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

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Write $\mathbb{P}^3_{\mathbb{C}}$ as a union of disjoint lines

Is there a set $\Gamma=\{L \subseteq \mathbb{P}^3_{\mathbb{C}}: L \textrm{ is a projective line}\}$ such that every point $p \in \mathbb{P}^3_{\mathbb{C}}$ lies on exactly one line $L_p \in \Gamma$? ...
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1answer
33 views

Number of inflection points of an algebraic projective curve

I' m trying to prove that a curve in $\mathbb{P^{2}(\mathbb{C})}$ of degree $d$ has an infinite number of inflection points or it has at most $\le 3d(d-2)$ inflection points. Let be $C$ the curve and ...
3
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1answer
26 views

Are points in general position generic points?

In Harris' algebraic geometry book, $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$ are said to be in general position if no $n+1$ or fewer of them are dependent. I want to prove that, if ...
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1answer
31 views

A few questions regarding Pappus' Theorem

I am trying to understand a proof of Pappus' Theorem. This is taken from the book Geometry and Topology by Miles Reid and Balázs Szendröi. Definition 1 Let $PQ$ be the line through two points $P=(x_0 ...
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30 views

Generalization of Euler theorem for homogeneous polynomials

Euler's theorem for homogeneous polynomials is well known. If $F:\mathbb{R}^{n}\rightarrow \mathbb{R}$ is a homogeneous polynomial, then we have: $x_{1}\frac{\partial F }{\partial x_{1}} + ... + ...
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74 views

Multiplicity of point as a zero of polynomial.

Let $C \subset \mathbb{P}_2$ be a projective curve defined by a homogeneous polynomial $P(x, y, z)$, and let $L \subset \mathbb{P}_2$ be the line $\{z = 0\}$. Assume $L$ is not a component of $C$. If ...
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2answers
55 views

Manifold over a Finite Field

I'm trying to either associate a manifold with a finite field, or, ideally find a way of considering finite fields as manifolds, in a non-trivial manner. I hope to be able to use this to extend ...
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1answer
30 views

If $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$ are in $d$-general position, then they are in $1$-general position.

Let $\mathcal{L}_{d}^{n}$ be the $\binom{d+n}{n}-1$ dimensional projective space of hypersurfaces of degree $d$ in $\mathbb{P}^{n}$ and $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$. We denote by ...
2
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1answer
30 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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0answers
11 views

Calculate the plane angle from 2D plane

I am analysing a squared plane from a perfect cube. This plane is distorted by the perspective view of a camera. I would like to know ask please, some approaches of how could I get to know the ...
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20 views

Camera calibration from the image of the absolute conic

It's known that once the image of the absolute conic $\omega$ is identified in an image then one can find calibration matrix $K$. \begin{align*} \omega^{-1} = KK^T \end{align*} Given the images of the ...
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54 views

Literature request: Method for constructing projective manifolds

Currently (background: I'm preparing to write a thesis in mathematical physics) I'm quite often encountering a certain method for constructing projective manifolds, where the space is specified by ...
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45 views

Is the union of two projective curves in the projective plane a projective curve?

As the title suggests, is the union of two projective curves in the projective plane a projective curve? Any help would be appreciated, thanks.
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2answers
26 views

lines through a point of the projective plane

I'm having difficulty understanding a particular example of Mumford's "Red Book". In exemple D, first chapter, he considers the set of lines passing through a point of $\mathbb{P}_2$ (do we call it ...
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12 views

Scaling axes to reflect perspective plane of an image.

Suppose I have an image which contains a road in it, and I want to be able to pinpoint the locations of cars on that road using pre-calibrated distances. How to do that using formulas that map pixel ...
0
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1answer
30 views

Conditions of $f=a+bx+cz+dx^2+exz+fz^2+…$ such that its tangent line is $z=0$ and inflection point is at the origin.

Let $x,z$ be coordinates on $k^2$ and $f\in k[x,z]$; write $f$ as $$f=a+bx+cz+dx^2+exz+fz^2+...$$ Write down the conditions in terms of $a,b,c,...$ such that (a) $P=(0,0)\in C: (f=0)$; (b) the ...
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2answers
22 views

Why cannot the homogeneous coodinates be zero?

