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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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I don't understand how the Cross-Ratio works in Projective Geometry [closed]

Recently started studying a bit of Projective Geometry, but I think I'm a little stuck at the point of what the cross-ratio is, since it only seems to remain invariant under projections from 1D to 1D, ...
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Looking for an example of a real (hyper)cubic surface with special restriction

Need help looking for an example of a cubic surface over $\mathbf{R}$: it has two connected components, one of the components is convex. Here's my thoughts so far: The model in my head is the ...
Degenerate D's user avatar
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Visualisation of Projective quadrics

I'm stuck at extracting the geometric picture of $\mathbb{P}^1 \times \mathbb{P}^1$ in the following example from my textbook. Example 3.10: For $\mathbb{P}^1 \times \mathbb{P}^1$ the only nontrivial ...
Rowing0914's user avatar
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Double Contact Chained Ellipses Problem

A few years ago, when I played around with GeoGebra, I came up with the following conjecture. Conjecture Let $n\in\mathbb{N}, n\geq3$. Let $E$ be an ellipse, and let $E_{1}, E_{2}, \dots, E_{n}$ be ...
K. Miyamoto's user avatar
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Magnitude of sum of projections is invariant under rotation of picture plane.

Suppose you are looking at two orthogonal vectors such that they appear as a single horizontal line on the picture plane. You are standing L distance from a pivot point C in the space. If you move to ...
StimMarine's user avatar
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Consequences of intersection product equal to $0$

We work over $\mathbb{C}$. Let $X$ be a smooth projective variety, let $D$ be a nef prime divisor and let $C$ be a smooth irreducible curve. I know that, if $D\cap C=\emptyset$, then $D\cdot C = 0$. ...
konoa's user avatar
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Is this Correct about the Points in the Projective Space?

For each $i=0,…,n$, let $U_i=\{(x_0:…:x_n)\in \mathbb{P}^n(k)|x_i\neq 0\}$. Given $U_i\subseteq \mathbb{P}^n(k)$, I want to identify the points that are in the subsets $U_2$, $U_2\cap U_3$ and $\...
Mr Prof's user avatar
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Cross-ratio for more than 4 points on a line

It is known that the pairs $(\mathbb{P}^1,4 \mbox{ points})$ are classified by a $1$-dimensional family, parametrized by cross-ratio (up to an action of $S_3$). I would like to ask if the same is true ...
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Blow-up of a secant variety and its exceptional divisor

Let $X \subseteq \mathbb P^n$ be a smooth projectively normal variety. Let $Y := \mathrm{Sec}_1(X)$ be the (first) secant variety of $X$, i.e. the Zariski closure of the union of all lines in $\mathbb ...
Skadiologist's user avatar
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Stereographic Projection Using a One-Sheeted Hyperboloid

This is in reference to the stereographic projection of a one-sheeted hyperboloid, as detailed on page 199 of this book. The author visualises the inversive Minkowskian plane by using a stereographic ...
Anomander Rake's user avatar
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Projective Geometry Question on Proving Equality of Two Angles

I have been trying to solve question 3 from this handout but have not been able to make much progress. The question is: $AD$ is the altitude of an acute triangle $ABC$. Let $P$ be an arbitrary point ...
Chris Daniel's user avatar
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Matching the major and minor axis ends of the ellipse in the perspective projection of the circle

I mostly get the circle as an ellipse in perspective projection, but I could not understand exactly which parts of the circle in space correspond to the major and minor axis ends of this ellipse, I am ...
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Quadric equation in Projective Geometry

It is well known that 5 points in $\mathbb{R}^2$ define a conic. While studying the book Geometry II by Marcel Berger, I came across a theorem stating that 9 points in $P\mathbb{R}^3$ define a quadric....
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Showing the Graph of a Projective Transformation is Intersection of a Quadric Surface and a Plane

I'm trying to do Exercise 3.6 at the end of this pdf: Let $\tau: P^1(\mathbf{R}) \rightarrow P^1(\mathbf{R})$ be a projective transformation and consider its graph $$ \Gamma_\tau \subset P^1(\mathbf{...
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If the number of intersection of two conics is an odd number, the quadratic forms are not simultaneous diagonalizable

I'm trying to do Exercise 3.6 at the end of this pdf (in $\Bbb CP^2$): Show that the two quadratic forms $$x^2+y^2-z^2, \quad x^2+y^2-y z$$ cannot be simultaneously diagonalized. Attempt 1: Their ...
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Constructing the Lie Quadric Using v-Asymptotic Tangent Lines

I am studying the geometric construction of the Lie quadric as outlined in a text, and I need some clarification on the process. The description is as follows: '' The geometric definition is as ...
User0212's user avatar
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Projective varieties contained in dense open subsets

Let $X$ be a smooth irreducible projective variety over the complex numbers. Let $U$ be a nontrivial dense open subset of $X$. Does there exist a projective curve $C$ inside $U$? My attempt: Let's ...
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Shadow a Cube with body diagonal along y axis

We have a cube with its body diagonal (greatest diagonal joining two opposite vertices) aligned along the y axis. Thus, the cube stands on one vertex with the opposite vertex directly above it. The ...
BlueInfinite1729's user avatar
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Projective space as Grassmannian

I want to understand how (or rather, why) the schemes representing the functors of $n-1$-dimensional projective space and the Grassmannian of lines are equivalent. More concretely, for an integer $n &...
LurchiDerLurch's user avatar
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Projection and depth map of a parabolic cylinder.

