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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.

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Number of unit cubes meeting the boundary of a convex set

Suppose $C \subseteq [0,n)^k$ is a convex set, and $\partial C$ is its topological boundary: its closure minus its interior. Is it true that $\partial C$ meets at most $2k n^{k-1}$ unit cubes? By a ...
Andrew Marks's user avatar
3 votes
2 answers
31 views

Find a partial area of a trapezoid

The formula for the area of a trapezoid is $$A = \frac{(a+b)}{2}h$$ where a and b are the length of each base and h is the trapezoid's height. So I want to figure out the area of a portion of the ...
rdemo's user avatar
  • 317
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0 answers
5 views

Estimates on Covering Number for Convex Polytopes Partitioning a Convex Set

Consider a convex set $K\in\mathbb{R}^3$ and a collection of convex 3-polytopes $C^i\subseteq K$ of equal volume $\frac{1}{N}$ for $i\in(1,\ldots,N)$ that partition $K$ (a Voronoi or Laguerre diagram ...
Theo Lavier's user avatar
2 votes
1 answer
58 views

The "seashell constant": closed form for $\frac12\exp\int_0^1-\log(\sin(\frac{\pi}{6}+\frac{2\pi}{3}x))\mathrm dx$?

I am looking for a closed form for $R=\frac12\exp\int_0^1-\log\left(\sin\left(\frac{\pi}{6}+\frac{2\pi}{3}x\right)\right)\mathrm dx\approx0.6159$. Wolfram does not give a closed form for $R$. Wolfram ...
Dan's user avatar
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-1 votes
1 answer
22 views

The diagonals of a regular pentagon with a side length of 1 form a new, smaller regular pentagon. What is the side length of the smaller pentagon?

The diagonals of a regular pentagon with a side length of 1 form a new, smaller regular pentagon. What is the side length of the smaller pentagon? Attempt: I lebel eveything, but I am unable to find ...
hd1's user avatar
  • 59
2 votes
2 answers
50 views

Find the equation of the circle that is tangent to 2 lines

Let the lines $(d_1):4x+7y-37=0$ and $(d_2):7x-4y+49=0$, and the points $A(4;3)$ and $N(18;-5)$. Let $\mathcal C$ the circle that is tangent to $(d_1)$ and $(d_2)$, the intersection between $(d_1)$ ...
joshua's user avatar
  • 1,047
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0 answers
22 views

Find the minimum and maximum values of $f(P ) = \angle ABP + \angle BCP + \angle CDP + \angle DAP$

Let $P$ be a point inside or on the boundary of a square $ABCD$. Find the minimum and maximum values of $f(P ) = \angle ABP + \angle BCP + \angle CDP + \angle DAP$. calling the angles $a,b,c,d$ if ...
math_learner's user avatar
-2 votes
0 answers
21 views

prove the addition identity cos(x+y) using the law of cosines [closed]

I’ve drawn a triangle with sides a b and c and written down two equations for side c using the law of cosines with angle x+y and using the sine of angle x and sine of angle y. I don’t know how to ...
Ilove Math's user avatar
2 votes
1 answer
41 views

Shortest path on the surface of a cylinder between given points $A$ and $B$

Suppose you have the cylinder $ x^2 + y^2 = R^2 $ And points $A = (R, 0, 0)$ and $ B = (0, R, h) $. Find the parametric equation of the curve of shortest length connecting $A$ and $B$. My attempt: If ...
i don't know what i am doing's user avatar
0 votes
0 answers
23 views

Determine the rotation necessary to bring a plane in contact with an ellipsoid

Given the ellipsoid $ (r - C)^T Q (r - C) = 1 \tag{1}$ And the plane $n^T (r - r_0) = 0 $. I want to determine the angle of rotation about an axis whose unit direction vector is $a$ and passes ...
i don't know what i am doing's user avatar
0 votes
1 answer
36 views

Derivation of the volume of a half-sphere

I am trying to derive the volume of a half-sphere with a constant r using integration. Integration I first try to integrate 90 degrees from the top of the halfsphere down to the bottom plane of the ...
Jan F. S's user avatar
-2 votes
0 answers
16 views

Geometry question on area lemma or ratio and proportion theorem

Let ABC be a triangle and D, E are points on the segment BC, CA respectively, such that AE = λAC and BD = μBC. Let AD, BE intersects at F. Find, in terms of λ and μ, the ratio AF : FD.
Aarush Singh's user avatar
0 votes
1 answer
54 views

Is it possible to find r in terms of the other variables?

