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

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3
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1answer
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parametric equations of folium of Descartes

We know the function of the folium of Descartes is $x^3+y^3=3axy$. The problem is to show that the folium of Descartes has parametric equations $x=\frac{3at}{1+t^3}$, $y=\frac{3at^2}{1+t^3}$ (this ...
2
votes
0answers
146 views

Computing distances between hyperspheres and sides of a hypercube?

Suppose you are given the $n$ dimensional hypersphere: $$\left(x_1 - \frac{1}{2}\right)^2 + \left(x_2 - \frac{1}{2}\right)^2 +\ldots+ \left(x_n - \frac{1}{2}\right)^2 = \frac{n}{4}$$ And the ...
1
vote
2answers
198 views

Fitting circle into an angle

I've been struggling with this for quite some time now, anyone could help me perhaps with this? Given an angle of an arbitrary degrees, and a circle with radius r. And imagine I would try to push the ...
2
votes
3answers
5k views

project a point in 3D on a given plane

A point in a 3D space is given as $ P(x,y,z) $. I want to find the position of this point projected parallel to the normal on a plane Q defined by $3$ non-collinear points $ Q1(x1,y1,z1), Q2(x2,y2,z2)\...
2
votes
3answers
207 views

Nature of a triangle with vertices $z_1, z_2$ and $-1$ such that $|z_1|=|z_2|=1=z_1+z_2$ [closed]

If $z_1$ and $z_2$ are distinct complex number such that $|z_1|=|z_2|=1$ and $z_1+z_2=1$, then the triangle in the complex plane with $z_1,z_2$ and $-1$ as vertices must be: equilateral. right ...
3
votes
4answers
346 views

Finding the area, general case with angle $\theta$.

Inspired by this question, I am curious to know the more general case. Given the radius of the large circle as $R$ and the angle $\theta \le \pi$, what is the area of the colored section? My ...
2
votes
1answer
230 views

annihilator of an intersection in infinite dimension

Given two subspaces of an infinite dimensional Banach space, is the sum of their annihilators dense in the annihilator of their intersection?
4
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5answers
192 views

Minkowski sum of two disks

An open disk with radius $r$ centered at $\mathbf{p}$ is $D(\mathbf{p}, r)=\{\mathbf{q} \mid d(\mathbf p, \mathbf q) < r\}$, and the Minkowski sum of two sets $A$ and $B$ is $A \oplus B=\{\mathbf p ...
3
votes
2answers
601 views

Proof: Invariant angle measure - same result for any circle drawn.

Below I have quoted Wikipedia. I am particular interested in the statement: The value of $\theta$ thus defined is independent of the size of the circle: if the length of the radius is changed ...
2
votes
1answer
484 views

2D calculate position of a point relative to 4 known points

I have 4 known points (a square) in 2D space: A: {x:0, y:0} B: {x:100, y:0} C: {x:100, y:100} D: {x:0, y:100} Then I have a point inside the square. I don't know its location, but I do know the ...
7
votes
3answers
434 views

find area of dark part

let us consider following picture we have following informations.we have circular sector,central angle is $90$,and in this sector there is inscribed small circle ,which touches arcs of sectors ...
2
votes
2answers
2k views

Ellipse circumference calculation method?

Actually I know how to calculate the circumference of an ellipse using two methods and each one of them giving me different result. The first method is using the formula: $E_c=2\pi\sqrt{\dfrac{a^2+b^...
3
votes
2answers
99 views

What's the symbol m in this sum?

I'm supposed to write some code to calculate the inertia moments of a shape, but I am afraid I have been given too little information. The matrix that I must obtain is this one: $$ \begin{vmatrix} ...
8
votes
5answers
3k views

Proving the length of a circle's arc is proportional to the size of the angle

How can I prove that: The length of the arc is proportional to the size of the angle. Every book use this fact in explaining radians and the fundamental arc length equation $s = r\theta$. ...
0
votes
1answer
282 views

Prove that in an obtuse triangle the orthocentre is the excenter of the orthic triangle

Consider an obtuse angled $\Delta ABC$ with altitudes $AD, BE, CF$ concurrent at $H$. Consider the orthic triangle $\Delta FED$. Extend $ED$ to $D'$ and $EF$ to $F'$. Prove that $\angle FDH = \angle ...
3
votes
1answer
439 views

Shortest Path and Minimum Curvature Path - implementation

Let's say we are given a race track, which may be described as a closed curve of given width (it may differ along the curve). My task is to implement an algorithm which finds two kinds of trajectories ...
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4answers
6k views

