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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Inverse uncertain homography

Suppose x'= Hx, with x' and x homogeneous coordinates in an image and H a homography matrix. However, H is the result of calculations that take into account the uncertainty of x' and x, meaning H~{$\...
Boimans's user avatar
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Finding geodetic coordinates of the plane-ellipsoid intersection ellipse

Problem I want to find the geodetic coordinates (longitude, latitude) of the ellipse defined by the intersection between a plane and ellipsoid. Point $P$ is an observer away from Earth, and the ...
Hunter's user avatar
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How many unique unit squares do the 2 diagonals of rectangle of size 3x5? And nxm (where n and m are relatively prime)

If I have a rectangle of size $3 \times 5$, how many unique unit squares (1x1) do the 2 diagonals of this rectangle cross? And what about if we generalize the problem to an $n \times m$ rectangle ...
Sean Nguyen's user avatar
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Is the derivative of a path on a manifold always on the tangent space?

Let $a$ and $b$ in $M$, let $\gamma: [0, 1]\rightarrow M$ a path which is differentiable coordinates by coordinates. Do we have $\dot{\gamma}(t)=(\frac{d}{dt}\gamma_1(t), ..., \frac{d}{dt}\gamma_n(t))\...
Didier's user avatar
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How special is $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$?

The well-known identity for complex points forming an equilateral triangle reads $$z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$$ I have a doubt concerning the uniqueness of this identity Is $...
rgvalenciaalbornoz's user avatar
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is there a set of great circles on a hypersphere analogous to Buckminster Fuller's 31 in 3 dimensions?

I am addressing points on a sphere using great circles, and am investigating using Buckminster Fuller's "31 great circles." I am considering the viability of doing the same on a 3-sphere (in ...
Travis Well's user avatar
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Prove using Thales

"I've been searching for what I'm missing in this proof that should be very straightforward. Could you guide me step by step on how to demonstrate it?"
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Determine the angles of quadrilateral that make it concyclic

Inspired by a recent post, consider the following problem. You are given the four sides lengths of a quadrilateral $ABCD$. Let these sides be $a = AB , b = BC, c = CD, d = DA $. I want to determine ...
of course's user avatar
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solve for t that minimizes/maximizes z(t) within an ellipsoidal cone

Context: My current work involves ellipsoidal cones - the intersection of a cone and ellipsoid where the origin of each is at $(0,0,0)$ and where I say it is in 'canonical position' when the cone axis ...
William M.'s user avatar
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Without calculus, prove that sine is constant regardless of triangle size

This may be trivial, but I couldn't find an answer. Let's imagine that we live in ancient Greece, and there is no calculus available. The greek mathematicians accepted as trivial that the formula for ...
user2445828's user avatar
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silly question (trig, geom, alg)

I'm working through ch 13 of Dummit and Foote (namely 13.3.3). It seems that during the problem, it is supposed to be obvious that b=2k, but I don't see why. (original construction: Drawing a circle ...
ness's user avatar
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Where will a plane intersecting a cone maximize the distance to inscribed spheres? (V. Arnold)

A cone is cut by a plane, making a closed curve. Two spheres, each inscribed into the cone, are tangent to the plane, at points $A$ and $B$ respectively. Find the point $C$ on the cut line (i.e. the ...
SRobertJames's user avatar
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What is $b$ in $r=b\theta$ of Archimedean spiral?

This Wikipedia entry says Equivalently, in polar coordinates $(r, θ)$ it (Archimedean spiral) can be described by the equation $r = b\theta$, with real number $b$. Changing the parameter $b$ controls ...
User's user avatar
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Smallest axis-aligned regular hexagon for a set of hexagonal lattice points [closed]

Given a set of points on a hexagonal lattice, is there an efficient way to compute a center point $c$ and a minimal integer radius $r$ such that all points are within Manhattan distance $r$ of $c$? ...
Ian H's user avatar
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Why is the group of rotations in $\mathbb{R}^n$ not $n$-dimensional?

I have a rather basic mis-understanding about Lie groups and Lie algebras. Consider the Lie group $SO(N)$ for $N>3$ of rotations on $\mathbb{R}^N$. On the one hand this Lie group has dimension $N(N-...
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Can adjacent points exist in geometric space?

My question is going to focus on quite a counterintuitive thing. A couple of preliminaries. I understand geometric space as a set of points. A point, in turn, is an abstract idealization of an exact ...
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i want a general equation to calculate the change in slope angle of tangent lines between two points on an arc [closed]

I have an arc in the xy coordinates, I can calculate everything regarding the second point using this post, but I want the simplest general equation to give me the slope angle of the tangent line at ...
Ahmed Abdallah Fouad's user avatar
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What shape a point with a constant distance along a parabola will trace?

