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Questions tagged [geometry]

For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.

3
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1answer
33 views

The definition of the tangent vector of a manifold

In many textbooks of differential manifold, authors usually define the tangent vector on a manifold as below: Definition: Suppose $m$ is a $n$-dimensional smooth manifold, $x\in M$, the tangent ...
1
vote
1answer
30 views

Gluing $2n$-gons

I am confused about this how I would visualize/prove this. Given that we have a regular $2n$-gon that preserves orientation (the surface is orientable because the $2n$-gon's opposite sides are glued ...
1
vote
2answers
40 views

Optimization path avoiding a set of points

I have a random polygon, convex or non-convex, defined by its vertices and two random points outside of the polygon (A and B) all of them defined in ${\rm I\!R}^{2}$, how can I get an optimized path, ...
0
votes
1answer
21 views

Why rotation of dodecahedron corresponds to an even permutation of inscribed five tetrahedra?

I’m reading Arnold’s Abel’s Theorem in Problems and Solutions, where it says: We now prove that the alternating group $A_5$ is not soluble. One of the possible proofs uses the following ...
2
votes
0answers
33 views

What's the name of the surface resulting from rotating the exponential curve?

In three dimensions, if we rotate the curve $(x,0,e^x)$ for $x\in\mathbb{R}$ around the axis through the points $(0,0,1)$ and $(1,0,0)$ by the angle $2\pi$, what's the resulting surface called? It ...
0
votes
1answer
41 views

Can anyone help me to find what is the similarity between these two triangles Just check the picture)? [closed]

Can anybody show me how does the red triangle (bigger triangle) is similar to the green triangle (smaller triangle). I need to know how they are similar so that I can do a proportional between them.
1
vote
0answers
22 views

Consequences of the axioms of connection and order

How do I prove that: Theorem 4. If we have given any finite number of points situated upon a straight line, we can always arrange them in a sequence A, B, C, D, E,. . ., K so that B shall lie between ...
0
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2answers
62 views

Compute numerically the angle $x$ in the triangle without trigonometry

Be $\triangle CAB$ right in $A$ such that $AB=a$ and $\angle CBA = \alpha$. Extend $BC$ to $D$ such that $\angle CAD=2\alpha$ and $\angle ADC=x$. If $M$ is a point $\in BC$ such that $BM=MC$ and $MD=...
5
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2answers
114 views

Geometry high school math competition question

Let $ABC$ be an equilateral triangle with side length $2$. Let the circle with diameter AB be $\gamma$. Consider the two tangents from $C$ to $\gamma$, and let the tangency point closer to $A$ be $...
0
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0answers
38 views

How to prove that the statement “there exists a constant, X, such that the sum of all angles in a triangle is equal to X” is equal to postulate 5 [closed]

I have this as a homework question, and didn't know if it would be the same proof as when the angles of a triangle sum to 180. I was also unsure if I should be using theorems like exterior angle that ...
1
vote
0answers
28 views

Given a set of points $S$ in the plane $\mathbb{C}$, is there a way to define “the smallest Euclidean shape that $S$ makes”?

For example, given $S := \{(0, 0), (0, 1), (1, -1), (-1, -1)\}$, I want to say, precisely, that the "smallest shape that $S$ makes" is something like the Star Trek symbol (https://1000logos.net/star-...
3
votes
0answers
27 views

Kind of hard(?) geometry proof [duplicate]

I am struggling with this proof. Consider the circle $\Gamma$ with a diameter $AB$. Another circle $\Lambda$ is drawn such that it is internally tangent to the $\Gamma$ and tangent to the diameter $AB$...
0
votes
1answer
22 views

Proving that a point P is on a circle and a line

Given a circle $C$ with center $A$ and radius $r$. Given a line $D$ with a vector ${u}$ passing through point $P_0$. Knowing that $P$ is on $D$ only if $P = P_0 + t u$ Knowing that $P$ is on $C$ if $\...
0
votes
1answer
31 views

Explicit formula for the projection from the line to an arbitrary circle

Overview: Given an arbitrary point $t$ on the horizontal axis of a Cartesian plane and a point $\textbf{p}$ on a circle, I would like to find the point $\textbf{t}'$ located at the intersection of the ...
0
votes
1answer
24 views

Mean Q of all points q on a cone whose origin vectors 0q are perpendicular to a given point P.

Consider a point $P~(P_x,P_y,0)$ which lies somewhere in the Cartesian region $x > 0. $ Consider a simple 3D cone surface with half angle $\gamma$, originating at point $\mathcal{O}$ $(0,0,0)$ ...
1
vote
2answers
53 views

How to generate pair of 2D random points whose average distance will be some given value?