I've never posted before and I'm sorry if this kind of question isn't suited to this forum. In the image I've attached, I'm wondering if it's possible to find the length r in terms of the lengths a ...
user avatar
1 vote
1 answer
22 views

How to find the shortest line segment connecting two skew lines, with constraint that line is parallel to some plane?

I've found lots of examples for how to identify the shortest line segment connecting two skew lines. For example, here: Distance between Skew lines However, for purposes of a simple game engine I am ...
BitPusher16's user avatar
-1 votes
0 answers
33 views

Intersection of planes in a 3d space WITHOUT vectors [closed]

We are learning about finding the intersection of planes / lines in 3D space by drawing diagrams, and I don't get it at all. The problem is we aren't doing anything with vectors or graphical diagrams, ...
paintedwolf's user avatar
0 votes
1 answer
45 views

Center of gravity of slanted cylinder

Trying to solve the problem of the buckling of a water storage tower. As the storage tank displaced, the center of gravity/mass moves. Water which once occupied a cylindrical volume is now occupying ...
victor's user avatar
  • 145
3 votes
1 answer
77 views

Strange substitution made in a paper to find asymptotics

In the quoted section from this paper, why is the author able to "substitute this result into Eq. (2.1)"? This should hold for $z$ large. But not everything on the contour is large. Why can ...
Sam Kirkiles's user avatar
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2 votes
0 answers
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Three diagonals are concurrent in a hexagon with an angle condition.

This is a practice problem from Polish mathematician Waldemar Pompe as following: A convex hexagon $ABCDEF$ with six equal sides satisfies the following condition: $$ \angle A+\angle C+\angle E=\...
shanmumu's user avatar
  • 127
1 vote
1 answer
52 views

How to show equation of function from a different parameterisation

$p,q \in [0,1]$ I have the equation: $(x,y) = pq(2,1) + p(1-q)(-1,-1) + (1-p)q(-1,-1) + (1-p)(1-q)(1,2)$ I want to show that the parabolic boundary connecting $(2,1)$ and $(1,2)$ is given by $5(x-y+1)^...
John Smith's user avatar
-5 votes
1 answer
27 views

a RIGHT PRISM has a triangular cross section. Find the area of this cross section [closed]

Two of its rectangular faces are contained in the planes $r\cdot (\hat{i}-2\hat{j})=0$, $r\cdot (3\hat{i}-\hat{j}+\hat{k})=4$. The two edges of the prism which are parallel to the intersection of ...
mah moud's user avatar
0 votes
1 answer
32 views

Equation in space of complex numbers

Giving an equation: $$z^2-(m+2)z+4(m-1)=0$$ I need to find the number of all the integer $m$ such that this equation has two complex solutions $z_1$, $z_2$ that satisfy: $$\vert z_1^2-m(z_1-4)\vert=\...
Lê Trung Kiên's user avatar
0 votes
0 answers
19 views

How to calculate distance of diameter on rounded rectangle? [closed]

I'm sorry that my reputation is not enough on this site, it is not available to upload images. I would like to know the calculation which calculates diameter of rounded rectangle. Width, length and ...
Goniiee's user avatar
  • 11
0 votes
1 answer
42 views

Definition of a hyperbola

Definition: A hyperbola is a set of points in the Euclidean plane, such that for any point $P$ of the set, the absolute difference of the distances to two fixed points $F_1,F_2$ is constant $2a$, $a&...
user1099762's user avatar
0 votes
0 answers
24 views

Are the eigenvectors of the generalised Laplace operator always periodic?

In $\mathbb{R}$ the eignevectors / eigenfunctions of the Laplace operator yield the fourier series, which, among other things, is made up of exclusively periodic functions. If you have a Riemannian ...
Makogan's user avatar
  • 3,359
1 vote
0 answers
34 views

Fundamental Group of a Surface of genus $g$ can be realized as a deck transformation of isometries on its universal cover.