Define a parabola where it's directrix will be a line over the Y axis and the focus point will be a point in the X axis. Example parabola Define a distance n and grab a point D in the parabola where ...
Gabriel Quintino's user avatar
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1 answer
153 views

Proof of Triangle Inequality for $d(g; x, y) = \left(|x-y|^4 + g\,| x \times y |^2\right)^{\frac{1}{4}}$

I am seeking assistance in proving that a function, denoted as $d(g; x, y)$, defined on $\mathbb{R}^2 \times \mathbb{R}^2$ and parameterized by the non-negative real number $g$, may satisfy the ...
roiban12096's user avatar
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solution-verification | Prove that the triangles $AMD'$ and $ACB'$ have the same center of gravity.

the question Let the cube $ABCDA'B'C'D'$ and $M$ be a point on the semi-right $(AB$ so that $BM=AB$. Prove that the triangles $AMD'$ and $ACB'$ have the same center of gravity. the drawing the idea ...
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Prove that the distance $d$ from point $A$ to plane $(DNC)$ verifies the relation $d< \frac{AB+3AD}{6\sqrt{2}}$

the question In the triangle $ABC$ we consider $(AM$ the bisector of the angle $\angle A$ so that $MB=3MC, M\in (BC)$ and $N\in (AB)$ so that $BN=2NA$. On the plane of the triangle $ABC$, the ...
IONELA BUCIU's user avatar
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Show that quad $ABDE$ is cyclic

We're given a random red line and a point $A$ outside of it. From the red line, we take a random point $B$ such that the circle $\omega = \odot (A,AB)$ cuts the red line again at $C \neq B$. Let $D = ...
hellofriends's user avatar
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Regarding solving for the coordinates of reflection points in 3D space

In space, there exists a smooth plane, and additionally, there are two points, denoted as $A (x_a, y_a, z_a)$ and $B (x_b, y_b, z_b)$. A beam of light is emitted from point A, reflects off the mirror ...
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Can anyone help with this geometry problem? [closed]

enter image description here Thanks!
Techno23's user avatar
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Show that $\angle PCX=45$. if and only if $\angle (CP,YN)=30$

the question Consider the regular quadrilateral pyramid $VABCD$ with the vertex in $V$ and the points $M, N, P$ the means of the segments $AD, BC, VA$. Show that the angle of the line $CP$ with the ...
IONELA BUCIU's user avatar
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18 votes
3 answers
592 views

Why would the triangles join up to a rhombus?

The question I am going to present may as well sound very dumb. But this is becoming a hell of a confusing thing for me. The question is from ISI B.Math-B.Stat entrance exam 2022 UGA question paper. ...
Rounak Sarkar's user avatar
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I am stuck on a picture geometry problem [closed]

I am trying to get the altitude of both the isosceles triangles (with circle $3r$ and $1r$). (How isosceles?, Well the two triangles with circle having radius $2r$ are congruent because both share ...
user1150809's user avatar
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0 answers
16 views

How to constrain a circle's center to move on a defined function [closed]

I am trying to write the equation of a circle whose center moves on the graph of a function. (any function, e.g.- sin(x)). Is it enough to just write (x-t)^2+(y-f(t))^2=r^2 ?
Ajitesh Sophat's user avatar
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Proving Angles are Equal in a Parallelogram with Specific Outer Circle Intersections

Hello Math Stack Exchange community, I'm grappling with a geometry problem involving a parallelogram and some interesting outer circle properties. I hope someone can provide insight or a solution. ...
Wismar Günther's user avatar
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1 answer
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Simple questions on the parametrization of the surface $\left\{(x;y;z)\in\mathbb{R}\times\mathbb{R}^{+*}\times\mathbb{R}:x^2+z^2=1/y\right\}$

I am studying alone and I would like to have a feedback on my two first answers in "a)" and "b)" and get help on "c)". Question: We have the following parametric surface ...
OffHakhol's user avatar
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dihedral angles of non-regulare icosahedron given its coordinates

I have the coordinates (3D) of the vertices of an icosahedron (not a regular one) and I want to find all dihedral angles. I am fine with just equations to which they need to fulfill as I would think ...
koen ruymbeek's user avatar
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Prove that $\left\{ (x;y;z)\in\mathbb{R}\times\mathbb{R}^{+*} \times\mathbb{R}:x^2+z^2=1/y\right\}$ is a submanifold and justify his dimension

In introduction I would like to say that I just begin to study (alone) what is are submanifold so I need your help in order to be sure that my understanding of those concept is correct. Question: Let $...
OffHakhol's user avatar
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What is happening in this 'multi'-sinusoidal function?

