Questions tagged [projective-geometry]
Projective geometry is closely related to perspective geometry. These types of geometry originated with artists around the 14th century.
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Finding the matrix $A$ of a projective transformation $\pi_A$
I'm struggling to understand this question about projective transformations:
Define the $3 \times 3$ real matrix
$$
A = \begin{pmatrix}
1 & r & s \\
2 & t & 1 \\
u & -1 & v
...
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Term for bivariate homogeneous tensor
This is a homogeneous coordinate matrix:
$
\begin{bmatrix}
x_0 & y_0 & 1 \\
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
...
\end{bmatrix}
$
Is there standard nomenclature for a bivariate ...
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I have a drawing where I have a sloping line and a point not on that line. How can I draw a line through that point that meets the line. [closed]
I have a drawing that has a sloping line and a point not on that line.
I want to draw a line passing through the given point that intersects the given line in a point outside of the paper.
I believe ...
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Proof of embeddibility of projective smooth $k$-scheme with dimension $d$ in $\mathbb{P}^{2d+1}_k$ ( Part 2, Gortz, Wedhorn )
I am reading the Gortz, Wedhorn's Algebraic Geometry, proof on Theorem 14.132 and stuck at some statements. ( Can anyone who have the Gortz, Wedhorn's book help? )
EDIT : This post is not duplicate. ...
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What is the conic $D$ in this case?
Let a Steiner conic $D$ be obtained from two pencils of lines $\check{A}$ and $\check{B}$, between
which, as lines in $\check{\mathbb{P}^2}$, a projective map is given $f:\check{A} \to \check{B}$. Let ...
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Parallelogram with an external point connected to vertices
Let $DBC$ be a triangle and $A'$ be a point inside the triangle such that $\angle DBA'$ is equal to $\angle A'CD$. Let $E$ be such that $BA'CE$ is a parallelogram. Show that $\angle BDE$ is equal to $\...
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Mathematics for Perspective Drawing
I am an undergraduate math major who likes to draw, and I would like to learn the math behind perspective drawing.
I recently watched this video: Everything about Perspective & Correct ...
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Minimum degree of a holomorphic map from an algebraic curve of genus g to the Riemann sphere
This is a problem I found on the Rick Miranda's book.
Problem
What is the minimum integer $k$ such that for every curve $X$ of a fixed genus $g$ there is a holomorphic map $F: X \rightarrow \mathbb{P}^...
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How to sort the cards in Dobble/Spot it!
The game "Dobble" ("Spot it!" in the USA; see wikipedia for some details) is a card game. Each card has 8 symbols printed on them; each pair of card has exactly one common symbol. ...
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Can we define division by 0 in an extended algebraic structure? [duplicate]
Body:
In the usual field of real or complex numbers, division by 0 is undefined because no x satisfies 0x=1.
However, various extensions (projective arithmetic, wheels, non-standard analysis, or ...
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Dual statements in Projective Geometry: Why sometimes two points becomes one line?
The principle of duality in projective geometry, from wikipedia:
In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by ...
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Proof that point-tangent pairs on circles reciprocate into tangent-point pairs on conics
On page 144 of Coxeter's $\textit{Geometry Revisited}$, in the chapter on Projective Geometry, in the course of extending the Euclidean plane to the projective plane, Coxeter has this paragraph:
"...
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A rational map from a conic to a line through its tangent intersection
Let $C$ be a nondegenerate conic in $\mathbb{P}^2$, and fix a line $t$.
For each point $P\in C$, let $\ell_P$ denote the tangent to $C$ at $P$.
Define the map
$$
\Phi_C:\; P \longmapsto \ell_P\cap t.
$...
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In $PGL(n,\mathbb{R})$, is every real element strongly real?
Let $A\in GL(3,\mathbb R)$ and let $[A]\in PGL(3,\mathbb R)$ be its projective class.
Assume $[A]$ is real in $PGL(3,\mathbb R)$, i.e. $[A]$ is conjugate to its inverse $[A^{-1}]$.
Geometrically (via ...
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Intersection of stabilizers in $PGL(3)$ of a smooth doubly ruled quadric surface in $\mathbb{P}^3$ and two skew lines on it
Let
$$
Q = \{[X:Y:Z:W]\in \mathbb{RP}^3 \mid XW - YZ = 0\},
$$
a smooth quadric surface. Using the Segre embedding
$$
\Sigma:\ \mathbb{P}^1\times \mathbb{P}^1 \to \mathbb{P}^3,\quad
([x:y],[z:w]) \...
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All all embeddings of $\mathrm{PGL}(2,\mathbb{R})$ into $\mathrm{PGL}(3,\mathbb{R})$ stabilizer of some smooth conic?
The automorphism group of a smooth conic in $\mathbb{RP}^2$ is isomorphic to $\mathrm{PGL}(2,\mathbb{R})$. We get an embedding
$$
\mathrm{PGL}(2,\mathbb{R}) \;\longrightarrow\; \mathrm{PGL}(3,\mathbb{...
