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

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Show that the line $KD$ bisects $\angle{EKF}$

Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each ...
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Set of roots of quadratic form $B(x,y,z,t)$ on the line $z=t=0$ is nonempty.

This is a proof from Section 7.1 of Undergraduate Algebraic Geometry by Reid. Suppose $S\subset \mathbb{P}^3$ is a nonsingular cubic surface, given by a homogeneous cubic $f=f(x,y,z,t)$. Consider ...
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24 views

Distance between projections

Let $x,y,z \in \mathbb R^2$ such that $||x|| = ||y||= ||z|| = 1$. Project $z$ onto the lines defined by $x$ and $y$ as follows: \begin{equation} z_x = (z^\text{T}x) x, \ \ z_y = (z^\text{T}y) y, ...
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35 views

Kernel of a map $\phi: \mathbb{C^*}^3 \to \mathbb{C^*}^2$

I cannot understand which is the kernel of the following map $\phi: \mathbb{C^*}^3 \to \mathbb{C^*}^2$ with $$ (t_1,t_2,t_3) \mapsto \left(\frac{t_2}{t_1}, \frac{t_3}{t_1}\right) $$ In other words I ...
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116 views

Exact Expression for numerical Solution 0.9595767

I need you to do just what any math genuis in a shallow Hollywood movie does: looking at big tables of numbers and seeing exact structure! These $3 \times 3$ matrices are solutions to a well-posed ...
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1answer
28 views

Two new conics associated with two triangles inscribed a conic

1-Let $ABC$ and $A'B'C'$ be two triangle inscribed a conic. Let $P$ be a point on the conic. $PA$ meets $B'C'$ at $A_1$, define $B_1, C_1$ cyclically. $PA'$ meets $BC$ at $A'_1$ define $B'_1, C'_1$ ...
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32 views

Area of piece of paper folded around straight line of orientation $\theta$

Imagine drawing a straight line $l$ through the center of a square piece of paper with area $1$. Now fold the paper along that line. Q: What is the function for the area covered by the folded ...
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43 views

The set of polynomials which “cut out” smooth subsets of projective space is open and dense

Let $k[x_0,x_1,...,x_n]_d$ be a space of all forms (in other words, homogenous polynomials) of degree $d$ of variables $x_0, x_1,...,x_n$ over algebraically closed field $k$. Let's think of ...
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61 views

Image of a line or conic on Veronese surface.

This is part of Exercise 5.13 from Undergraduate Algebraic Geometry by Reid: Consider the Veronese surface $S$ defined by the map: $$\phi: \mathbb{P}^2\rightarrow \mathbb{P}^5$$ where ...
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How to determine that the 3 points given in homogeneous coordinates are collinear? [closed]

How do I prove that the 3 points given in homogeneous coordinates are collinear? $$A=(1,3,2)^T, B=(0,6,8)^T, C=(3,3,-2)^T$$
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19 views

Transform in eigenvetor space

Hello I have a squared matrix C C = 0 2.2361 63.7887 2.2361 0 61.6117 63.7887 61.6117 0 and I calculate its ...
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1answer
53 views

Is the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ birational?

This is Exercise 5.3 (a) in Undergraduate Algebraic Geometry by Reid. Does the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ define a rational map? Determine ...
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What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed ...
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1answer
28 views

Irreducibility of an affine variety in an affince space vs in a projective space.

Proposition 5.5 in Undergraduate Algebraic Geometry by Reid says (I only write down a brief idea since the proposition is long and involves some other notations to define): The affine variety $U$ ...
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20 views

Homography with line correspondences

When calculating a homography with line instead of point correspondences, what is the derivation of the formula: $$ l_i = H^T\cdot l^{'}_i $$ I know that: $$ l^T\cdot x = 0 \quad\text{and}\quad ...
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Diffeomorphism between $\mathbb{P}^n$ and the submanifold of $\mathbb{R}^{(n+1)^2}$ consisting of certain matrices?

