# Tagged Questions

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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### 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: z_x = (z^\text{T}x) x, \ \ z_y = (z^\text{T}y) y, ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $OABCD$ 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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### 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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### 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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### 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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### 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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### $V\subset \mathbb{P}^n(k)$ 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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### 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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### 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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### 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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### 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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### 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 ...