If $\Sigma_g$ is a closed surface of genus $g\ge2$, then there exists a deck transformation of isometries on its universal cover $\mathbb H^2$ isomorphic to $\pi_1(\Sigma_g)$. I’m comfortable with the ...
EarnDaleheart's user avatar
1 vote
1 answer
58 views

Ruler and compass construction of an inscribed quadrilateral

Suppose that I have a (convex) quadrilateral $ABCD$. In the interior of it I have 4 distinct points $P,Q,R,S$ in general position (i.e. are vertex of a quadrilateral). The question how to construct a (...
quantum's user avatar
  • 1,667
0 votes
1 answer
41 views

Finding area using green's theorem

A friend of mine gave me a variant of the goat problem, which is the following: If a goat is tied to a circular fence of radius $10$ feet with a rope of $20$ feet, how much land can the goat roam? I ...
Kamal Saleh's user avatar
  • 6,542
0 votes
0 answers
20 views

Maximum Line Segments in an n × n Grid Without Loop formation

Exploring Proof for Maximum Line Segments in an (n * n) Grid Without Loop Formation Hello Math SE community, I am investigating how to maximize the number of line segments in an $(n * n)$ grid ...
omkar tripathi's user avatar
-1 votes
0 answers
81 views

AI capable of mathematical creation? (reference request) [closed]

I was studying philosophy of AI when a question came to my mind. I am a bachelor Mathematics student and I'd like to write an essay (both informative and argumentative) about the state of the art in ...
Amanda Wealth's user avatar
0 votes
0 answers
97 views

Intersection of $\mathbb{R}^n$ [closed]

I read in a comment that $$\mathbb{R}^2 \cap \mathbb{R}^3 = \emptyset$$ Why can't we say that $\mathbb{R}^2 \cap \mathbb{R}^3 = \{ 0 \}$ that is the zero point, the origin I mean? Is it just because ...
Heidegger's user avatar
  • 3,261
0 votes
2 answers
71 views

Surface area of a sphere in the first octant

I am trying to solve an optional geometry problem in MIT's edx calculus course, but I can't figure out why my attempt at this problem is incorrect. The problem is: The four-sided solid shown is the ...
JohnT's user avatar
  • 1,428
0 votes
0 answers
9 views

Real Valued 2D fourier series over a complex domain (and 2D manifold)?

This answer does a great job of explaining the fourier basis for $\mathbb{R}^2$, however, the answer assumes a square domain. Is it possible to generalise the answer to mor complicated domains? Like a ...
Makogan's user avatar
  • 3,359
1 vote
0 answers
34 views

Recognize geometric pattern in natural form

UPDATED Although my question arises from biology, it’s about geometry. I’m interested in various natural structures: fractals, packing, hyperuniformity etc. Here is photo of pores of tinder fungus or ...
lesobrod's user avatar
  • 784
1 vote
1 answer
81 views

Tangent problem involving orthocenter and circumcenter in a triangle

Let $ABC$ be an acute triangle inscribed the circle $(O)$ with three altitudes $AD, BE, CF$, orthocenter $H$. The tangent of $(O)$ at $A$ intersects $BC$ at $S$. Let $M$ be the midpoint of $BC$, $K$ ...
anonimo's user avatar
  • 441
1 vote
1 answer
57 views

Can $5$ persons in an $8$ft $\times$ $8$ft room obey the $6$ft social distancing rule?

The context of this problem may be dated. Suppose you have five persons in an $8$ ft $\times$ $8$ ft square room. Is it possible to spread them out so that there is no violation of a $6$ ft social ...
Marc Reyes's user avatar
-1 votes
0 answers
82 views

Ideas for a PhD in Mathematics [closed]

I will soon start studying maths, but I will do so at a moderate pace. My background is in philosophy and humanities, but I have always had an eye on maths. I wanted to ask what ideas might be good ...
Ian 's user avatar
  • 1
0 votes
0 answers
53 views

how was geometry founded? [closed]

how did euclid and the ancient greeks know what to prove in geometry, was there a higher goal for them? I don't understand how the Pythagorean theorem was just "thought of" one day and ...
thatpithere's user avatar
0 votes
0 answers
20 views

Local convexity and $\lim_{s\rightarrow t} \frac{|\gamma(s)-\gamma(t)|}{L(\gamma|_{[t,s]})}$

I have to prove the following lemma: Let $\gamma:[a,b]\rightarrow \mathbb{R}^2$ be a locally convex curve with constant speed $C$. Then one sided derivatives of $\gamma$ exists at all points and it ...
Mathemann's user avatar
13 votes
1 answer
288 views

A remarkable fact about the unit circle; looking for a shape with an even more remarkable fact.

