n$

I'm solving an exercise about the complex projective space, and during a step of the solution I'm asked to find a surjective map $D^{2n} \to \mathbb{CP}^n$. I defined the map in this way $$ ...
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Project triangle from $\mathbb{R}^3$ into $\mathbb{R}^2$ with to fixed vertex texture coordinates

I have a triangle made out of the three vertices p1, p2, p3. I know the positions of the vertices in $\mathbb{R}^3$, called $x_i, y_i, z_i$ for $i=1,2,3$. I now want to assign each vertex a texture ...
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32 views

Conic through 4 points

Let $p_1,\ p_2,\ p_3,\ p_4$ be any 4 different points on $\mathbb{CP}^1$ and $x_1,\ x_2,\ x_3,\ x_4$ are 4 different points on $\mathbb{CP}^2$. How can I show that there is unique conic $Q$ passing ...
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22 views

Can projective spaces be given structure of a linear space.

Let $\mathbb{RP^{n-1}}=\mathbb{R^n}/ \sim $ where x ~ y iff $\exists \ \lambda \in \mathbb{R} \ s.t \ \lambda x=y$ Can $\mathbb{RP^{n}}$ be given the structure of an $\mathbb{R}$-module . Moreover, ...
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Bounding the cohomology of a smooth projective variety

Let $X/\mathbb C$ be a smooth projective variety. Suppose it is smoothly embedded in $\mathbf P^n$ as the zero locus of an ideal generated by homogeneous polynomials $f_1, f_2, \dots, f_r$ in $n+1$ ...
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72 views

CP(2) = SU(3)/U(2)?

In my understanding the complex projective line $CP^1 = \mathbb{C}^2/\mathbb{C^*}$ where $\mathbb{C^*}$ is $\mathbb{C}$ without $0$, i.e. just 2 complex coordinates and a homogeneous factor. And the ...
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Why is $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$ constant if dim $\phi(\mathbb{P}^n)<n$?

Let $\phi:\mathbb{P}^n\rightarrow \mathbb{P}^m$, $n\leq m$. I want to demonstrate that if dim $\phi(\mathbb{P}^n)<n$ then $\phi(\mathbb{P}^n)=pt$ (ex. 7.3(a), ch.II from Hartshorne). It's well ...
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45 views

Hyperplanes as dual projective spaces

I was reading through Harris's Algebraic Geometry book, and was slightly perplexed by the following paragraph: "Note that the set of hyperplanes in a projective space $\mathbb{P}^{n}$ is again a ...
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42 views

Module over an associative ring with unity and axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
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Module over a ring which satisfies Whitehead's axioms of projective geometry

I asked this question(Characterization of a vector space over an associative division ring). It was pointed out that the answer was negative. So I reconsidered the problem and have come up with the ...
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Whitehead's axioms of projective geometry and a vector space over a field

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
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26 views

Whitehead's axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
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1answer
19 views

Application of Desargues' theorem for constructions

I found this interesting document (german) on the internet. On page 8 it says: "Draw a line segment between two given points only using compass and ruler, while the distance between the two points is ...
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1answer
14 views

What is the plane gradient?

My professor recently used the following phrase "the unknown 3D point is in a plane whose gradient is $(a,b,c)^T$". I can't seem to place his terminology anywhere on the internet. What does he mean by ...
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23 views

Determining 3D position of point from 2D projection.

Say I have a 3D point $p$ and I project this point (using perspective projection) onto the image plane at 2D point $u$. Knowing that $p$ is on a plane with gradient $(a,b,c)^T$, how can I express the ...
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Showing angles are preserved by isometry.

Im trying to show that a rigid transformation (isometry) preserves angles. Here is my approach so far. Let $x,y \in \mathbb{R}^n$ and $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ be a rigid motion ...
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Description of real projective spaces in various contexts

What I want to know is : What is the description of real projective spaces (specially $RP^0$, $RP^1$, $RP^2$) respectively in context of topology, geometry and algebra? I'm searching for simple ...
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29 views

Polar correlation and conics in $\Bbb RP^2$

I'm stuck on a small detail in Proposition 1.2.8 in Geiges' Introduction to Contact Topology. Let $C$ be a conic in $\mathbb{R}P^2$ given by $q^tAq=0$, where $A$ is a nonsingular, symmetric 3x3 ...
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“projective maassbestimmung” in Automorphic Functions by Fricke + Klein

I was reading a copy of Fricke and Klein's Theory of Automorphic Forms, and I came across the phrase projective maassbestimmung in the first chapter. Google translate returns: maßbestimmung as ...
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44 views

Is it true that ${\mathbb P}^1_{(1,2)} \cong {\mathbb P}^1$?

In a class on Algebraic geometry, we learnt the following - ${\mathbb P}^1_{(1,2)} \cong {\mathbb P}^1$ over the field ${\mathbb C}$. I'm not sure I followed the entire argument exactly. I'll ...
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If all I have is this image of a projection, how can I get screen coordinates for a object of a certain size?

All I have is this image and I want to figure out the coordinates relative to the 0,0 in the depicted coordinate system for an planar object of any size. Eg. (1m x 1m or 2.31m x 3.23m) Think a rug ...
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How to multiply a vector and matrix when the matrix includes a translation?

What is the proper way to right multiply an $N$ x $N$ matrix $H$ by an $N$ x $1$ vector $\mathbf{v}$, if $H$ includes a translation vector? For example, say $$H=R-\mathbf{tn}^T$$ where $R$ is a ...
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25 views

When are two 3D Lines parallel in Plücker matrix form?

When are two lines in 3 dimensional space parallel, when the lines are both represented by Plücker matrices $L$ and $L'$. I'm trying to prove the solution to this question: ...
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15 views

Is a Spread Unique?

Let $V$ be a vector space of dimension $n$. It is well known that when $r | n$, there is a set of disjoint $r$-dimensional subspaces of $V$, which covers $V$, called Spread. My question is that is a ...
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31 views

3D Vector projection on a Plane

I want to Project a Vector on to a Plane. Assume, you have a Central Point (1,1,1) and you want to move (0,0,3) in z-direction. How can I project the end of this movement (point) on a plane with ...
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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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37 views

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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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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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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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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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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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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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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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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136 views

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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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 ...