The problem: I've been working on a project involving 3D graphics and camera projections, and I came across a problem that I need some help with. I have a parabolic cylinder described by the equation $...
Surzilla's user avatar
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Sum of pullbacks of SNC divsors again SNC?

Let $X$ be a smooth projective variety (over $\mathbb C$), $\Delta$ an effective SNC (simple normal crossings) divisor on $X$. Consider moreover $\mathbb P^1_{\mathbb C}$ and a SNC divisor $a+b$ on $\...
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Self-polar triangle with two sides a secant has third side a secant?

This is from Coxeter Projective Geometry, see http://voutsadakis.com/TEACH/LECTURES/PROJECTIVE/Chapter10.pdf page 13 , an incidence table of a projective plane with 31 points and lines ($PG(2,5)$). A ...
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Extension complexity - Equivalent definitions

I'm confused by two, apparently equivalent, definitions of extension complexity. See the attached screenshot: From: Sparse sums of squares on finite abelian groups and improved semidefinite lifts ...
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Polar curve of a non singular projective cubic curve with respect to inflection point is union of two distict lines.

Hey guys I am currently struggling with a question that goes as follows. Let $C$ be a non-singular projective cubic and let $p \in C$ be an inflection with tangent line $T$. Show that the polar curve ...
Dorelanië's user avatar
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Generic planar sections of a projective variety

In an article by Broberg and Salberger, it is stated that The set of pairs $(\Lambda,F)\in\mathbb G(k,n)\times \mathbb P\left(\mathbb Q_d[X_0,\ldots,X_n]\right)$ for which $\Lambda\cap V(F)$ is ...
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Axis of a projection with center $Q$ on a bisector of one of the angles formed by $l_1,l_2$

Let $l_1,l_2$ be distinct lines intersecting at the point $P$. Let $m_1$ be the bisector of one of the angles formed by the two lines $l_1$, $l_2$ and $Q \neq P$ a point on $m_1$. Let $Q^{\vee}:l_1 \...
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$P,Q \in CP^1$, |P|=|Q|=4. $\exists \phi s.t. \phi(P) = Q \iff f(s) = f(t)$, with $f(x) = \frac{(x^2-x+1)^3}{x^2(x-1)^2} $, s and p the cross-ratios

I am solving the following question: $\mathcal{P}$ = $\left\{P_1, P_2, P_3, P_4\right\} \subseteq \mathbb{C}P^1, \mathcal{Q} = \{ Q_1, Q_2, Q_3, Q_4\} \subseteq \mathbb{C}P^1$. Prove that there exists ...
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Are all continuous functions that send conics to conics in $\mathbb{R}^2$ a generalized linear fractional transform?

Motivation: Consider an arbitrary conic section in $\mathbb{R}^2$ given by $$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$ Now consider the map $$\phi: \mathbb{R}^2 \rightarrow \mathbb{R}^2, \phi_{\begin{...
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Defining the real projective line via equivalence relation

I'd just like to know if my approach is correct. Let $\sim$ be an equivalence relation on $\mathbb R^2$ defined as: $$(x_1,y_1)\sim(x_2,y_2) \iff \exists\lambda\in\mathbb R\smallsetminus\{0\} : x_1 = \...
Elvis's user avatar
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Regular conic $\mathcal{C}$ in $\mathbb{P}_{2}({\mathbb{C}})$ with equation $XAX^T$; tangent lines to conic

Given a non-null row matrix $D$ the line of homogeneous equation $DX^T=0$ is tangent to $\mathcal{C}$ iff $DA^{-1}D^T=0$; where $XAX^T=0$ is the matrix form of the regular conic. Furthermore, given $P ...
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Complete quadrangle $ABCD$; line $r$

Consider the complete quadrangle $ABCD$ and a line $r$ that doesn't pass through any of these points. $X= r \cap AB; Y= r \cap AC; Z=r \cap AD; T=r \cap BC; U=r \cap BD, V= R \cap CD$ and $X',Y',Z',T'...
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About the projection of a variety from a generic plane being birational

Take $Z\subseteq \mathbb P^N$ an irreducible subvariety of dimension $m$. For $\Lambda$ a projective subspace of dimension $N-m-2$ not intersecting $Z$, define $$\rho_\Lambda:Z\rightarrow\mathbb G(N-m-...
Simon Pitte's user avatar
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Bertini's Theorem for a linear system with a single base point

Let $(X,H)$ be a polarised non-singular projective variety over $\mathbb C$ with $\dim X \geqslant 2$. Suppose that the base locus $\operatorname{Bs}|H| = \{p\}$ is a single point. Then Bertini's ...
Skadiologist's user avatar
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Question about Fulton’s “Intersection Theory”, example 8.4.6

