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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 think I ve found a solution for "c)" in all the cases I will be happy to have your feedback on what I ve writen. c) To prove that $\vec{\phi} (r ; \theta) $ is a bijection from $r > 0 , ...
OffHakhol's user avatar
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1 vote
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$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 $\Bbb{R}^3$

If you define $ F: \mathbb R^3 \to \mathbb R$ as the map $F(x_1, x_2,x_3) = x_1^2 +x^2_2 - x_3^3$ then $S$ can be viewed as $S=F^{-1}(\lambda)$. By a standard result in Differential Geometry, $S$ is ...
Federico T.'s user avatar
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Why is radian so common in maths?

The radian has certain special properties that facilitate analyzing trigonometric functions. $$\frac{d}{dx} \sin(x rad)=\cos (x rad)$$ Now let us look at quadrants. (1 quadrant is a right angle) $$\...
Michael Ejercito's user avatar
1 vote
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geometry problem where the solution involves the use of some properties of complex numbers in geometry

a) You have already written a proof using $$s(M)=9 + 2m\bigg(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\bigg) + \frac{2}{m}(a+b+c) + \frac{a+c}{b} + \frac{a+b}{c} + \frac{c+b}{a}\tag1$$ b) You can use $(1)$ ...
mathlove's user avatar
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-1 votes

The position of a ladder leaning against a wall and touching a box under it

Just to throw my 2 cents in, I prefer a purely algebraic solution as well. It seems there are multiple ways to attack this problem, here is mine: Screenshot, instead of typing everything out again
user2257698's user avatar
2 votes

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

Let's call the vertices of the quadrilateral $A,B,C,D$ in counter clockwise direction. Let $a = AB , b = BC, c = CD , d = DA $. We can can let $A = (0,0)$ on the cartesian plane, and $B = (a, 0) $. ...
of course's user avatar
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2 votes

Can space be augmented with a plane at infinity so that parallel planes intersect at a line at infinity?

Yes, you can extend Euclidean 3-dimensional space to projective 3-space by adding a plane at infinity. A projective plane at infinity, to be precise. The conflict you observed is due to an incorrect ...
MvG's user avatar
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1 vote

Set of all points that are of equal distance to any specific straight line and any specific point

Well, a parabola is in fact defined as you said, the set of points that are equidistant from a given point and a given line, see Wikipedia. As for your second question, you can use the definition you ...
blomp's user avatar
  • 101
0 votes

Simple projective geometry (in the projective plane) question

Let $W$ be a point on $ABF$ and $\sigma_{W}$ be the perspectivity with centre $W$ that maps the points of $ADE$ onto the points of $DCF$. $\pi_2 \circ \sigma_{W} \circ \pi^{-1}_1$ is a projectivity ...
Romanda de Gore's user avatar
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Placing an ellipse with known dimensions and orientation such that it is tangent to two other disjoint ellipses

I've implemented the above equations for an example. The results are shown below. The black and blue ellipses are the fixed ellipses, the orange ellipse has two locations to be placed. Following is ...
of course's user avatar
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0 votes

How do I triangulate a point from 3 segments, rather than from 3 points?

First, I am going to replace the notation of the vectors $a_1, b_1, c_1$ with the vectors $ u_1, u_2, u_3 $ (that I am going to assume are unit vectors) along which the sources and the mics are ...
of course's user avatar
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How do I triangulate a point from 3 segments, rather than from 3 points?

Heuristically, if you have 4 mics, there might be just barely enough information to determine the placement of the points. With 5 mics, heuristically, I would expect that a unique solution should ...
D.W.'s user avatar
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0 votes

Is There a Conceptual Connection Between the 3D Winding Number and Ray Casting Algorithms?

I've resolved my query regarding the connection between the 3D Winding Number and Ray Casting methods. Traditionally, Ray Casting counts the number of times a ray crosses a polygon's edges to ...
K.R.Park's user avatar
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1 vote
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Can a triangle up to isometry shatter seven points?