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Hartshorne's proof of Theorem 7.2, the Projective Dimension Theorem.
Hartshorne has the following theorem:
Theorem 7.2. (Projective Dimension Theorem) Let $Y$, $Z$ be varieties of dimensions $r$, $s$ in $\mathbb{P}^n$. Then every irreducible component of $Y\cap Z$ has ...
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Are non-singular projective curves necessarily irreducible?
I've been working on the following Hartshorne problem (I.6.4):
Let $Y$ be a nonsingular projective curve. Show that every nonconstant rational function $f$ on $Y$ defines a surjective morphism $\...
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Generalization of fundamental theorem of projective geometry
Let $M$ and $N$ be complete, simply connected, nonpositively curved Riemannian (i.e. Hadamard) manifolds of dimension greater than $1$. Is a bijection from $M$ to $N$ taking lines of $M$ to lines of $...
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Characterization of harmonic pairs via an $O$-independent cotangent ratio
Let $C,D,A,B,A',B'$ be six distinct collinear points, with $O$ any point not lying on $CD$.
Suppose that the following quantity has a finite limit, independent of the horizontal position of $O$, as $O$...
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Harmonic Conjugates and Orthogonal Circles
I have been working on a geometry PDF relating to orthogonal circles (The Mathematical Gazette, Vol. 5, No. 86 (May, 1910), pp. 282-283) but I can't understand a part of a proof provided there, here ...
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Is the Mandelbrot set a projection of a 3D object?
Is the Mandelbrot set a projection of a 3D object? Some of the images I have seen look like there is depth if you allow yourself to look at it that way. See attached image for example. Some views look ...
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Intuition behind projective frame
I've really tried to desperation before asking here. In a few words, I simply can't think visually about a projective frame. I don't know what to imagine.
Vector spaces
With a basis of a vector space, ...
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Movable points on a conic that retain the same Pascal line.
Pascal's theorem defines the associated red concurrent straight line when $(A,B,C,a,b,c,I)$ points on a conic are all fixed at start.
Now points $(A,b,I)$ are kept fixed as before but moving other ...
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Conics and tangents
Conic $\mathcal{G}$ is inscribed in $\triangle A B C$ and conic $\mathcal{K}$ tangent to lines $A B, A C$ at points $B, C$. Let conics $\mathcal{G}$ and $\mathcal{K}$ intersects at two points $P, Q$. ...
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Intersection of moving curves with known base points in complex projective plane
If you have two curves $\mathcal{C}_1, \mathcal{C}_2$ of homogeneous degree $m$ and degree $n$ in the complex projective plane, along with a parameter $[s:t] \in \mathbb{P}^1$ parameterizing the ...
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Inversive geometry problem
Let $\Omega$ be a circle with center $O$, let $A$ and $B$ be two points on a diameter $XY$ of $\Omega$. Let $X_1$ and $Y_1$ be the points of intersection, on opposite sides of the line $AB$, of the ...
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Prove that $U$ lies on the perpendicular bisector of $AI$ using cross ratios
Let $I$ and $O$ be the incenter and circumcenter of $\triangle ABC$. Let $D$ be on $BC$ such that $ID \perp BC$. Let $E = AO \cap BC$, and let $T$ be a point such that $AT \perp BC$ and $IT \parallel ...
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Prove that a point on the polar of $A$ is the incenter of $\triangle ABC$.
Let circle $\omega$ be internally tangent to $\Omega$ at $D$. Let $O$ and $O'$ be the centers of $\Omega$ and $\omega$. Let $A$, $B$, and $C$ be on $\Omega$ such that $AB$ and $AC$ are tangent to $\...
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Prove that O,P,E collinear if $E=AC\cap BD$, O is the circumcenter, P in ABCD s.t. $\angle PAB+\angle PCB=\angle PBC+\angle PDC=90^\circ$.
A convex quadrilateral $ABCD$ on a circle $\omega$ with center $O$ has $AC \neq BD$ and $AC\cap BD = E$. $P$ is inside $ABCD$ such that $\angle PAB+\angle PCB = \angle PBC + \angle PDC = 90^\circ$. ...
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Lines divide plane into areas with at least three sides
Here is an exercise in Polynomial Methods in Combinatorics.
Suppose that $\mathfrak L$ is a finite set of lines in $\mathbb{RP}^2$. We say the lines of $\mathfrak L$ are concurrent if there is a ...
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What math subjects are relevant for someone wanting to learn how and why plane and solid geometry, projective geometry work in the context of art?
everyone! I am hoping to get some direction and book recommendations. I am an artist and have been learning from an art teacher a little about the role that geometry played for the Old Masters and the ...
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Sparse Cyclic Sum sets
At Perfect Golomb Circular Rulers, there are examples of circular rulers that measure all arc lengths up to a given value. Here are circular rulers with 3, 4, 5, 6, 7, 8, 9 and 10 marks that measure ...