Let $\mathbb{P}^n$ denote the set of all lines through the origin in the coordinates space $\mathbb{R}^{n+1}$. Define a function$$q: \mathbb{R}^{n+1} - \{0\} \to \mathbb{P}^n$$ by $q(x) = \mathbb{R}x ...
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1answer
31 views

Differential geometry of projective bundles

Can someone give me a reference about projective bundles from a differential-geometric point of view? I am not very familiar with algebraic geometry. I would like, for example, some theory about when ...
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25 views

Projective Transformation best fit for multiple 4-point sets

I have a photo that shows some (for example 10) jumbled paper sheets, each with known size, situated on a plain large table. The distances between these sheets are unknown. So I would have 10 ...
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1answer
14 views

Circle equation in homogeneous coordinates

Can someone give me a derivation why the circle equation is expressed in homogeneous coordinates like this (as described in Hartley): $$ (x-a\cdot w)^2 + (y-b\cdot w)^2 = r^2\cdot w^2 $$
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1answer
28 views

Number of points on a line in a finite projective plane

I've been reading some proofs regarding finite projective planes of order n, and often they start out by assuming that each line contains n+1 points. Is this a fact that follows from the axioms for ...
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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
374 views

Projected Area of Circle onto the Side of a Cylinder

I have encountered a problem that requires me to find the projected area of a beam of light (circular cross-section, with radius R1) vertically onto the side of a cylinder with R2 as its ...
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13 views

Project 4 cones onto a sphere

I have four cones. The angle of each cones is 140 degree. I need to project it onto a sphere(place it ) such that, the cones cover the maximum area with minimum overlap. I initially thought that ...
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20 views

$OABCD$ tetrahedron with $OA ⊥ OB ⊥ OC ⊥ OA$

I've got stuck at this problem: Let $OABCD$ be a tetrahedron with $OA ⊥ OB ⊥ OC ⊥ OA$. If $OH$ is the orthocentre of triangle $ABC$, show that $OH$ is perpendicular on plan $(ABC)$. Then ...
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1answer
24 views

Group law on elliptic curves.

Let $k$ perfect field. If we have a cubic non-singular projective curve $C(k)$ (over a field $k$), take two diferent points $P_1,P_2 \in C(k)$ and consider the line through the points, by Bezout ...
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1answer
38 views

exercize about the foundamental group of $\mathbb{P}^n(\mathbb{R})$

Let $p$ be a point in $\mathbb{P}^n(\mathbb{R})$ and $\Sigma$ the set containing all the projective lines passing through $p$. Given $s\in \Sigma$ we can define a continous closed path (let's say ...
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The intersection of two parallel lines

My teacher told me that two parallel lines have a point of intersection, it is called Point at infinity. But I I can't understand how it can be true or how was it proved, can someone explain this to ...
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22 views

Finite geometry - how to determine parallel classes

I try to learn a little about finite geometry and I have now encountered the following exercise: Exercise: Construct the affine plane $\mathrm{AP}(\mathbb{Z}_3)$. Determine it's parallel classes ...
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275 views

There is some intuitive idea of Pascal's 's theorem in Projective Geometry?

In projective geometry, Pascal's theorem (formulated by Blaise Pascal when he was 16 years old) determines that a hexagon inscribed in a conic, the lines that contain the opposite sides intersect in ...
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Projection along an axis

I have problem with understanding what projection along an axis means in practise. For example I have two paraboloids $H=\{(x,y,z)\in \Bbb{R}^3:z=2xy \} $ and $E=\{(x,y,z)\in \Bbb{R}^3:z=x^2 +y^2 \} $ ...
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Isomorphism between semi-orthogonal group $O(2,2)$ and direct product of projective general linear group $PGL(2,\mathbb{R})$ with itself

Is there any natural isomorphism between semi-orthogonal group $O(2,2)$ and direct product of projective general linear group $PGL(2,\mathbb{R})=GL(2,\mathbb{R})/R^+$ with itself?, where $O(2,2)$ is ...
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What was the genesis of Hua's identity?

Many resources I have read prove Hua's identity more-or-less mechanically. I have seen there is more than one raison d'être for Hua's identity: e.g. its connection to the fundamental theorem of ...
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24 views

$V\subset \mathbb{P}^n(k)$ is irreducible, prove that $\dim(V)$ is the transcendence degree of $k(V)$ over $k$.