You may have heard of the following remarkable fact about the unit circle: If $n$ equally spaced points are drawn on a unit circle, and line segments are drawn from one of the points to each of the ...
Dan's user avatar
  • 23.7k
2 votes
0 answers
70 views
+50

Shapes with simple distance functions.

Given a set $A$ in $\mathbb{R}^2$, the distance function (DF) of $A$ is defined as $$ \delta_A(\mathbf{x}) = \inf\{\|\mathbf{x}-\mathbf{y}\|: \mathbf{y} \in A \} $$ Some sets $A$ have a nice tidy ...
bubba's user avatar
  • 43.5k
1 vote
3 answers
151 views

Finding the height of a skyscraper

I am trying to solve an optional self-assessment geometry problem in MIT's 18.01x course, but am struggling somewhat with the geometry. I don't know how to fully convey the problem without a picture, ...
JohnT's user avatar
  • 1,428
1 vote
2 answers
189 views

$D$ inside $\triangle{ABC}$ such that $\angle{ABD}=\angle{CBD}=6^{\circ}$,$\angle{BCD}=12^{\circ}$,$\angle{ACD}=18^{\circ}$, find $\angle{BAD}$ [closed]

FAQ: Why post answer in problem? If you read my profile intro, you will know that (1) I post question because I had not seen any answer in pure geometric approach. (b) I promised to post answer when i ...
Adventitious Angles Qs Poster's user avatar
-3 votes
0 answers
41 views

How to prove that b1c1 is perpendicular to h1h2 [closed]

CA=AB,CA⊥AB, O is the center of the circle that is tangent to the hypotenuse of the isosceles right triangle АСВ. Prove that B1C1⊥H1H2
Interlocutor's user avatar
2 votes
0 answers
28 views

Avg. no. of random points on d-sphere needed to cover the center in their convex hull?

If I draw points on the d-dimensional sphere (i.e., on the hyper-surface of unit-length vectors in d+1-dimensional Euclidean space) uniformly at random until their convex hull contains the center of ...
user1513911's user avatar
-1 votes
0 answers
20 views

The maximum area of a circle to fit in an arbitrary cuboid(in Euclid space) [closed]

I was attracted by this problem 2 years ago and worked out a solution then. However, this solution often confuses me, and I can't find any work on this issue on the Internet. I would like to ...
RedQuark's user avatar
1 vote
0 answers
22 views

Area of Intersection of Hyperbolic Disks

I am trying to find the area of intersection of two hyperbolic circles using integrals. The problem: Let $D_1$ be the hyperbolic disk of radius $R$ centred at $O$. Consider a point $X \in D_1$ at ...
Algebro1000's user avatar
-1 votes
0 answers
19 views

Find VC dimension of segments in $\mathbb{R^3}$ [closed]

I need to find VC dimension of all segments in $\mathbb{R^3}$. The only thing I can use is the definition of VC dimension. Unfortunately, I have no idea how to solve the task and how to even start it. ...
Jane Doe's user avatar
  • 119
0 votes
1 answer
53 views

Maximum length of a boat able to turn at a junction of two orthogonal canals a turn [closed]

Let's say you are making a boat, and the goal is to navigate down a canal of width $a$. But not only that, you want to make a turn into another canal that is perpendicular to the one you navigate of ...
MiguelCG's user avatar
  • 275
-4 votes
1 answer
44 views

A surface passing through two different surfaces [closed]

Suppose I have two surfaces $f_1=k_1$ and $f_2=k_2$ in 3D. Then, how do I find the equation of a surface passing through (intersecting) the two surfaces $f_1,f_2$? Like, does $f_1-f_2=k_3$ help? But, ...
vidyarthi's user avatar
  • 7,065
0 votes
1 answer
23 views

Outward pointing normal Tetrahedron

For this tetrahedron I need to write down the order of the vertices such that the normal vector points out of the tetrahedron. For the base DAC, I have drawn the normal vector pointing outwards ...
Dam's user avatar
  • 259

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