# 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 ...
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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 ...
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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 ...
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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 ...
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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 $D\cdot C = 0$. ...
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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 ...
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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 ...
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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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### 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 ...
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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 $m$ ...
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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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### 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 ...
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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 ...
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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 $G = EF \cap BC$, ...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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; ...
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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 ...