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

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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 ...
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Can one prove Brianchon's theorem using Ceva's theorem?

Can I prove Brianchon's theorem using Ceva's? I am also wondering if parabola and hyperbola can be inscribed in a hexagon?
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Injective and continuous function that is an embedding

Consider $n,d\in \mathbb N$ and $N= {n+d\choose d}-1$, then the well known $d$-uple embedding: $$\rho_d: \mathbb P^n(\mathbb C)\longrightarrow\mathbb P^N(\mathbb C)$$ is a continuous (respect to ...
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Blow-up in Projective Space: Choosing the Appropriate Chart

Consider $x^2+y+\alpha=0$ (made-up example), where $(x,y)\in\mathbb{C}^2$ and $\alpha\in\mathbb C$ is a free parameter. The equation defines a family of curves. The curves have no common points in ...
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Geometries (Euclidean and Projective)

We can think of Euclidean Geometry and Cartesian (Coordinate) Geometry as equivalent, in the sense that some proposition is true in Euclidean Geometry iff it's true in Coordinate Geometry. It makes ...
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Area of projection of cube in $\mathbb{Z}^3$ onto a hyperplane

A cube with vertices $(\pm 1, \pm 1, \pm 1)$ gets projected into the plane perpendicular to vector $\mathbf{n}\in S^2$. The projection is a hexagon, how do I find the area? I think I can just ...
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21 views

Projective roots of a homogeneous polynomial

Suppose that $f(X,Y)\in\mathbb C[X,Y]$ is a homogeneous polynomial of degree $n$, then we can consider it as a function on $\mathbb P^1_\mathbb C$. It has at most $n+1$ projective roots (points of ...
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39 views

Proving a theorem using Pappus' theorem

I need some help. I want to prove Desargues' theorem via using Pappus' theorem. And I don't know how. Please, help me!
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5d Basis Vectors of Penrose's Tilings

I have been writing some software to display/render Penrose tilings. I was hoping to use the approach of projecting a 5-dimensional lattice into 2d and apply some coloring based on regions etc. I ...
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Packing infinitely many ellipses into a circle

Given a circle $C$, and an infinite set $S$ of mutually disjoint ellipses which are inside and tangent to $C$, prove that there must exist a disk $D$ which lies inside $C$ but outside every ellipse. ...
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16 views

Preserving incidence relation proof

How can one prove via analytic method that projective map preserves incidence relation?
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The cross-ratio of four points is a real number exactly when the four points lye on a line or a circle

Let $z,z_1,z_2,z_3$ be four points on the extended plane. Their cross-ratio $(z,z_2,z_3,z_4)$ by definition is the image $Tz$ of $z$ under the Möbius transformation $T$ that sends $z_1,z_2,z_3$ to ...
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While extending the $\pi_0$ configuration, shouldn't it end at $\pi_3$?

Kindly refer to pg. 9 of Foundations of Projective Geometry by Hartshorne. $\pi_0$ contains points such that there can only be $1$ line between two points. There many not be lines joining all pairs ...
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37 views

A doubt in the proof of Desargues' Theorem.

I have a question regarding the proof of Desargues' Theorem. When the traingles $ABC$ and $A'B'C'$ are assumed to be lying on the same plane. A point $X$ is taken outside that plane, and the lines ...
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Area of an ellipse proportional to integral of cross-ellipse distances?

I am curious if the area of an ellipse can be shown to be proportional to the integral of all cross-ellipse distances. Before I define cross-ellipse distance, I will give a motivating example from a ...
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A little problem about $4$ points in $\mathbb P^1(\mathbb C)$

I have to solve the following problem: Consider a set $D\subseteq\mathbb P^1(\mathbb Q)\subset \mathbb P^1(\mathbb C)$ such that $|D|>3$. Then there exists a Moebius transformation $M$, and ...
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A question from “Foundations of Projective Geometry” by Hartshorne.

"Foundations of Projective Geometry" by Hartshorne says the following: The completion of the affine plane of four points is a projective plane with 7 points. The affine plane of $4$ points is ...
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18 views

Vertices and indices of a tesseract before and after projection

how to define the vertices and indices of a tesseract before and after projection into 3D ,is the way in which vertices are connected to form lines " Wireframe " in 4D remains the same after ...
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21 views

Origin in homogeneous coordinates [closed]

In homogeneous coordinates in projective geometry, is the origin given by: (0,0,w) for all w?
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Enumerative projective geometry

I am wondering whether for any two lines $\mathfrak{L}, \mathfrak{L'}$ and any point $\mathfrak{P}$ in $\mathbf{P}^3$ there is a line having nonempty intersection with all of $\mathfrak{L}, ...
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117 views

Blow-ups in Projective Space

This is in regards to a question (no solutions or comments thus far :-() I asked earlier in regards to the blow-up of an elliptic curve: Question Let $f(x,y)=y^2-4x^3+ax+b$, where $(x,y)\in\mathbb ...
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22 views

A question about three collinear points

This video (at 44:00) says that in a projective space if three points are collinear, and two of those points lie at infinity, then the third point will also have to lie at infinity. I wonder why ...
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A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
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179 views

Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
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Movements in the complex projective plane.

My textbook denotes movements in the Euclidean plane by $P(a,b,\alpha):\mathbb R^2\to\mathbb R^2$. Each movement depends on three numbers $a,b,\alpha\in\mathbb R$ and is given by $(1)$ $$(1)\quad ...
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How to define a “distance” from point to line in 3D projective space which is projectively invariant?

