Questions tagged [geometry]
For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.
52,570 questions
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So we know about the Cartesian coordinate system, but how to naturally arrive at it when going on a quest to identify a location?
You would need to use the concept of distance and direction in order to derive the coordinate system with the three perpendicular axes from the first principles.
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Counting disjoint $k$-tuples of lines in $\mathbb F_q^n$
Let $k$ be a positive integer. How many $k$-tuples of disjoint lines are there in $\mathbb F_q^n$? Here two lines are disjoint if they do not share a point in $\mathbb F_q^n$.
I was wondering because ...
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Cutting a Möbius strip in thirds. Why are the resulting strips interlinked?
It is well known that cutting a Möbius strip "in half" down the middle results in a band with two twists, homeomorphic to a cylinder. See this question for example.
If instead, one begins ...
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The only area–preserving linear symmetry of a strictly convex body fixing a boundary point is the identity?
If $K\subset\mathbb R^2$ is strictly convex, $T\in SL(2,\mathbb R)$ is linear with $T(K)=K$, and $T$ fixes some boundary point $p\in\partial K$, must $T$ be the identity?
Without strict convexity, we ...
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Can a hemispherical surface be flattened if tearing is allowed but no stretching is allowed in the interiors
Can a hemispherical surface be flattened if tearing is allowed but no stretching is allowed in the interiors ?
A flattening, in this case, would be a continuous mapping from the hemispherical surface ...
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Completeness of the Euclidean Postulates
In The Elements, Euclid states the postulates of geometry. The Euclidean postulates originally state that:
A line is defined by two points as the points whose difference in distances to the two ...
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How to find the side lengths of a right triangle when you only have one length and the angles? [closed]
I know that it's a 30-60-90 triangle, and that one side length (not the hypotenuse) is 8√3. It's the side connecting the 30 degree and 90 degree angle. I'm trying to find the lengths of the other two ...
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Fourier-Motzkin Elimination for Single Vertex Computation
Given a convex polytope $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$, I want to know whether we can use Fourier-Motzkin elimination (or an adaptation therefore) to compute one vertex of $P$ (or to ...
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Reference for the formula for conformal change of sectional curvature
Let $(M, \tilde{g})$ be a Riemannian manifold and let $g:=e^{2u}\tilde{g}$ be a conformal change. I'm trying to find a resource (ideally a book, or a paper where it's mentioned or derived) for the ...
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Can any superellipse $|x|^p+|y|^p=1$ arise as the envelope of a one-parameter family of curves?
I am interested in representing $C_p$ as envelopes of curves $F(x,y,a)=0$, where\begin{align}
C_p=\{(x,y)\in\mathbb{R}^2:|x|^p+|y|^p=1\}.
\end{align}
“Interesting” means it is not something like $F(x,...
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Book from Bertsimas suddenly switches from $(m+1)$ basic points to $m$ basic points in a problem that requires $(m+1)$ basic points
In the section $3.6$ of the book Introduction to Linear Optimization (Bertsimas and Dimitris), the column geometry of the simplex method is explored.
It starts with the fact that any bounded ...
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geometric hypercube defining vertices on a grid
Given $2^d$ points in n dimensions. $P \subset \mathbb R^n$, what is the most "meaningful" map $\varphi :\{0,1\}^d \to P$ so that $P$ can be interpreted as the vertices of a d-hypercube.
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Proving triangle congruence for a plane embedded in 3D
In Hilbert's "Foundations of geometry", the SAS congruence criterion for triangles is assumed as an axiom (IV, 6.) (basically - actually, Hilbert assumes something a bit weaker). From there, ...
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Why is this angle always less than $49°$?
I was drawing this configuration in GeoGebra, repeating it dozens of times, always considering any triangle $ABC$ with the centroid $G$, while maintaining the $25°$ angle.
An observation then occurred ...
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Calculating the intercept course/point of two objects on the surface of a sphere - target moving on a rhumb line, interceptor on a great circle
My question is very similar to How to find the launch direction to intercept an object moving on a sphere? But the answer there assumes both objects move on great circles. My problem involves a target ...
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Why does Titu’s lemma lead to the circumcenter, but the minimum of the sum of squared distances occurs at the centroid?
Let $A = 0$, $B = 3$, and $C = 6i$ be three points in the complex plane.
Define
$$F(z) = |z|^2 + |z - 3|^2 + |z - 6i|^2.$$
My reasoning:
Using Titu’s lemma (Engel form of Cauchy–Schwarz), we can write
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Trying to use complex bash to solve this geometry problem [closed]
Let ABC be an acute triangle with AB = AC. Let D be the point on BC
such that AD is perpendicular to BC. Let O, H, G be the circumcentre, orthocentre and
centroid of triangle ABC respectively. Suppose ...
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Find the exact path to get the ball in the hole with 2 bounces
What I was thinking was that the first bounce could hit the ball to the left wall so that it would then hit the top wall. At the top wall it could reflect to the hole? I don't think I'm thinking about ...
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IMO $2024$ Problem $4$ (Using Method of Moving Points)
I was attempting to solve Problem $4$ of the $2024$ IMO using the The Method of Moving Points.
Let $ABC$ be a triangle with $AB < AC < BC$. Let the incentre and incircle of triangle $ABC$ be $I$...
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What is the minimum side length of a cyclic polygon so that it has three sides which satisfy the triangle inequality?
We define:
Triangular Polygon. A polygon with sides $a_1,a_2,\ldots,a_n$ is called triangular if there exists at least one triple of sides $\{x,y,z\} \subset \{a_1,a_2,\ldots,a_n\}$ which satisfies ...
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Vector tangent to meridian
Let there be two different points $ \vec{p_1}, \vec{p_2}$ on a unit sphere.
I need to get unit vector $\vec{t}$ at the point $\vec{p_1}$ tangent to the meridian (big circle) connecting these points.
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How to rotate a 3d object drawn along the z axis so that it lies along some other arbitrary axis
There are several very similar questions to this one, and I have read them all, but they are all a generalized version of the problem, and full of Math language. I don't speak Math, so the answers to ...
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Formal definition of the surface-ordered exponential $\mathcal{P}\exp(-\int_S F)$ and its validity for non-curvature 2-forms
I have a question regarding the precise mathematical definition of the surface-ordered exponential, often seen in the context of the non-abelian Stokes' theorem.
Let $P \to M$ be a principal $G$-...
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What are the holosnubs of the regular polyhedra?
According to polytope wiki, if P is a regular polyhedron, and H is the holosnub of P, then H will be a uniform polyhedron, if H is not degenerate or a polyhedron compound.
The stella octangula is the ...
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Find the radius of the circle. Hence, or otherwise, find the length $AB$.
The diagram shows a circle which passes through points $B, C, E$ and $D,$ and two straight
lines $ABC$ and $ADE$ which intersect at right angles at $A$.
$AD=4cm, DE=40cm$ and $BC=14cm.$
Find the ...
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