Questions on the circle, a curve composed of points that are at a fixed distance from a fixed point.

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6
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197 views

What's the average distance between two discs in the plane?

Consider two discs in the plane of radius $r$ and $s$, with centers separated by a distance $l$. If we choose a point uniformly at random from each disc, what is the expected distance between the two ...
6
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0answers
624 views

The number of circles that will fit inside the area of larger circle?

Let's say circle $\omega_1$ has a diameter $X$. Let $X>Y$; $Y\in \mathbf{R}^{+}$. How many circles with diameter $Y$ will fit inside $\omega_1$? Is there a formula for this?
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63 views

Does $T$ map a circle to a circle?

I have something to ask about the following map on in $\mathbb{C}$ $$ T \quad : \quad \longmapsto \frac{-2}{\bar{z}+i} -i $$ The map is well-defined whenever $z \neq i$. I have shown that it maps the ...
4
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0answers
339 views

Pinwheel- perimeter of semicircular region

Above, we have a larger circle of $r=16$ with 8 equally spaced semicircles of radius=8. Each semicircle has one end on the larger circle's center and the other on the circumference of the larger ...
3
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0answers
49 views

Rotation number of inverse maps on the circle.

I'm still a bit lost in my studies of rotation numbers. Any help is much appreciated! Let's say we have a homeomorphism $F: \mathbb{R} \rightarrow \mathbb{R}$ which is a lift of a homeomorphism ...
3
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0answers
35 views

unit circle trigonometry where angles is greater than 90

how is possible to have sin of angle greater than 90. if sin is ratio of opposite side and hypotenuse in right angle triangle then triangle with one of the angle greater than 90 can not be right angle ...
3
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0answers
23 views

How does this polar function behave?

I came across this question in my textbook for Nonlinear Optimisation and I don't know what to do: Consider the function: $$ f(x_1,x_2)=(r-1)^2-\frac{1}{2}(r-1)^2\cos \left( \frac{1}{r-1}-\phi ...
3
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0answers
109 views

Ellipses touching a circle

Given a circle and two points $A$, $B$ in the plane, how do I find an ellipse with focal points $A$ and $B$ that touches the circle? How many such ellipses are there (at least/at most)? Can I ...
3
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0answers
82 views

Characterizations of a linear fractional transformation

Consider the function $$ g(t) = \frac{1+it}{1-it} = \frac{1-t^2}{1+t^2} + i \frac{2t}{1+t^2}. $$ (The second equality holds except when $t=i$.) It seems to be widely known that this function is the ...
3
votes
0answers
47 views

Topology of a 3D wired Mandala?

There is a so called 3D-wired Mandala, based upon $2$ large circles each flowered symmetrically on its circumference by two sets of each $8$ half-circles. The circles are interconnected together by ...
3
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0answers
519 views

Circle Packing Algorithm

I have question related to circle-packing. The problem is to find the circle of minimum radius enclosing four non-overlapping circles of arbitrary radius. I have to write a program in C for this ...
2
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0answers
42 views

What is the curve's name for the “reciprocal” equation of a circle?

The equation of a unit circle is $$(x-a)^2+(y-b)^2=r^2$$ When the origin $$(a, b)=(0,0)$$ the equation becomes $$y=(1-x^2)^{1/2}$$ Naturally when this equation is plotted on graph paper we get a ...
2
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0answers
39 views

Packing circles in circle vs semicircle vs quarter of circle

Consider $N$ disjoint circles with radius $1$ packed into a larger circle $C$. Let $R$ be the smallest possible radius of $C$, allowing the best packing density. Now take the $N$ unitary circles ...
2
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0answers
56 views

To find a fifth degree equation by using circles and lines that cannot be solved by radicals

An example quintic whose roots cannot be expressed by radicals is $x^5 - x + 1 = 0$. I asked a geometry question about a fifth degree equation long time ago . I had an equation in the question. It ...
2
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0answers
311 views

Rounding Corners: How to calculate Fillet radius?

