shape, congruence, similarity, transformations, properties of classes of figures, points, lines, angles
7
votes
2answers
961 views
Inscribed kissing circles in an equilateral triangle
Triangle is equilateral (AB=BC=CA), I need to find AB and R.
Any hints?
I was trying to make another triangle by connecting centers of small circles but didn't found anything
7
votes
1answer
429 views
Comprehensive compilation of conic section formulae
My frustration started after hours of searching failed to turn up a formula for the vertex of a parabola in the general form
$$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$
As is already well known, the discriminant ...
6
votes
3answers
363 views
How is it that this shape can converge to what looks like a triangle but has a different perimeter?
I had this strange notion some time ago, and I recently wrote a blog post about it, as a mere curiosity. I don't really consider it a "serious" mathematical question; but out of interest, I wondered ...
6
votes
3answers
700 views
What is the proper geometric description of a the oval used for a horse racetrack?
I'm talking about the shape made up of a rectangle with a semi-circle at each end. Does it have a particular name?
5
votes
1answer
176 views
Uniform distributions on the space of rotations in 3D
I believe on moral grounds that the following three definitions are equivalent, and determine "the" uniform distribution on rotations in three dimensions.
The Haar measure on $SO(3)$.
The uniform ...
4
votes
1answer
61 views
Angular alignment of points on two concentric circles
I have two concentric circles $C_1$ and $C_2$ with radii $r_1,r_2$ such that $r_1< r_2$and a set of finite points $P=\left \{ p_1,p_2...p_n \right \}$ and $Q=\left \{ q_1,q_2...q_n \right \}$ are ...
4
votes
2answers
327 views
Geometric intuition behind gradient, divergence and curl
I learned vector analysis and multivariate calculus about two years ago and right now I need to brush it up once again. So while trying to wrap my head around different terms and concepts in vector ...
4
votes
2answers
1k views
How to transform a set of 3D vectors into a 2D plane, from a view point of another 3D vector?
I googled around a bit, but usually I found overly-technical explanations, or other, more specific Stackoverflow questions on how 3D computer graphics work. I'm sure I can find enough resources for ...
4
votes
9answers
5k views
how to find center of an arc given start point, end point, radius, and arc direction?
Given an arbitrary arc, where you know the following values: start point (x0,y0), end point (x1,y1), radius (r) and arc direction (e.g. clockwise or counterclockwise from start to end), how can I ...
4
votes
1answer
455 views
Interesting Taxicab Problem?
I came up with this problem after discussion of taxicab geometry in math class... I thought it was a simple problem, but still pretty neat; however, I am as of yet unsure of whether my answer is ...
3
votes
1answer
947 views
Matrix for rotation around a vector
I'm trying to figure out the general form for the matrix (let's say in $\mathbb R^3$ for simplicity) of a rotation of $\theta$ around an arbitrary vector $v$ passing through the origin (look towards ...
1
vote
3answers
292 views
Inscribing a rhombus within a convex quadrilateral
I was wondering if it is possible to inscribe a rhombus within any arbitrary convex quadrilateral using only compass and ruler? If possible, could you describe the method?
If not could you give an ...
1
vote
1answer
161 views
clock related challenge
A person leaves his house between 4.00 and 5.00 pm. He carefully notes the position of the minute hand and hour hand when he leaves the house. He returns back between 7.00 and 8.00 pm.He notices that ...
1
vote
2answers
194 views
Rotation around a point?
I know that rotation can be understood by simple complex transformation (as shown on 758)
$$\begin{align*}y_{1}+iy_2
&= \left( \cos(\alpha) + i \sin(\alpha) \right) \left( x_{1}+ix_{2} \right) ...
1
vote
1answer
56 views
Reconstruct shape of a body from rationality of its projections
There is a closed convex body $S$ in $\mathbb{R}^3$. Areas of its projections on all planes (not only those normal to axes $x,y,z$) are rational numbers. Can we deduce that $S$ is a ball?
Replace ...
14
votes
4answers
535 views
Gerrymandering/Optimization of electoral districts for one particular party
I'm asking this on behalf of Zach Weiner (actually it's my own initiative in order to promote this site). Original text is here, and is as follows:
Hey--
This is Zach from SMBC, and I have a math ...