This is a problem from Ideals, Varieties, and Algorithms by Cox et. al. $V\subset \mathbb{P}^n(k)$ is irreducible, prove that $\dim(V)$ is the transcendence degree of $k(V)$ over $k$, where $k(V)$ ...
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1answer
48 views

Hypersurfaces meet everything of dimension at least 1 in projective space

The following exercise is taken from ravi vakil's notes on algebraic geometry. Suppose $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension at least $1$, and $H$ is a nonempty hypersurface in ...
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Can 3 transformations (V, Σ, U) of SVD to describe a perspective transformation?

As known SVD (Singular value decomposition) is a factorization of the form M = UΣV∗. https://en.wikipedia.org/wiki/Singular_value_decomposition SVD of the linear map T can be easily analysed as a ...
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Is it possible to project orthogonally an ellipse with major and minor axes $2a$,$2b$ so that its image is a circle with diameter $2b$?

Problem: Prove that the area of an ellipse with major axis and minor axis of lengths $2a$ and $2b$,respectively, is $ab \pi$ . Proof: We do this by projecting the ellipse into a figure whose ...
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Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
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Why are orthogonal projection matrices not … orthogonal?

I know that given an orthogonal matrix U, then orthogonal projection onto the column space of U is represented by the matrix $UU^t$, which is again orthogonal. I've computed these types of matrices ...
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15 views

Projection of an oblique circle on XZ plane

While going through an exercise of surface integration, I got confused in this problem.The surface is the intersection of sphere $S:x^2+y^2+z^2-1=0$ and the plane $P:y-x=0$. Clearly, the curve of ...
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36 views

Why is cot(a) function used in perspective projection?

I'm working with the different space projections. But I wonder about the perspective projection. Let me remind you one template, which may be used in 3D rendering software: ...
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Projecting a sphere from inside

I am trying to make a renderer for a programming project, and yet I am having trouble projecting the points onto the screen (the way it works so far, the camera can't look down on a face because the ...
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123 views

Two twisted cubic curves in $\mathbb P^3$ intersect iff they lie in a common cubic surface

Let $C_1$ and $C_2$ be twisted cubic curves in $\mathbb P^3$. I want to prove that they intersect if and only if they lie in common cubic surface, perhaps singular. The second condition can be ...
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What is the point of having at least three points on every line of a projective plane?

A common definition of "projective plane" includes the following axioms: every pair of distinct points lies on exactly one line every pair of distinct lines meet in exactly one point every line has ...
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Tangent Space to Grassmannian

I have a second question today. In Harris' "Algebraic Geometry: A First course" he constructs (on page 200) an isomorphism between the tangent space of the Grassmannians and some homomorphisms: He ...
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Why do we have to normalize the eigen-vectors before orthographic projection?

Given such a matrix about the grades of 6 students in Maths, Computer Sciences and French: \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 2 \\ 2 & 2 & 1 ...
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“General position” in a poset

Does anyone have a reference for this notion of "general position" in a poset: A set $S$ in a poset $(X,\le)$ is said to be in general position if for any $A,B\subseteq S$, $\{x:\forall y\in ...
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How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of tangent $2$-planes? [duplicate]

A manifold $M$ is said to admit a field of tangent $k$-planes if its tangent bundle admits a subbundle of dimension $k$. How do I see that $\mathbb{RP}^4$ and $\mathbb{RP}^6$ do not admit fields of ...
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22 views

Metric Image Rectification using Camera Angle and Focal Length

I'm trying to measure the size of an object in millimetres from an close-range image of the object captured with an angled camera. The application is intended to be from a smartphone, so we can't ...
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12 views

Transformation Matrix for particular problem

I have a question regarding transformation matrices. I have two images both showing a table. I have coordinates of the corners of the tables, and now I want to apply a transform to 1 of the images so ...
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674 views

Understanding cross ratio and harmonic conjugates

I'm studying projective geometry and I'm really having trouble with ''grokking'' what's it all about. Is there an easy/intuitive/visual way to understand cross ratio? I understand that it's ...