This is indeed possible. The points $(0.95,0.4)$, $(0.63,0.13)$, $(0.37,0.12)$, $(0.25,0.44)$, $(0.35,0.83)$, $(0.46,0.96)$ and $(0.91,0.76)$ are shattered by copies of a triangle with squared side ...
joriki's user avatar
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1 vote
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Is it possible to translate points on a shape using a scaling transformation?

The class of transformations you are considering are called dilations. A dilation is by definition the composition of a homothety (of the form $\vec{x}_i' = \lambda(\vec{x}_i-\vec{x}_0)+\vec{x}_0$, ...
DarkLordOfPhysics's user avatar
1 vote

Can a oblique antiprism be constructed?

I've updated the OP with sketches, but am baffled why I can find almost no reference, discussion, or models (when far more complicated polyhedra models abound), and have still not found a proof or ...
SRobertJames's user avatar
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1 vote

Parametric equation of inward pointing half sin waves

First, to help work out how we might create something like this, we will think more with respect to polar coordinates. If you don't know what polar coordinates are, all you need to understand for this ...
mrepic1123's user avatar
0 votes

Kaehler form in coordinates

First notice that the complex structure acts on 1-forms with its transpose map $J^* : \theta \mapsto \theta \circ J$. Then you can easily check that, for $k = 1 \ldots n$ $$ J^* \theta_{2k-1} = -\...
Federico T.'s user avatar
1 vote
Accepted

Conditional distance distribution between 2 points given the distance to another point

The trick to use here should be this one: conditionally on $X$, the two random variables $|X-Y|$ and $|X-Z|$ are iid. Consider the notation $|X-Y|=\overline{XY}$. Let's face the problem in terms of ...
framago's user avatar
  • 411
0 votes

Determine the equation of the right circular cone whose intersection with the $xy$ plane is a known ellipse

As mentioned in the update in the question, I'll assume that the given ellipse has the form $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \tag{1}$ where $a \gt b$. We know that the cone with such a section ...
of course's user avatar
  • 21k
1 vote

How to calculate the area of a projected path onto a plane? (Bee traveling problem)

Here I'll develop an analytical approach that hopefully will address the theoretical aspects of the original question. As the bee moves in 3D space, and easst/west, norht/south, and up/down are ...
cjferes's user avatar
  • 2,191
3 votes

Question on counting the number of triangles formed by 1999 points in a square

For a much simpler verification, double count the angle sum. Consider all of the vertex angles of the triangles. What is their sum? Each triangle has an angle sum of $180^\circ$. The 1999 points each ...
Calvin Lin's user avatar
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0 votes

Determine Intersection between a Plane and a Line Rotated about an Axis of Rotation

The line segment is given as $ \ell(t) = P_0 + t \ d , 0 \le t \le 1 $ Then you have an axis of rotation specified by the unit vector $k$, which passes through the point $C$. And finally you have the ...
of course's user avatar
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0 votes

Determine Intersection between a Plane and a Line Rotated about an Axis of Rotation

Too long for a comment. Here can be used an approach involving optimization procedures, but a simpler approach to the problem is to obtain a graphical checking. Calling $R(\theta)\cdot\left(\begin{...
Cesareo's user avatar
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2 votes
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Inclusion map in Mayer-Vietoris sequence for $\mathbb {RP}^2$ with 3 points removed

To get a deformation retraction of $X$ onto the bouquet of circles, you should arrange things as in this answer. To use your decomposition to compute homology (without knowing $X\simeq S^1\vee S^1\vee ...
FShrike's user avatar
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0 votes

The triangle $ABC$ is right-angled in $A$. Prove that the inequality $(1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2}$ holds.