Edit: Thanks to those commenting, I fixed my issue and now have reached the below function, as I had aimed. My remaining question would be how I might describe this function. I have taken the function ...
cadasc_56's user avatar
-5 votes
0 answers
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3d geometry hvvtvgvvkvuv [closed]

Consider the tetrahedron formed by the planes y+z=0,z+x=0,x+y=0, x+y+z=a . The direction cosines of the shortest distance lie between the planes y+z=0 and z + x = 0 is:
Dev Patel's user avatar
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0 answers
30 views

How to Maximize the Area of a Semicircle With a Fixed Perimeter of 50 cm [closed]

I cannot differentiate area formula for semicircle Area = $1/2π r^2$. I cannot use the constraint equation $50=πr+2r$ as the area is already in terms of radius. How can I do this without limits and ...
math student's user avatar
-1 votes
0 answers
82 views

Precalculus-level math help: is this solvable? [closed]

$\triangle{A_0B_0C_0}$ has $\angle A_0 = 40^{\circ}$, $\angle B_0 = 60^{\circ}$, and angle $\angle C_0 = 80^{\circ}$. It is rotated about vertex $A_0$ clockwise by some angle $\alpha$ to get $\...
Mintylemon66's user avatar
1 vote
1 answer
58 views

$S = \left \{ (x_1,x_2,x_3) \in\mathbb{R}^3 : x_1^2+x_2^2-x_3^2 =\lambda\right \} $. Find the $\lambda$ for which $S$ is a submanifold of $\Bbb{R}^3$

I am learning alone and I have just start studying about manifolds and below a question. I want to be sure that my answer and so my understanding is correct. Question: Let $ \lambda \in \mathbb{R}$ ...
OffHakhol's user avatar
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4 votes
4 answers
145 views

Can I determine the angles of a quadrilateral if I know the lengths of the sides and the difference between the diagonals?

I know the lengths of the four sides of a quadrilateral and the difference between the diagonals (but I do not know the actual lengths of the diagonals). My instinct is that this information ought to ...
Dave R's user avatar
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1 answer
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Set of all points that are of equal distance to any specific straight line and any specific point [closed]

So I know parabolas are like this, but only for line parallel to the $x$-or $y$-axis. Is it possible to do it for any line? Taking it even further, what about the set of all points that are equal ...
jojo_mark's user avatar
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0 answers
15 views

Function for equal surface area of overlapping circles when adding more circles [closed]

So I was doodling and found this interesting problem and I'm not sure how to generalize this for more circles. The answer for 2 circles seems relatively easy but I'm unsure how to go further. Could we ...
Helmut Baumgartner's user avatar
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0 answers
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Hardest complex number question to exist is here

I had a doubt related to this question which was previously asked whether we would even consider the locus as a locus for λ^2 > 3 . As this question was asked for all real λ then I think such ...
Aayush Sethia's user avatar
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1 answer
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Is There a Conceptual Connection Between the 3D Winding Number and Ray Casting Algorithms?

The 3D winding number provides a numerical answer to whether a point is inside or outside a closed surface, with its definition arising from surface integration. In my recent journey through ...
K.R.Park's user avatar
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How do I convert an engine's rpm to vehicle speed (with transmission ratios)

I'm currently writing code for a vehicle that can convert the rpm, wheel diameter, and total gear ratio into speed that can be displayed D = Wheel Diameter in source units (1 source unit = 3/4 of an ...
metroFIRE's user avatar
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1 answer
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Is it possible to translate points on a shape using a scaling transformation?

Say you have a triangle with the following verticies on a 2D plane: Tri = [ (0,0), (0,1), (0.5, 0.5) ] You wish to move V3 from (0.5,0.5) to (2,2), but you can not directly interact with this point. ...
David W's user avatar
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2 votes
1 answer
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Can space be augmented with a plane at infinity so that parallel planes intersect at a line at infinity?

The real plane can be augmented with a line at infinity such that two parallel lines intersect at a point at infinity, and the set of all such points forms a line. In space (3 dimensional solid ...
SRobertJames's user avatar
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Coordinates of two points near a circle [closed]

Could someone help me find the locations of the points A and B? Let me know if you need more information. Thanks ;)
value1's user avatar
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2 votes
1 answer
45 views

Can a oblique antiprism be constructed?

Can a oblique antiprism be constructed? Intuitively, it would seem oblique antiprism exist: Take any right antiprism and translate one of the parallel faces within it's plane. I'm baffled, though, ...
SRobertJames's user avatar
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Parametric equation of inward pointing half sin waves

Parametric equation of inward pointing half sin waves I can create a circle in red and I can create a sin wave that goes around a circle in green. Parametric equation: ...
Rick T's user avatar
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Volume of water in a tilted paraboloidal bowl

Suppose that you initially have a container which has the shape of a paraboloid with equation $ z = a x^2 + b y^2 $ where $ 0 \le z \le h $ Now you tilt this paraboloid by rotating it about any point (...
of course's user avatar
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Kaehler form in coordinates

Consider $M^n$ a $n$-dimensional Kaehler manifold and $\omega(X,Y) := g(JX,Y)$ its Kaehler 2-form. Let $\{e_0,\dots,e_{2n-1}\}$ be a orthonormal hermitian frame, i.e, $e_{2i+1} = Je_{2i}$ and $\{\...
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