Why is volume of cylinder > volume of cube

If we have a cylinder with radius 1 and height 1 and cube with side lengths equal to 1 Volume of cube = $r^3 = 1^3 = 1$ Volume of cylinder = $\pi r^2h = \pi 1^2\times1 = \pi$ clearly $\pi > 1$, ...
0
votes
1answer
84 views

Find angle in another view of image

Is it possible to find angle in some "virtual" view from scaled view? (yes it is :] ) What i mean, for example: You got table and you look at it from some position (you know that the left top corner ...
1
vote
1answer
143 views

find area of square part

let us consider following picture where $ABCD$ is square, and using $A$ and $C$ as center,there is drawn arc,we should find area of dark part.we know that length of square is $a$,as i see the ...
1
vote
3answers
3k views

Find the equation of a line given the bounded area [closed]

Find the equation of the line through $(2, 2)$ and forming with the axes a triangle of area $9$.
0
votes
1answer
280 views

find minimum length of triangle

suppose that we have $ABC$ triangle,with $AB=28$ and $C=120$,we should find minimum length of triangle,if it is know that $AC:BC=3:5$,it is clear that minimum side is $AC$,also because sides are ...
1
vote
2answers
137 views

find angle in triangle

Let us consider problem number 21 in the following link http://www.naec.ge/images/doc/EXAMS/math_2013_ver_1_web.pdf It is from georgian national exam, it is written (ამოცანა 21), where word "...
0
votes
1answer
2k views

How to find rotation angles along X,Y,Z axes with a known vector to bring the axes to correct situation

I am working with 3d point data. When I checked the data I realized that there is some error on my data and need to do some kind of rotational rectification because the points which should be lie ...
1
vote
1answer
69 views

Trapezoid, find the sides

I have a right tangential trapezoid. I know the radius of the circle inscripted and the perimeter of the trapezoid. How can I find the sides?
0
votes
1answer
458 views

What is the height of a regular polygon?

I have three small circles forming a pyramid. I would like to centre that group in a square but have spent a couple of hours trying to calculate the height of the pyramid. I just can't seem to get ...
2
votes
3answers
1k views

Visual proof of the addition formula for $\sin^2(a+b)$?

Is there a visual proof of the addition formula for $\sin^2(a+b)$ ? The visual proof of the addition formula for $\sin(a+b)$ is here : Also it is easy to generalize (in any way: algebra , picture ...
-1
votes
1answer
81 views

Internally diving of vectors

Given the vectors $$\begin{eqnarray*}A&=&i+j-k\\B&=&i-j+2k\\C&=&j+k\end{eqnarray*}$$ How do I find the position vectors which divide BC AC internally in the ratio of 3:2?
2
votes
3answers
1k views

prove that minor arc of a great circle is the shortest distance

How to prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere? The key to solving the problems is that we must take all curves connecting ...
1
vote
5answers
2k views

Finding perpendicular bisector of the line segement joining $ (-1,4)\;\text{and}\;(3,-2)$

Find the perpendicular bisector of the line joining the points $(-1,4)\;\text{and}\;(3,-2).\;$ I know this is a very easy question, and the answer is an equation. So any hints would be very nice. ...
2
votes
2answers
320 views

Calculate the inner angles of the triangle $A(2,-3,5),B(0,1,4),C(-2,5,2)$

I want to calculate the inner angles of this triangle. $$A(2,-3,5),B(0,1,4),C(-2,5,2)$$ I know that for calculate the angle I need to do the following thing: $$\cos(\alpha)=\frac{A\cdot B}{|A||B|}$$ I ...
1
vote
1answer
90 views

Will a point moving on a sphere always at an angle x (0 deg. < x < 90 deg.) to the “equator” reach a “pole”?

Formulating my question seems to have given me the answer: that the point will continue getting closer to the pole but never reach it. Am I correct? Edit in response to Martin Argerami: I see ...
4
votes
2answers
191 views

A proof in circles.

I need help proving this problem: $AB$ is a diameter of a circle. $CD$ is a chord parallel to $AB$ and $2CD = AB$. The tangent at B meets the line $AC$ produced at $E$. Prove that $ AE = 2AB $. ...
0
votes
1answer
58 views

Generalizing a statement about points in the unit square

What is the three-dimensional version of this statement: Any $n$ points in the unit square can be labeled $x_1,\ldots,x_n$ to satisfy the inequality $$\|x_1-x_2\|^2 +\|x_2-x_3\|^2+\cdots+\|x_n-x_1\...
3
votes
3answers
2k views

Surface Area of a Hypercube

I am interested in computing the surface area of an $n$-dimensional hypercube and am interested in a reference or an answer which defines the notion of surface area for higher dimensional polytopes as ...
2
votes
0answers
169 views

Is it possible to draw any 2D shape using just lines and arcs?

