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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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 ...
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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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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 $\...
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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 ...
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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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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 ...
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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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