12
votes
1answer
281 views
Density of randomly packing a box
I want to throw alot of copies of an object of nonzero volume, randomly into a large box. Ignoring boundary effects of the box, with which type of object will the expected packing density be the ...
12
votes
2answers
432 views
Do infinitely many points in a plane with integer distances lie on a line?
Someone posted a question on the notice board at my University's library. I've been thinking about it for a while, but fail to see how it is possible. Could someone verify that this is a valid ...
10
votes
2answers
265 views
Elementary Geometry Nomenclature: why so bad?
A long-ish wall of text, and I apologize.
Some background: when I was a first-year university student, my chemistry professor was lecturing and was trying to find the word to describe a shape. A ...
9
votes
4answers
694 views
Two paradoxes: $\pi = 2$ and $\sqrt 2 = 2$ [duplicate]
Possible Duplicate:
Is value of $\pi = 4$?
Can anyone explain how to properly resolve two paradoxes in this YouTube video by James Tanton?
9
votes
4answers
1k views
How to understand dot product is the angle's cosine?
How can one see that a dot product gives the angle's cosine between two vectors. (assuming they are normalized)
Thinking about how to prove this in the most intuitive way resulted in proving a ...
9
votes
1answer
1k views
Why does GPS require a minimum of 24 satellites?
From Wikipedia,
The GPS design originally called for 24 SVs, eight each in three
approximately circular orbits, but this was modified to six orbital
planes with four satellites each. [...] The ...
8
votes
5answers
1k views
A circle with infinite radius is a line
I am curious about the following diagram:
The image implies a circle of infinite radius is a line. Intuitively, I understand this, but I was wondering whether this problem could be stated and ...
8
votes
2answers
249 views
If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear
If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear.
I really can't seem to get anywhere on this problem, ...
8
votes
1answer
607 views
Find points along a Bézier curve that are equal distance from one another
I'm trying to figure out a generic way of determining a series of points on a Bézier curve where all the points are the same distance from their neighboring points. By distance I mean direct distance ...
7
votes
4answers
905 views
Drawing a thickened Möbius strip in Mathematica
I would like to have Mathematica plot a "thickened Möbius strip", i.e. a torus with square cross section that is given a one-half twist. Ideally, I would like this thickened Möbius strip to be ...
7
votes
2answers
252 views
Are rotations of $(0,1)$ by $n \arccos(\frac{1}{3})$ dense in the unit circle?
Under which conditions will successive rotations of $(0,1)$ by an angle $\theta$ guarantee that given $\delta > 0$ and some point $p$ on the unit circle, there exists some $n$ such that rotating ...
6
votes
3answers
2k views
How to make a sphere-ish shape with triangle faces?
I want to make an origami of a sphere, so I planned to print some net of a pentakis icosahedron, but I have a image of another sphere with more polygons:
I would like to find the net of such model ...
6
votes
2answers
426 views
Deriving an implicit Cartesian equation from a polar equation with fractional multiples of the angle
Usually, applying the conversion formulae
$r^2=x^2+y^2$
$\cos\;\theta=\frac{x}{r}$
$\sin\;\theta=\frac{y}{r}$
to transform an equation in polar coordinates to an implicit Cartesian equation is ...
5
votes
2answers
341 views
How to derive the law of cosines without the pythagorean theorem
To me, it seems that the Pythagorean theorem is a special case of the law of cosines. However, all derivations that I can find seem to use the Pythagorean theorem in the derivation. Are there any ...
5
votes
1answer
533 views
Analytical Expression to find the Shortest Distance between Two Ellipses?
If I have the Keplerian elements for two orbits, how do I compute the shortest distance between these two orbits in 3D space? Is there any analytical expression to compute that?
4
votes
1answer
41 views
Optimal rotation to align a circle with external points
I have a circle $C$ with radius $r$ and a set of finite points $P=\left \{ p_1,p_2,\ldots,p_n \right \}$ are identified external to the circle $C$. These points may lie on the exterior or the interior ...
4
votes
1answer
150 views
what are some isometries of S^2 without fixed points?
Spherical geometry question involving isometries.