How do I find the maximum rounding I can apply to either corner for any amount of rounding on the other corner? The all circles are perfect circles, but I can't figure out the max size of the ...
2
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0answers
114 views

Can you explain the solution of this geometric problem

A year ago IBM research posted an interesting geometrical problem: A gardener plants a tree on every integer lattice point, except the origin, inside a circle with a radius of $9801$. The trees ...
2
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0answers
42 views

Bijection from the plane to itself that takes a circle to a circle must take a straight line to a straight line

Prove, that a bijection from the plane to itself that takes a circle to a circle must take a straight line to a straight line. There exists an elementary proof? I know this question can be found here ...
2
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0answers
103 views

Ellipse radius interpolation with different radiuses

I am writing a library for graphical LCDs and I want to incorporate a function to draw a circle on the screen. I have already succeeded in drawing simple circles, however, I want to be able to pass a ...
1
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0answers
23 views

The Biggest Smallest Piece to Smallest Biggest Piece ratio of a circle cut by n chords with maximal number of regions

It is well known that a circle cut by n chords gives at most (n^2 + n + 2 )/2 regions eg. http://mathworld.wolfram.com/CircleDivisionbyLines.html Questions:- How close to equal area regions can we ...
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0answers
14 views

Trilateration question help

Kind of stuck n this question, I just got the circle equation written down for the robot don't know what to do from here. A bicycle robot is travelling on a circle centred at the origin and with a ...
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0answers
37 views

arrange div elements in circle and square

I n number of divs which are arranged in a circle using javascript. Right now i set the dimension of each div to 40*40. Below is what i am able to achieve so far. This is how i find X & Y of each ...
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0answers
47 views

Family of circles touching a line

I found this in a book but I am not able to understand how they got this result. It goes the equation family of circles touching a given line $(y-y_1)=m(x-x_1)$ at $(x_1,y_1)$ for any value of $m$ is ...
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0answers
50 views

angles subtending arcs at the circumference and centre

$A$ and $B$ are two points on the circumference of a circle center $O$. $C$ is a point on the major arc $AB$. Draw the lines $AC$, $BC$, $AO$, $BO$, and $CO$, extending the last line to a point $D$ ...
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0answers
46 views

Circle geometry problem with overlapping circles

I want to find the length of the red line. The angle between radii is the same for both sectors and radii of both of the circles are known.
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0answers
35 views

Grid overlay on an annulus. Move n squares to create a sector that is closest to the area of the original.

I want to create an image in photoshop, and need to break an annulus, pictured below, into smaller segments. I can use other methods to find the solution, but I'm interested to see how mathematicians ...
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0answers
46 views

Is there a relation for when a circle intersects more than half the perimeter/circumference of another circle?

Is there some nice formula or algoritm for determining when a circle "hides"/intersects more than half of the perimeter of another circle, in a circle-circle interaction. Example image: Two example ...
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0answers
147 views

Problems with Circles and Lines on a Cartesian Plane

(a) Find the equations of the two circles each of which touches both coordinate axes and passes through the point $(9,2)$. (b) Find the coordinates of the second point of intersection of the two ...
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0answers
45 views

Optimization and derivatives homework

Find the dimensions of a right circular cylindrical can with both a top and a bottom that holds 8 cubic cm and is constructed with the least amount of material possible. Radius of can= cm Height ...
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0answers
24 views

Best path for finding within a radius of x units from this point

Say i am standing at a point and knew there is one thing within a radius of x units from this point. What is best path to find that thing. Best can mean shortest, but the discussion can be more open. ...
1
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0answers
84 views

Circle in a simplex

Let $T$ be a $2$-dimensional simplex in $\mathbb{R}^2$. A circle $C(x,y,r) \subset \mathbb{R}^2$ is given by its center $(x,y) \in \mathbb{R}^2$ and radius $r\ge 0$. Show that the set of circles in ...
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0answers
74 views

Circle Geometry and Conic Section textbook

I seek a textbook for good conic section and circle geometry questions. Slightly above introductory level. - slightly. But I wouldn't mind introductory level questions to consolidate my knowledge. I ...
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0answers
68 views

Probability of a certain circular configuration

Pick each of $n$ angles , $\theta_1$ through $\theta_n$ , uniformly randomly in the range $[0,2\pi$]. Define the distance $d_{i,j}$ between $\theta_i$ and $\theta_j$ by $d_{i,j} = \min(|\theta_j - ...
1
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0answers
127 views

family of circles in bipolar coordinate system

I don't get the idea how the equation for this family of curve is $\displaystyle y^2 + (x - a \coth v)^2 = \frac{a^2}{\sinh ^2v}$ from this article on Wikipedia. Suppose, the equation is ...
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0answers
66 views

Is there a continuous version of $tan^{-1}(\frac{y}{x})$ for the entire unit circle?