Everything written below supposes 2D world. Suppose any rectangle area. I would like to randomly generate 2 points from this area whose distance (on average) is some given value d. In other words I ...
0
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0answers
32 views

What is the name of this surface? (resembles Boy's surface with holes)

Does anyone know the name of this surface? I feel like it has a particular name but can't seem to remember it for some reason.
2
votes
4answers
61 views

If the median AM of a triangle ABC bisects the angle $\hat{A}$, then the triangle is an isosceles.

Can we solve the above problem using only the criteria for congruent triangles (i.e., without using the fact that the sum of the angles of a triangle is $180^\circ$)?
1
vote
0answers
50 views

How to get legal diagonal count for (x,y) In only one direction?

In my previous question I get all legal diagonals for $(x,y)$ point on coordinate plane, Now I'm trying to get the diagonal for only one direction. e.g: $(3,4)$ on board length 5, the top right would ...
0
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0answers
26 views

English version of Arnold's “Geometry and Dynamics of Galois Fields”?

I wonder if there's an English version of Geometry and dynamics of Galois fields by V. I. Arnol'd?
2
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2answers
57 views

Minimum value of $PA+PB$ is

If $P(x,y,z)$ lie on line $\displaystyle \frac{x+2}{2}=\frac{y+7}{2}=\frac{z-2}{1}$ and $A(5,3,4)$ and $B(1,-1,2)$ . Then minimum value of $PA+PB$ is what i try let $\displaystyle \frac{x+2}{2}=\...
-4
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0answers
23 views

Is there any proof that corresponding angles are equal? [closed]

Can you prove that corresponding angles are equal?
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0answers
24 views

Generating function of cuts of some graph

Consider the graph $G$, each edge of which has weight $T$. Also consider some cut of $G$. We call the weight of the cut - product of all edges included in the cut. Generating function of cuts of $G$ ...
8
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1answer
72 views

Is it true that if inclusion of a boundary component is a homotopy equivalence then the manifold deformation retracts onto it?

In topological spaces, if $A\subset X$ and the inclusion is a homotopy equivalence, that doesn't imply that $X$ deformation retracts onto $A$. For example take one of those examples of a contractible ...
0
votes
1answer
46 views

What is the induced orientation on a 1-manifold with boundary that is the image of closed interval under a smooth immersion?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. I have been trying to wrap my head around this for about 2 hours (...
1
vote
1answer
59 views

Chess : how to count all legal moves on diagonals for a queen placed in $(x,y)$

I'm trying to get a formula that counts all legal spaces over diagonals for a given point $(x,y)$, e.g.: In the above image, you can see there are $13$ legal squares over diagonals. Is there a ...
0
votes
1answer
37 views

Is there a better name for a “spherical annulus”?

For example, the set $\{x\in\mathbb{R}^d: 1\leq|x|\leq2\}$. For $d=2$, I'd call this an annulus. What do we call it in higher dimensions?
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votes
1answer
23 views

An equilateral triangle has a perimeter of 810 cm. Calculate the area of ​an octagon that has the side equal to 4/9 of the side of the triangle. [closed]

An equilateral triangle has a perimeter of 810 cm. Calculate the area of ​​an octagon that has the side equal to 4/9 of the side of the triangle.
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votes
1answer
44 views

Proposition 3.14: Geometry

Please help prove the following proposition: Proposition 3.14 Supplementary angles of congruent angles are congruent. Is this right? (1) Suppose angle ABC is congruent to angle DEF (given) (2) We ...
2
votes
3answers
126 views

Introduction to Geometry Books

I am looking for a book that covers introduction to geometry. Currently, I am reading "Geometry: A metric approach with models", by R.S Millman. I like the book but I would like to read another highly ...
5
votes
3answers
117 views

Picture proof that the area of a right triangle is $xy$

I stumbled on the following result by accident: Let $A, B, C$ be the vertices of a right triangle, with opposite side lengths $a, b, c$ respectively, where $\angle C = 90^\circ$ and $a^2 + b^2 = c^...
0
votes
0answers
17 views

Proof of a basic geometrical aseveration [closed]

I'm starting to study geometry from 0%, and I want help to prove it: "All pair of adjacent angles are too supplementary" Thanks you.
1
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0answers
23 views

How to understand intuitively the geometry of a pantograph

I have built a pantograph, driven by a pair of small motors - I hope the arrangement is clear enough to understand: In my software to drive the machine, I can handle the geometry very nicely now; it'...
0
votes
1answer
18 views