Given a point (x, y) on the Euclidean plane, for any non-zero real number Z, the triple (xZ, yZ, Z) is called a set of homogeneous coordinates for the point. Why can't Z be zero?
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1answer
37 views

Find the image of circular points by fitting conics

According to Single Axis Geometry by Fitting Conics by Jiang et al., one can compute the image of the circular points in a picture from conics which are the images of circles. Fit two conics to ...
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2answers
90 views

How can hypersurfaces “know” the degree of their defining polynomials?

I'm currently trying to learn some complex and projective geometry. There is one issue bugging me again and again, from different perspectives, and I just can't get my head around it. One incarnation ...
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1answer
20 views

Question in geometry on Fano Plane

Hello friends I have a geometry homework question asking me to do the following: I need to prove all projective planes of order two are isomorphic by showing they are all isomorphic to the Fano Plane. ...
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9 views

Generalizing all singular cubic curves in the projective tropical plane with genus zero

How can I qualitatively classify all these curves under the condition that they must all be fully supported with no double lines touching infinity?
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1answer
39 views

How to calculate true lengths from perspective projection?

Suppose that I have a single point perspective drawing like . and suppose also that I know some of the real horizontal distances and distances along lines converging to vanishing point. E.g if i know ...
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0answers
53 views

Is projective space really a moduli space for lines through the origin?

The Wikipedia page for Moduli spaces states that real projective space $\mathbb{RP}^n$ is a moduli space which parametrizes the space of lines in $\mathbb R^{n+1}$ passing through the origin. ...
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18 views

Vanishing points from three collinear points

I would like to find the 2D vanishing point from a three collinear points as is shown in "Multiple View Geometry in Computer Vision" Example 2.19 (see here). What I did so far: 1 - I've extracted ...
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1answer
30 views

What do the conics $x^2=-1$ and $0=1$ correspond to in $P_{\mathbb{R}}^2$?

I read this in Reid's book Undergraduate Algebraic Geometry. I don't even know whether my question is worded correctly. It says "In a suitable coordinate system, any conic in $P_{\mathbb{R}}^2$ is one ...
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1answer
42 views

Lines on a singular cubic surface

How many lines the cubic surfaces $xyz=w^3 \in \mathbb P^3$ has? I found only three: $x=w=0$, $y=w=0$ and $z=w=0$. How to prove that there are no other lines? Also, this surface is singular, is it ...
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0answers
24 views

Connection between Chladni Plates and Projective Geometry?

Does anybody know of a connection between Chladni Plates and Projective Geometry? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be ...
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0answers
10 views

Pascal and Brianchon's theorems for hyperbolic paraboloid

How would one formulate a version of these two theorems for the hyperbolic paraboloid, and what would be a simple proof? How are the classical formulations of these theorems related to this quadric?
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7 views

Prove quadrics are rational algebraic surfaces.

I have to prove that an irreducible quadric in RP^3 is a rational algebraic surface, ie, the homogeneous coordinates of any point can be expressed as polynomials in two variables. My idea was to do ...
0
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1answer
13 views

Existence of a projective transformation.

Suppose we have nine distinct points of $\mathbb{P}^2$ not all on the same line, and such that any straight line passing through two of these points also passes through a third. I want to show there ...
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1answer
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General versions for quadrics of Pascal and Brianchon theorems

I am looking for a generalization to quadrics (with proofs) of Pascal's and Brianchon's theorems. It´s for Three dimensional analytical geometry. I would be very thankful if you could point me ...
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14 views

Quadric generalizations of Pascal and Brianchon's theorems

I am looking for a generalization to quadrics (with proofs) of Pascal's and Brianchon's theorems. It´s for Three dimensional analytical geometry. I would be very thankful if you could point me ...
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2answers
106 views

Tangential space to the rational normal curve

Exercise 15.5 (Harris, Algebraic Geometry: A First Course): Describe the tangential surface to the twisted cubic curve $C \subset \mathbb P^3$. In particular, show that it is a quartic surface. What ...
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25 views

What seems to be the standard notation for Study quadric?