I've heard that using just a french curve and straight edge, an architect can sketch any 2-dimensional shape on his blueprint. This makes me wonder: is there a theory that proves that any 2-...
3
votes
1answer
259 views

Help show angles at corner of a pyramid add up to more than $\pi$. (Picture included)

How can I prove that $\delta_i + \gamma_{i + 1} + \beta_{i + 1} \ge \pi$? Intuitively it seems clear because if you flatten the edge of the pyramid, you are going to have to make either $\delta_i$ or $...
5
votes
3answers
270 views

Points in unit square

Let $n$ points be given in the unit square. How to prove or disprove: the points can be labeled $x_1,\ldots,x_n$ to satisfy the inequality $$\|x_1-x_2\|^2 +\|x_2-x_3\|^2+\cdots+\|x_n-x_1\| ^2 \le 4,$$ ...
0
votes
1answer
74 views

Can the triangle inequlity extened to show the distance inequlity of a trapezium

$AB // CD$. What are the angle conditions (acute, obtuse or right angle) of $a,b,c,d$ to be satisfied the inequality $ |AB+BC| > |CD|$? $AB,BC,CD$ are distances.
1
vote
1answer
58 views

Term for changing properties in higher dimensions

Somewhat simple question, but it's the following. Consider D-volumes (that is, the equivalent volume measurement in D dimensions) of spheres of ever-higher dimensions. The percent of D-volume ...
1
vote
4answers
4k views

how to find arc center when given two points and a radius

I am a math-illiterate, so I apologize if this doesn't make sense... I am working on trying to draw a custom interface using the iOS Core Graphics API. In a 2D space, I need to create a "rounded" ...
2
votes
1answer
1k views

coordinates of icosahedron vertices with variable radius

I was looking on the wikipedia page about icosahedrons and it says that for edge length $a$ the radius of the circumscribed sphere around the icosahedron is given by $r = a \times sin(\frac{2\pi}{5})$....
3
votes
0answers
248 views

largest empty sphere or rectangle

In N (~ 500) dimensions, I wish to find out the largest sphere or rectangle such that the sphere/rectangle does not contain already existing points. The entire set of points is bounded in an axis-...
0
votes
1answer
55 views

Modification of the triangle inequality

We know from the triangle inequality that $X+Y \geq Z$, My question is under what conditions of $a,b,c$ (acute, obtuse or right angle) that $Z >X $ and $Z \geq Y $
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votes
0answers
384 views

Slices of a hypercube

Take the unit $d$-cube with vertices $\{0,1\}^d$, and restrict to the vertices that lie between (or within one of) a pair of parallel hyperplanes. These vertices form a graph whose edges are the edges ...
5
votes
2answers
110 views

How can we prove that this triangle is Equilateral Triangle?

This is a problem which was sent to me by a friend , but i couldn't solve it , in particular , i don't have ideas for that . I hope you can help by hints or any thing . Here is the problem in the ...
0
votes
1answer
446 views

Equal area rectangles in a piece of paper

If there is a paper that is split into $2$ sections (not to be assumed equal) with one section being laterally divided into $4$ equal subsections and the other $5$ (lateral means horizontal, or ...
2
votes
3answers
214 views

index free proof of dot product of two wedge products

I am learning geometric algebra, and meet an identy of (edited according to Andrey's comments below) $$ (a\wedge b)\cdot(c\wedge d) = (a \cdot d)(b\cdot c) - (a \cdot c)(b \cdot d)$$ as in wiki ...
13
votes
4answers
4k views

Is it possible to divide a circle into $7$ equal “pizza slices” (using geometrical methods)?

Or is it possible to divide a circle into n equal "pizza slices" (I don't know how to call these parts, but I think you'll know what I mean), where n hasn't got a common divider with $360$? Or are the ...
0
votes
1answer
48 views

Largest angular distance from an orthonormal basis

Let $S^{n-1}$ be the unit sphere in $\mathbb{R}^{n}$. Let $\{e_1,\ldots,e_n\}$ be an orthonormal basis for $\mathbb{R}^{n}$. Let $\Sigma=\{e_1,-e_1,\ldots,e_n,-e_n\}$ be the set of $2n$ points ...
3
votes
2answers
327 views

Geometric Deformations

There are geometric transformations such as translation, rotation and uniform scaling (Affine transformations). I am interested in knowing whether there is a separate class of transformations that ...