In Fulton’s aforementioned book, after stating Bezout’s theorem, he states that a classical application of it is to show that an irreducible projective variety $X\subseteq\mathbb P^n$ of dimension $m$ ...
Simon Pitte's user avatar
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Classification of isotropic connected Riemannian manifolds

This answer mentions a classification of all isotropic connected Riemannian manifolds up to isometry. I am looking for a reference with a proof of this classification.
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Image of region under a projective transformation

Problem goes like this Given the region ${F} = \{ (x, y) \in \mathbb{R}^2 \mid 0 \leq x \leq 1, y \geq 0 \}$ and the matrix $M$ of the projective transformation $f$ of the extended affine plane, $$ M =...
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Harmonic conjugate in a perspective of center $C \notin r$

Let $f: P_{2}(\mathbb{R}) \to P_{2}(\mathbb{R})$ be an involutive perspective with axis $r$ and center $C \notin r$,i.e, the fixed points of $f$ are those that belong to $r$ and also $C$. Given $P \...
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Projective transformation- Perspective with center $C$ on the line $r$

Suppose that the homography/projective transformation $f:P_{2}(\mathbb{R}) \to P_{2}(\mathbb{R})$ that's not the identity, but it has a line $r$ of fixed points and doesn't fix any exterior point of $...
J P's user avatar
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Thurston's metric on $\widetilde{SL(2,\mathbb{R})}$ is twisted. So is this paper "wrong"?

$\widetilde{SL(2,\mathbb{R})}$ is the universal covering space of $SL(2,\mathbb{R})$, the group of $2\times 2$ real matrices with determinant $1$. This group acts via Möbius transformations on the ...
Derso's user avatar
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2 votes
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computing fixed points of a toric action

Let $W_1=\{ ([a : b], [x : y : z]) \in \Bbb{CP^1} \times \Bbb{CP^2}: ay=bx \}$ Given the action of $\Bbb T^2$ on $W_1$ defined as $(u, v) \cdot ([a : b], [x : y : z]) = ([ua : b], [ux : y : vz])$ How ...
darkside's user avatar
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3 votes
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Show that the polar line of the exincenter passes through this midpoint

We're given a triangle $\triangle ABC$ whose incenter is $I$ and its $A$-exincenter is $I_A$. Let line $EF$ be the polar of $A$ with respect to the incircle of $\triangle ABC$, let $G = EF \cap BC$, ...
hellofriends's user avatar
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Is there an isotomic analogous of circular points of infinity?

In isogonal pivotal (with pivot at the line of infinity) cubics with respect to a triangle $\triangle ABC$. By a suitable projective transformation, fixing $A$,$B$, and $C$, sending the incenter to ...
Curious's user avatar
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Incidence Geometry and projective plane

Consider an interpretation of incidence geometry $(P,L,I)$, where $P \in \mathbb{R} - \{0\}$, $L \in \mathbb{R} - \{0\}$ and $I = \{(p,l) \in P \times L : pl \in \mathbb{Z}\}$. Is this interpretation ...
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Which differential equations are invariant under change of camera projection?

For background, I am working in the plane $\mathbb{R}^2$. I know that the derivatives $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ are invariant under translation. I know that the ...
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Projective geometry: implications of unicity of conic through 5 points of which no 3 collinear (no coordinates)

This is in the projective plane. In several books, Coxeter proves that 5 points of which no 3 collinear determine a unique conic, or dually, 5 lines of which no 3 concurrent determine a unique conic ...
user643717's user avatar
2 votes
0 answers
12 views

Understanding homogeneous projective rational manifolds

Every complex flag manifold is a homogeneous projective rational manifold (HPRM). But are there examples of HPRMs that are not complex flags? Can somebody provide examples of such spaces? For some ...
user avatar
1 vote
1 answer
55 views

Does duality mapping preserve cross ratio?

I'm new to projective geometry. I learned the definition of cross ratio of 4 collinear points and that of 4 concurrent lines in $\mathit{P}\mathbb{R}^{2}$. The question is, by duality we can map 4 ...
LehrLukas's user avatar
1 vote
1 answer
71 views

A geometric algebra to represent lines and circles

I recently came across this video on Projective Geometric Algebra A Swift Introduction to Projective Geometric Algebra. I was surprised by how powerful it was in finding intersection of lines, lines ...
xyz1234's user avatar
  • 103
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1 answer
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Orthogonal projection of a double-sided cone onto a plane

I would like to solve the following problem: a double cone with vertex at the origin of coordinates (0, 0, 0) is given (See fig. 1). The blue cone is symmetrical to the yellow one w.r.t. the origin; ...
Gino's user avatar
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4 votes
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Projective plane over the octonions (Cayley plane)

You can use octonion algebra's $\mathbb{O}$ over a field to coordinatize projective planes. They are called Cayley planes as far as I know. You can't use the usual approach with homogeneous ...
Vincent Batens's user avatar

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