The study quadric is defined on $\mathbb RP^7$ with the following homogeneous quadric: $$ a_0b_0+a_1b_1+a_2b_2+a_3b_3=0,~(a_0:a_1:a_2:a_3:b_0:b_1:b_2:b_3)\in\mathbb RP^7 $$ Would it have a standard ...
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9 views

Parametrization of a Conic to Compute Multiplicity of Intersection

Given the two curves $C: y=x^2$ and $D: y=2x^2$ how can I use the parametrization to show that the multiplicity at (0,0) is 2?
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1answer
42 views

Question about $\mathbb{P^1}$ - without loss of generality doubt

I want to show that: Given three distinct points $P_1,P_2,P_3 \in \mathbb{P}^1$ and three distinct points $Q_1,Q_2,Q_3 \in \mathbb{P}^1$, there is a unique isomorphism $f: \mathbb{P}^1 \rightarrow ...
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1answer
32 views

Calculate area with cross ratio

Suppose that I am given a drawing of a table, on which a book lies in one of the corners. The measures of the book are known, how can I find the measures of the table using cross ratios?
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28 views

Construct sum of points on projective line

In a course in Geometry we where asked to geometrically construct the sum of two points $x$ and $y$ on a projective line by help of "the theorem on complete quadrilaterals". This theorem (as stated in ...
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49 views

Gluing construction of the projective space scheme.

When constructing the projective space scheme $\mathbb{P}_R^n$ for a ring $R$, we may take the subrings $$ A_i = R\left[\tfrac{X_0}{X_i}, \ldots, \widehat{\tfrac{X_i}{X_i}}, \ldots, ...
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2answers
38 views

Basic Projective Geometry Question

Can someone help me to see why any two points in $\mathbb P^1$ are linearly equivalent as divisors? If this is true, how come two points on a smooth projective cubic curve are not linearly ...
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1answer
30 views

Calculating points of intersection and their multiplicities

p=(0,0), C: y=x^2, D: y=2x^2. Using Bezout's theorem and symmetries, show that ip(C,D)=2. Show this one more time using a parametrization of the conic. First off, Bezout's Theorem says there are 4 ...
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1answer
31 views

Why $f$ has to be invertible?

Let $V,W$ be two $(n+1)$-dim. vector spaces. Proposition: Let $P(V), P(W)$ be $n$-dim. projective spaces. $\tau \colon P(V) \to P(W)$ is a projective transformation, if it comes from an ...
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1answer
42 views

What this notation R^3 ∖ (0, 0, 0) means?

I was reading a "Projective Space" article on Wikipedia, when I came across this line "equivalent definition is the set of equivalence classes of $\mathbb R^3 \setminus (0, 0, 0),$ i.e. 3-space ...
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1answer
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poles and zeros of function field of $\mathbb{P}^1$.

In which condition: an element of function field of $\mathbb{P}^1$ has zero or pole or no-zero&no-pole. I am thinking that: since $\mathbb{P}^1$ and $\mathbb{A}^1$ is birrationally equivalent ...
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1answer
29 views

Slice, projection, contour: A terminology question.

Consider a multivariate function, say $y=f(x_1,x_2,\dots,x_n)$, and suppose that $z=f(x_1,x_2,\dots,x_{n-1},g(x_1,x_2,\dots,x_{n-1}))$. What do we call $z$ with respect to $y$? Projection, level set, ...
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1answer
37 views

Why 9 points determine a quadric

The books I have state this redult as obvious from the definitions, but it is not clear to me why this is so.
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1answer
40 views

$d+1$ distinct points of a rational normal curve in $\mathbb{P}^{d}$ are linearly independent

Let $X\subset\mathbb{P}^{d}$ be a rational normal curve. After a change of coordinates, it is the image of the map: $\nu:\mathbb{P}^{1}\rightarrow\mathbb{P}^{d}, (a_{0}:a_{1})\mapsto ...
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Weil does not imply Cartier on variety $X$.

Show that the divisor $D$ defined by $a = b = 0$ in the variety $X \subset \mathbb{A}^4$ defined by $ad - bc = 0$ $($the cone on a smooth quadric surface$)$ is not locally principal. My attempt ...
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23 views

blocking set of projective planes

Have a quick question, considering the Desarguesian projective plane of order q, what is an upper bound of the minimal blocking set? Wikipedia says the size of the blocking set is bounded below by ...
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71 views

Is there a projective metric on a projective space induced by a p-norm?

A projective metric on a projective space is a metric on the underlying set such that shortest path with respect to this metric are parts of entire projective straight lines. The 2-norm induces the ...