Particularly looking for isometries with no fixed points.
4
votes
4answers
458 views
Constructing a triangle given three concurrent cevians?
Well, I've been taught how to construct triangles given the $3$ sides, the $3$ angles and etc. This question came up and the first thing I wondered was if the three altitudes (medians, ...
4
votes
3answers
666 views
Find a plane perpendicular to a plane passing by point
In $\mathbb R^4$ I have: $$\pi: \begin{cases} x+y-z+q+1=0 \\ 2x+3y+z-3q=0\end{cases}$$
I have to find $\pi' \bot$ $ \pi $ and passing by $P=(0,1,0,1)$. How can I do that? Thanks a lot!
4
votes
2answers
177 views
Computing the projection of an infinite 3D grid of points
Consider the subset $S$ of $\mathbb{R}^3$ consisting of points whose coordinates are integers (compare Gaussian integers, Euclid's orchard). The view from the origin has interesting structure; it has ...
4
votes
2answers
360 views
Determine angle x using only elementary geometry
Using only elementary geometry, determine angle x.
You may not use trigonometry, such as sines and cosines, the law of sines, the law of cosines, etc.
4
votes
4answers
561 views
Find the centre of a circle passing through a known point and tangential to two known lines
I am trying to find the centre and radius of a circle passing through a known point, and that is also tangential to two known lines.
The only knowns are:
Line 1 (x1,y1) (x2,y2)
Line 2 (x3,y3) ...
4
votes
3answers
285 views
Decidability of tiling of $\mathbb{R}^n$
Given a polytope of dimension $n$, is there some general way to determine if it can tile $\mathbb{R}^n$?
4
votes
2answers
974 views
Prove that three points are enough to draw/define one and only one circle
Prove that three points are enough to draw/define one and only one circle, how would this be done?
4
votes
2answers
144 views
Tiling Posters on a Wall
I'm a noob, and I'm not a mathematician (Although I will be a Math major next semester). My question is:
I have 68 maps I would like to use as posters on my wall at home. They are all rectangles, ...
4
votes
2answers
857 views
Dot product in coordinates
Dot product of two vectors on plane could be defined as product of lengths
of those vectors and cosine of angle between them.
In cartesian coordinates dot product of vectors with coordinates (x1, y1) ...
4
votes
1answer
248 views
3
votes
2answers
2k views
Find the volume of the largest right circular cone that can be inscribed in a sphere of radius r?
I checked this question but didn't fully understand it.
I know that the volue of a right circular cone is $V = \frac{1}{3}\pi x^2h$
I know that I must take the first derivative and set it equal to ...
3
votes
3answers
283 views
Ellipse with non-orthogonal minor and major axes?
If there's an ellipse with non-orthogonal minor and major axes, what do we call it?
For example, is the following curve a ellipse?
$x = \cos(\theta)$
$y = \sin(\theta) + \cos(\theta) $
curve ...
3
votes
2answers
366 views
Recurrence for number of regions formed by diagonals of a convex polygon.
I've been having trouble with this particular problem, been thinking for it for a good hour or two, but I haven't gotten an explanation to the following question.
Suppose $a_n$ be the number of ...
3
votes
3answers
1k views
How many lines can be equidistant from 3 points?
How many lines can be drawn in a plane such that they are equidistant from 3 non-collinear points?
@John Bentin has shown below that there are at least 3. Why are there no more than 3?
3
votes
5answers
329 views
How do you parameterize a sphere so that there are “6 faces”?
I'm trying to parameterize a sphere so it has 6 faces of equal area, like this:
But this is the closest I can get (simply jumping $\frac{\pi}{2}$ in $\phi$ azimuth angle for each "slice").
I ...
3
votes
3answers
3k views
Orthogonal projection of a point onto a line
please give me a directions how to solve this:
find an orthogonal projection of a point T$(-4,5)$ onto a line $\frac{x}{3}+\frac{y}{-5}=1$
3
votes
1answer
567 views
Can you write a non-piecewise equation that describes an arbitrary shape?
This batman equation thing got me thinking: for an arbitrary curve drawn on the Cartesian plane, can you write a corresponding equation which is not piecewise? What about closed shapes, a la the ...