The fact that $tan^{-1}(\frac{y}{x})$ only "works" for the upper-right quadrant makes some calculations (for a physics simulator) impossible. I of course use $atan2(y,x)$ in the code, that's not what ...
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0answers
112 views

Given 2 outer points of a perfect circle, find the centerpoint

Alright, I hope this makes some sense. I am using a software that can create arcs. This arc is defined by: Begin point End point Center of "circle" The center is supposed to be the center of the ...
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0answers
353 views

Finding tangent points of circle inside a triangle

Hi, This is really a part 2 of a previous questions of finding intersecting points of a circle and triangle. I'd like to run my approach by you all to see if I'm thinking correctly. Maybe there's a ...
1
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0answers
215 views

Completelly cover area with minimum number of maxed circles NP-completeness (or harder) proof

everyone. I'm looking for paper with proof of NP-completeness following, or similar problem. Given: Area $S \subset \mathbb{N}^2$, let it be convex or rectangular, I believe it doesn't matter ...
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0answers
150 views

Area of ring section closed within a rectangle

I wish to find out the area of a section of a ring which can be acted on my a rectangle 100mm wide by 60mm height. I know the inner diameter,ID and outer diameter,OD of the ring and the width of the ...
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0answers
243 views

What “boundary conditions” can make a rectangle “look” like a circle?

I posted the question below in Stackoverflow but then realized that it perhaps would find a better audience here. I am solving a fourth order non-linear partial ...
0
votes
0answers
42 views

Convex optimization approximation

Consider the optimization problem $\mathcal{P}_0$ $$ \min_{x \in \mathbb{R}^2} \left\| x-p \right\|^2 $$ $$ \text{sub. to: } \ A x \leq b, \ \ x_1^2 + x_2^2 = 1 $$ where $p \in \mathbb{R}^2$ is a ...
0
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0answers
40 views

Finding circle which touches two functions

I wasn't sure whether my issue is with my Mathematica code or the actual way I am trying to figure out my problem so if it is a Mathematica issue I can ask it on that stack exchange. Firstly I have ...
0
votes
0answers
31 views

Translate vertical movement into radial movement?

I've tried all sorts of things, but I'm no mathematician and I've conceded defeat. So I come here for help. I don't know if I really worded the question correctly since I don't even know what I should ...
0
votes
0answers
23 views

Function where circle is special case with i=2

I saw this function some while back but cannot recall its name and therefore cannot find it and research it. Can someone please remind me its name? $|x|^i + |y|^i = 1$ where $i$ is ${1,2,3,4,5 ...}$ ...
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0answers
16 views

Coordinate geometry of circles; Circles through the points of intersection of 2 other circles

Here is my question. Let S=0 be the equation of circle 1 and T=0 be the equation of circle 2. There is a standard expression that, if you want the equation of a circle passing through the points of ...
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0answers
29 views

Find angle of an arc in the circle using 3 coordinates

I want to find angle of semicircle. I have 3 coordinates (center_a,center_b) , (pivot_a,pivot_b) and (last_point_a, last_point_b). From triangle , i can find angle using equation using cosine ...
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0answers
19 views

Rotation of point with infinite child objects. (Chain rotation)

More of a thought experiment here, knowledge for knowledges sake. Let's say you can create infinite points that rotate smoothly and at the same speed as each other through a full revolution - let's ...
0
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0answers
34 views

Computing the coordinates of a point, offset from a rotated point.

Good day. I have a question which should be easy but I have not been able to figure it out. The coordinates of a point on a unit circle, given an angle, is $$\begin{align} x &= \cos(\alpha) \\ y ...
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0answers
27 views

Outer tangent of two spheres in sphere surface

I have the necessity to draw on a sphere (earth) an air corridor. In the cartesian plane, this corridor is made by some circles and from outer tangents that connect these circles as you can see in ...
0
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0answers
23 views

Position vectors of sphere/circle touching central one

I am trying to understand the meaning of an expression describing the "kissing" number problem. On Wiki, it states the following: Let $x_n$ be a set of $N$ $D$-dimensional position vectors of the ...
0
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0answers
27 views

Area covered by multiple (possibly intersecting) circles on surface of sphere

I have a number of circles of same radius on surface of sphere (Google Maps API). I'm trying to calculate the total area covered by these possibly intersecting circles. My current solution is ...