Since the concept of distance in Euclidean space is not invariant in projective space, that is , distance is invariant under Euclidean transformations but not under projective transformations, is it ...
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Showing there are 6 possible values for the cross ratio

If we look at the cross ratio $(x_0 x_1:x_2 x_3) = \lambda$ of 4 points in projective space, I can see that by looking at all possible permutations (24 of them) of the points we can see that only 6 of ...
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Exponent of Projection (T^2=T)

T:V->V is a linear transformation, also it's a projection, i.e. T^2=T. Find e^T. I thought of using the fact that if T=T^2 then e^T=e^(T^2) but I guess that doesn't work because exponent is a sum of ...
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51 views

Projective variety minus hyperplane $=$ affine variety

Claim: Let $V \subset \mathbb{C}P^n$ be a non-singular projective algebraic variety of complex dimension $k$ and let $P \subset \mathbb{C}P^n$ be a hyperplane. Then $V \setminus (V \cap P)$ is a ...
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Projection matrix to project a point in a plane

How to determinate the 4x4 S matrix so that the P gets projected into Q, on the XZ (Y=0) plane? Q = S P
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What are geodesics in H$^2$?

Specifically, I am looking at a question that asks What axioms for a projective plane fail in the this space? Any two “points” are contained in a unique “line.” Any two “lines” contain a unique ...
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Chirality of a Möbius band without boundary?

In this answer it is remarked that the real projective plane minus one point is homeomorphic to the Möbius strip without boundary. A normal Möbius strip is topologically equivalent to a real ...
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On existence of a tangent line passing through a given point

Question Suppose $k$ is an algebraically closed field of characteristic $0$, and $C\subseteq\mathbb P^2(k)$ is an irreducible projective plane curve of degree $n>1$, and $P$ is a point on $\mathbb ...
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$PGL_d(F)$ is 2-transitive but not 3-transitive if $d > 2$

An exercise asks to prove that: If $d > 2$, then the projective general linear group $PGL_d(F)$ of dimension $d$ over a field $F$ is 2-transitive but not 3-transitive on the set of points of the ...
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line at infinity

I tried solving the following question, could you have a look at my answer and tell me whether it's right or wrong? All input is appreciated. Question: Let $ABCD$ be the vertexs of a parallelogram in ...
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Applications of Finite Projective Planes

Can someone point me towards some applications of finite projective planes that are approachable without too much background knowledge? So far, I have vector spaces, Latin Squares, and Sudoku, but I ...
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What exactly does a Mobius Transformation do?

From what I understand, a Mobius transformation is of the form f(z) = $\frac{Az+D}{Cz+B}$ where A,B,C, and D may be real or complex What is f(z) doing to z exactly? And what are some of the ...
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How does a linear fractional function behave like a $2\times 2$ matrix?

So I did the math for this and got \begin{align*} A &= a_1a_2 + b_1c_2\\ B &= a_1b_2 + b_1d_2\\ C &= c_1a_2 + d_1c_2\\ D &= c_1b_2 + d_1d_2. \end{align*} My book does not talk ...
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associativity on elliptic curves — Milne's proof

In the proof that the group law on an Elliptic curve is associative, Milne (http://www.jmilne.org/math/Books/ectext5.pdf, page 28) sets up 3 cubics, and claims that they all contain the $8$ points ...
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What does it mean to call horizontal lines through O the “points at infinity” in real projective plane $RP^2$?

This is a picture from my book. I extended the line M to get a better idea of where $p_n$ is. It says the following: It is natural to call the horizontal lines through O the "points at infinity". ...
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What are Straight lines in the Gans Disk model of the Euclidean plane?

The answer of Blue ( http://math.stackexchange.com/a/1464/88985 ) to Hyperbolic critters studying Euclidean geometry made me interested in the Gans Disk model of the euclidean plane. Blue writes: ...
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classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with Fubini-Study metric, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\overline w, \overline z] \qquad \eta ...
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Please help me understand what “the alignement of epipoles with an axis” in some paper means

I have trouble understanding this paper. The goal is the estimation of the relative pose of two cameras $C_1$ and $C_2$, given five point correspondences. The coordinate system for the geometry is ...
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Intersection of two quadrics

How to understand (maybe, informally) why the intersection of two quadrics in general position in $\mathbb{CP}^3$ is an elliptic curve? It is obvious that it is a compact 2-manifold, i.e. a sphere ...
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Intuition behind using projective geometry for defining the addition on an elliptic curve

We already had the chord-and-tangent construction that can be used to define a way of "adding" points on an elliptic curve. Also this addition satisfies all the group laws. Still why one needs to ...
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Problem about $\mathbb{P}^3(K)$

Show that four skew lines in $\mathbb{P}^3$ have two transversals in common. I know that exist a quadric which contains three of the four lines....but i'm stuck EDIT: If the skew lines are ...
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Are maps locally preserving collinearity homographies?

Question Suppose $D\subseteq\mathbb R^2$ is an open disc (or generally a simply connected domain with a good boundary), and $f\colon D\to D$ is a bijection such that the images of collinear points ...
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elliptic curve group law

Let $C$ be an elliptic curve over a field $k \supset \mathbb{Q}$. Then given $P$ and $Q$, we can draw the line between $P$ and $Q$ (call this line $L$) and then "find the third intersection point", ...
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homogenous coordinate system vs cartesian coordinate

“The homogeneous coordinate system is used in projective geometry as much of the math ends up simpler in homogeneous coordinate space than it does in a regular Cartesian space.” Excerpt From: Haemel, ...
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covering of projective curve by affine parts

For $\mathbb{P}^n$ we can let $U_i = \{(x_1:\cdots:x_i:\cdots:x_{n+1}) : x_i \neq 0\}$. Then let $C \subset \mathbb{P}^n$ be a projective plane curve. We can decompose $C$ into a union of affine plane ...