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When is the projection of optimal solution of an optimization problem into a lower subspace also optimal?
I have a convex vector set over which we need to maximize the Frobenius norm (therefore is not a convex optimization).
I have the optimal solution of the problem over the convex set. Now if I consider ...
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Definition of a homogeneous polynomial of degree $\mathit d$ and theorems involving scalar properties
Let $\mathit K$ be a field, and let $d \in \mathbb{Z}_{\geq{0}}$. Define $f$ to be homogeneous of degree $d$ if and only if every monomial of $f$ with a nonzero coefficient has degree $d$.
The ...
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Blow up of projective space as incidence variety?
Blow ups are often described intuitively as "pulling apart via normal directions". One way in which this is precise is that the exceptional divisor is the projectivised normal cone. I'm ...
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Projective space - lines meeting each other
I am learning projective geometry and have 3 questions, highly related to each other, so I will ask all of them here. Thank you in advance as I'm really struggling to grasp all this.
Assume on $z = 1$ ...
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Is the hessian of a smooth planar projective complex cubic again smooth?
I have a planar complex projective cubic, let’s call it $F$. I’ve proven that it’s nonsingular and I’m now asked to prove that $D=\det(H(F))$ is again a smooth cubic. ($H$ is the hessian matrix of $F$....
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Hint for determining the size of incidence tables in Hall's free extension process
In Finite Geometries by György Kiss and Tamás Szőnyi, Exercise 1.10 reads:
(Kárteszi) Determine the size of the incidence tables obtained in the
rounds of Hall’s free extension process.
I've been ...
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How to proof with projective geometry
I'm studying for an upcoming test about projective geometry with old tests and one question that always pops up is to proof something using projective geometry, without using any calculations, but I'm ...
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Number of affine open subsets covering a locally closed subset of projective space
Let $X\subset \mathbb{P}^m$ be a quasi-projective variety of dimension $n$ over an algebraically closed field $k$. This is, $X$ is an open subset of its (Zariski) closure $Y=\overline{X}$.
I want to ...
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Prop III.1.5: the invariant differential w associated with WF is s.t. div(w) = 0.
In pp.48, Proposition III.1.5 of Silverman "The Arithmetic of Elliptic Curves," we consider a curve $E$ in Weierstrass form and the differential
$$\omega = \frac{dx}{F_y} = -\frac{dy}{F_x}$$
...
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2
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Constructing the structure sheaf of $\operatorname{Proj} S_{\bullet}$ by gluing sheaves
$\DeclareMathOperator{\Proj}{Proj} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\deg}{deg}$Ravi Vakil's book The Rising Sea defines the structure sheaf of $\Proj S_{\bullet}$ (where $S_{\...
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4-th degree homogene curves in projective space
I m thinking a lot about the following statement "In $\mathbb{C} P^2$, for every 3 points $X,Y,Z \in \mathbb{C}P^2$ there exists a quartic curve such as it passes through these points and for ...
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Why is the projection map $\mathbb{P}^n\times \mathbb{P}^m\rightarrow \mathbb{P}^n$ closed?
Question
Let $k$ be an arbitrary field, $\mathbb{P}^n$ be the projective space over $k$. We identify $\mathbb{P}^n\times \mathbb{P}^m$ as a closed subset of $\mathbb{P}^{(n+1)(m+1)-1}$ via Segre ...
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Why is $\{[x_0:\cdots : x_n]\}\times \mathbb{P}^m$ homeomorphic to $\mathbb{P}^m$?
Quesotin
Let $k$ be an arbitrary field, $\mathbb{P}^n$ be the projective space over $k$. We identify $\mathbb{P}^n\times \mathbb{P}^m$ as a closed subset of $\mathbb{P}^{(n+1)(m+1)-1}$ via Segre ...
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What is the Geometric Interpretation of Homogeneous Coordinates?
In linear algebra, given a vector space and a basis, the interpretation of a vector's coordinates are quite clear to me: a tuple of numbers telling how much one should move in the direction of each of ...
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Confusion about ideals of hypersurfaces in $\mathbb{P}^n$, as in Gathmann.
As in Ideals of hypersurfaces, Ex. 6.33, we consider a hypersurface $X \subset \mathbb{P}^n$. Then we know that $\dim X = \dim \mathbb{P}^n-1 = n-1$. Then it is said that "without loss of ...
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Set of real points of the projective variety $V_p(x_1^2-x_2^2-x_0^2) \subset \mathbb{P}^2$
I am confused by Gathmann, Example 6.15, and would appreciate anyone pointing out misconceptions about the following referenced example:
Consider $f:= x_1^2-x_2^2-x_0^2 \in \mathbb{C}[x_0,x_1,x_2]$. ...
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Is there a formula for the linear distance between the two limits of an Euler Spiral?
Can anyone provide a general overview of the geometry of the proportions of an Euler Spiral to a novice? I've been curious especially about the linear distance between to the two limits, i.e. if a ...
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