In a triangle $\triangle ABC$ $$\cos A+\cos B+\cos C=1+4\sin \tfrac A2\sin\tfrac B2\sin\tfrac C2.$$ Source. Sign mistake. When $A=90^\circ,$ $$\sin\tfrac B2\sin\tfrac C2=\frac{\cos B+\cos C-1}{2\sqrt2}...
Bob Dobbs's user avatar
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2 votes

Question on counting the number of triangles formed by 1999 points in a square

The Euler's formula tells that $v-e+f=2$, where $v$ is the number of vertices, $e$ is the number of edges, $f$ is the number of faces (including the outer infinite face). We know that $$v=4+1999=2003.$...
Aig's user avatar
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1 vote
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Prove that the triangles $VAC, EAV$ are similar if and only if $\angle EVO=30°$.

Case 1: Well done; you could have just specified that $VC=VA(=2x\sqrt3)$ Case 2: Well done; you could have quoted the angle bissector theorem and perhaps write more classicaly $\frac{V\color{red}A}{V\...
Stéphane Jaouen's user avatar
2 votes
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The triangle $ABC$ is right-angled in $A$. Prove that the inequality $(1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2}$ holds.

WLOG the hypotenuse is 1. Let the legs be $a,b$. Denote $ab = x$. $$(1-a)(1-b) = \frac{a^2b^2}{1+a+b+ab} \le \frac{a^2b^2}{1+2\sqrt {ab}+ab} = \left(\frac{x}{1+\sqrt x}\right)^2$$ Now, consider: $$\...
D S's user avatar
  • 3,601
2 votes

The triangle $ABC$ is right-angled in $A$. Prove that the inequality $(1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2}$ holds.

As it was pointed out in other answers and comments, this amounts to determining the maximum value of $f(x)=(1-\sin x)(1-\cos x)$ for $x \in [0, \frac{\pi}{2}]$. Since $f$ is differentiable, the ...
PierreCarre's user avatar
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1 vote

The triangle $ABC$ is right-angled in $A$. Prove that the inequality $(1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2}$ holds.

$\sin B = \dfrac{b}{a} , \sin C = \dfrac{c}{a} $, so $ (1 - \sin B) (1 - \sin C) = \dfrac{(a - b)(a - c)}{a^2}$ We can take $a = 1$ without loss of generality because we talking about angles here not ...
of course's user avatar
  • 21k
2 votes

The triangle $ABC$ is right-angled in $A$. Prove that the inequality $(1-\sin B)(1-\sin C)\leq \frac{{(\sqrt{2}-1)}^2}{2}$ holds.

Hint : It is a right angle triangle so, $B + C = 90^{\circ}$. So, this can be re-written as $$(1-\sin B)(1-\cos B)$$ I hope you will be proceed from to get maxima at $B=45^{\circ}$ and then the final ...
Mahendra Varma's user avatar
0 votes

Determine position of sound source from arrival times of a blip

I think this is similar to your idea but using the concept of circles. Since we do know the distance of the source from each receiver, we can write the eqn of 3 circles such that each receiver is the ...
Mahendra Varma's user avatar
0 votes

The equation of a line reflected about another line

Let $l$ be the line $ax+by+c=0$. Let $m$ be the line $dx+ey+f=0$. Let $r$ be the reflection of $m$ in $l$. When we reflect an equation in $l$, to get the new equation, simply replace each $x$ with $x-...
user182601's user avatar
5 votes

How to calculate the area of a projected path onto a plane? (Bee traveling problem)

I think this problem is intended to be a purely logical problem with very little vector geometry. For starters, the problem states (without proof) that there IS a plane that passes through the ...
Bobby Ocean's user avatar
  • 3,193
3 votes

How to calculate the area of a projected path onto a plane? (Bee traveling problem)

Adopt a Cartesian coordinate system such that the bee begins at the origin and east/north/up is in the direction of increasing $x/y/z$, respectively. Then the trajectory of the bee is a series of ...
DarkLordOfPhysics's user avatar
0 votes

Finding the coordinates of a moving point along the surface of a sphere.