Parametric equation of a parabola rotating on its axis

Write the parametric equation of the surface generated by a parabola rotating around its axis. I guess it's simply getting from the parabola equation to the parametric equations of a generic ...
0
votes
1answer
17 views

Parametrization of right circular cone

community! Write the parametric equations of a right circular cone of height $h$ and semi-aperture $α$, lying on the plane $z = 0$, contained in the first octant, so that the segment between ...
1
vote
1answer
27 views

Rectangle trapezoid

I would be very grateful if you can help me with this problem. I've constructed the median ON, N ∈ BC, and I was able to find that the triangle OCN is isosceles (height and median coincide). Probably ...
-1
votes
2answers
36 views

A question on convex quadrilateral [closed]

Suppose that $ABCD$ is a convex quadrilateral and the point $P$ is inside it. Given: the area of this quadrilateral is $168$ and $$  PA = 9 , PB = PD = 12 , PC = 5$$ What is the Perimeter of this ...
1
vote
1answer
53 views

Exercise to see some interpretation of the Lie bracket

Let $p \in M$ and consider a coordinate chart centred on $p$. Let $\varphi_{v_1}^t$ and $\varphi_{v_2}^t$ the flows of our two vector fields. Using the chart, we define $g:\mathbb{R} \to \mathbb{R}^n$ ...
0
votes
1answer
12 views

Averaging scalar fields across an interface

I have a 1D,2D,3D (line/square/cube) of unit dimension which is separated by an flat interface into two regions. A scalar quantity(p) with a jump discontiuity is defined in this domain. I know the ...
0
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0answers
17 views

how to prove that the radius of curvature is equal to the radius of the circle?

A material point M describes a circle of radius R and the velocity vo, k (v, a) = alpha - constant, whatever> = t0. It is shown that the radius of curvature is equal to the radius of the circle
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0answers
14 views

Linear Manifold - Parallel Space

I am studying a book on Convexity and Optimization and I found this statement. "Let $X_0$ be any point of a linear manifold M. Then the set: $$L := M - X_0 = \{X - X_0 | X \in M\} $$ is the unique ...
2
votes
2answers
31 views

Can 3D co-ordinates be transferred into 2D co-ordinates?

Is it possible to transform co-ordinates $(a,b,c)$ into $(x,y) $ such that $(x,y)$ is unique for each $(a,b,c)$ ? $a, b, c, x, y$ are in $\Bbb{R}$ .
4
votes
3answers
196 views

A geometric problem on number of points and lines

There are P points and L lines such that Each line contains 8 points Each point lies on 8 lines Any two distinct lines intersect in a unique point Any two distinct points lie on a unique ...
0
votes
2answers
15 views

Should a closed cone be regarded closed surface?

If i enclose the flat surface of the cone then is the cone a closed surface ? Ive read that cones aren't closed surfaces now if the flat isn't a vaccum then should the cone be regarded as closed ...
0
votes
0answers
19 views

Equation of line in $R^3$ - limited information

In $R^3$, for a line $L$, given knowledge of: (1) a point, position vector, $r(x_{0},y_{0},z_{0})$ belonging to $L$ (2) the momentum, $p(p_{x},p_{y},p_{z})$ at $r$, where $p$=$mv$; $v$=$d/dt(r)$, ...
0
votes
0answers
12 views

Finding normal vector of six node triangle

How to find the normal vector of six node triangle element (quadratic element) containing x,y,z. I know that if three points triangle (linear element) we just need to find it cross product. But how ...
1
vote
1answer
88 views

Verify proof for continuity of pointwise orientation

This has been asked here, but there is no answer. Prove that $[(X_1,...,X_n)]$ on a manifold $M$ is continuous if and only if every point $p$ in $M$ has a coordinate neighborhood $(U,\phi) = (U,x^1,...
2
votes
1answer
112 views

Exact Differential Equation Geometry

In a variety of contexts, I have noticed hints of a strong connection between exact differential equations and machinery from multivariable calculus. From another question, I have gathered that the ...
0
votes
0answers
21 views

Definition of constant-curvature curve embedded on an Ellipsoid of revolution

I am interested in identifying a type of curve so I can do literature review on it. What is the name of a curve embedded on an ellipsoid of revolution in which the curvature of the embedded curve is ...
0
votes
1answer
43 views

Prove that hyperbolic isometry with constant distance is the identity

Let $f$ be an isometry of the Poincaré half-plane model of two-dimensional hyperbolic geometry, denoted by $\mathbb{H}^2$. Prove that if the distance $d(z,f(z))=c$ for some constant $c\geq0$ for all $...