The standard directions at any point $P_1( R , \theta_0, \phi_0)$ on the surface of the sphere are $ u_1 = \begin{pmatrix} - \sin \phi_0 \\ \cos \phi_0 \\ 0 \end{pmatrix}$ $ u_2 = \begin{pmatrix} - \...
of course's user avatar
  • 21k
1 vote

How to calculate the area of a projected path onto a plane? (Bee traveling problem)

Following is the code I developed to solve this problem. The square of the area output by the program is $A^2 = 12$ ...
of course's user avatar
  • 21k
2 votes

Volume of a great icosahedron

First of all, @ChrisLewis is totally right. The area of a great icosahedron can be described as the area of a small stellated dodecahedron with $5\cdot 12=60$ congruent irregular tetrahedra carved out ...
jorisperrenet's user avatar
2 votes

geometry problem where the solution involves the use of some properties of complex numbers in geometry

Let me first replace this issue in a classical context. Let $t:=a+b+c \tag{1}$ Alignment relationships $2O+H_k=3G_k$ give : $$\begin{cases}h_1&=&a+b+m\\ h_2&=&b+c+m\\ h_3&=&c+a+...
Jean Marie's user avatar
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5 votes

Volume of a great icosahedron

Let us rescale the great icosahedron so that its true vertices have as coordinates cyclic permutations of $(\pm1,0,\pm\varphi)$ where $\varphi$ is the golden ratio $\frac{\sqrt5+1}2$, so that its true ...
Parcly Taxel's user avatar
0 votes

Volume of a great icosahedron

I'll find the point $B$ from the following illustration using the regular icosahedron from this answer. (sorry to every reader for the godawful "graphic design"...) Namely, it can be taken ...
Amateur_Algebraist's user avatar
17 votes
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What's wrong with this derivation of the volume of a hemisphere?

The thing to keep in mind with disk integration is the fact that the variable of integration represents the thickness of each disk. You are trying to integrate the volume of a half-sphere, using: $$ \...
DotCounter's user avatar
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0 votes

Investigating the shortest line segment connecting two disjoint ellipses

Just solve the following convex optimization problem: $\min \, \| r_1 - r_2 \|^2$ subject to: $ (r_1 - C_1)^T Q_1 (r_1 - C_1) \le 1$ $ (r_2 - C_2)^T Q_2 (r_2 - C_2) \le 1,$ which is classified as a ...
Amir's user avatar
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0 votes

Intersection of two ellipses at exactly 2 points

We can solve explicitly this problem with the use of tangency. Given two ellipses $$ \cases{ a_1 x^2+b_1 x y + c_1 y^2 - k = 0\\ a_2 x^2+b_2 x y + c_2 y^2 - l = 0 } $$ eliminating $y$ we have $$ p(x) =...
Cesareo's user avatar
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13 votes

What's wrong with this derivation of the volume of a hemisphere?

Your problem seems to be that you don't have a clear concept of how to integrate the volume of an object by stacking disks on each other. In order to compute a volume by stacking disks, you need the ...
David K's user avatar
  • 97.7k
1 vote

Does this spiral, formed from similar right triangles arranged around a point, have a name?

This is not just a class of logarithmic spiral, it is exactly a logarithmic spiral. If we call the angle of the triangle $\Delta\theta$, and growth rate, call it $q=1/\cos\Delta\theta$, the flair ...
Cye Waldman's user avatar
  • 7,224
2 votes
Accepted

How can I calculate position of a point after it moved towards another point?

Here is an image explaining everything. It's just the Pythagorean Theorem.
fleablood's user avatar
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2 votes
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What is the conditiones to have the plane $(ABC) $ and $(DEF) $ perpendicular?

I'll assume that $A$, $B$ and $C$ are pairwise different and that the same holds for $D$, $E$ and $F$ (otherwise the question is meaningless). We have that $$(ABC)\perp (DEF)\iff \textbf{u}\perp \...
Luca T. Castrillón's user avatar
28 votes

What's wrong with this derivation of the volume of a hemisphere?

You're computing the volume of a cone, not a hemisphere! More precisely, your integral gives the volume obtained by rotating the triangle $0 \le y \le x \le r$ about the $x$-axis. The hemisphere, on ...
Hans Lundmark's user avatar

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