For questions conserning circles. A circle is a curve composed of points in a plane that are at a fixed distance from a fixed point.

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256 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 ...
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83 views

Stacking circles

When I tried to stack 21 circles of radii $(30, 31, 32... 50)$ on top of each other in a tube (ID of $100$ wide), I thought they would reach the same height regardless of the order, however I was ...
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581 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 ...
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49 views

Archimedes Classic Proof for Area of Circle: Love it but can't grasp one aspect…

The proof assumes that:... The perimeter of any CIRCUMSCRIBED regular polygon is GREATER than the circumference of the circle. ie: !http://www.themathpage.com/atrig/Trig_IMG/eval1.gif Is this an ...
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99 views

The reverse pizza problem .

The pizza problem is a fairly well-known problem which sounds like this : You have a circular pizza and you need to cut it such that you and your friend would both receive half of the pizza . ...
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155 views

Find circles that completely cover a polygon minimizing the amount of space covered outside the polygon

I have an arbitrary polygon that I need to roughly represent using circles. Any point inside the polygon must lie inside a circle. There will be points outside the polygon that will fall under a ...
3
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29 views

Why does (h,k) generally represent the center of a circle?

Why are h and k generally used to denote the coordinates of the center of a circle? After a bit of research, we found that h may represent "horizontal shift" or "horizontal translation", but we're ...
3
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49 views

Spectrum of the circle - Clarification of certain points

I have a bit of difficulty as doing the problem Eigenvalues of the circle over the Laplacian operator. Since $g$ is a periodic function, do I have to use the Fourier series? If so, how could I do ...
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42 views

Prove that the circle contains the polygon.

Given a convex polygon. The circle is constructed for every triple of consecutive vertices of the polygon.We get the n circles. Select the circle with the largest radius. Prove that the circle ...
3
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49 views

Centroid and circumcenter — how close?

Suppose $R$ is some planar region, bounded by a curve. Let $C_1$ be the centroid of $R$, and let $C_2$ be the center of the "circumcircle" (the smallest circle enclosing $R$). Intuitively, it seems ...
3
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65 views

A question concerning radians and arc length

I was asked by a colleague yesterday about how the formula for the arc length of a circle is derived. I wanted to give them a correct answer, so I said I'd get back to them once I'd thought about it ...
3
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78 views

Problem about cyclic quadrilaterals

In cyclic quadrilateral ABCD, let E, F, G, H be the orthocenters of triangles BCD, CDA, DAB, ABC, respectively. Prove that EFGH is cyclic. Progress So far, found that if E is orthocenter of BCD ...
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119 views

Find the minimum radius of the circle which is orthogonal to two given circles

Problem : Find the minimum radius of the circle which is orthogonal to both the circles $x^2+y^2-12x+35=0$ and $x^2+y^2+4x+3=0$ . Solution : Let the equations : $x^2+y^2-12x+35=0.....(i)$ and ...
3
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162 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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28 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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198 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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93 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
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70 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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735 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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27 views

Volume of “the complex projective space” of a certain radius.

Consider the circle action on $\mathbb C^n$ given by $(e^{it},z)\to e^{it}z$. A moment map for this action is $J:\mathbb C^n\to\mathbb R:z\to -\frac{1}{2}|z|^2$. Let $M_l=J^{-1}(-\frac{l}{2})/U(1)$ ...
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28 views

How many different ways can a circle intersect a triangle N ways?

Consider a circle intersecting a triangle. The circle and triangle can have between 0-6 total intersection points. Is there a mathematical formula for the number of possible ways they can intersect ...
2
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28 views

concurrency of three lines on IO

Let $ABC$ be a triangle, and let $X$, $Y$, and $Z$ be the excenters opposite $A$, $B$, and $C$. The incircle of triangle $ABC$ touches $BC$, $CA$, $AB$ at points $D$, $E$, $F$, respectively. Finally, ...
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42 views

A circle can include all but one of n points, but which one can it be?

The answers to the question "Circle enclosing all but one of n points" demonstrate that, given $n$ points, it is possible to construct a circle such that all but one of the points is inside the circle ...
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56 views

Show that a complex equation represents a circle

I'm having troubling understanding the answer to a question. The question is: If $\ v=1+i$ and $\ z=x+iy$, for any real numbers x and y: Show that the equation $\left|z-v\right|= \left|vz\right|$ ...
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57 views

Equation for the points touching a circle.

In the plane $\mathbb R^2$, a point $P$, a point $M$ and the radius $r$ are given. Suppose, that $|\overrightarrow {PM}|>r$. Then, there exist two tangents from $P$ to the circle with mid point ...
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144 views

Calculate new pitch and roll after rotating about the z axis

I am wanting to know how to find out the new pitch and roll values when rotating around a circle. I have become a little stuck on how to achieve this, but hopefully someone will be able to point me in ...
2
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64 views

Triangle side-length problem

my problem is the following. A triangle ABC is given. P is a point on $\overline{AB}$. $k_1, k_2, k$ are the radii of the in-circles of APC, BPC, ABC. $s_1, s_2, s$ are radii of the ex-circles of ...
2
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80 views

Prove special case of Brianchon's theorem using inversion

Brianchon's theorem says: When a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point. From interactive demo: ...
2
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82 views

Find the Langitude and Longitude of the centre point of a circle given a point on the circumference.

I couldn't find a similar question! Given I have the latitude and longitude (x,y) of a point on the circumference of a circle, and I want the circumference to be 1000m. An example of a lat lang I ...
2
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0answers
90 views

Prove three chords of a circle are concurrent iff their poles with respect to a circle are collinear.

This probably would be a very simple problem if I could use any theorem I wanted about poles and polars, but in the book they give a definition and they say the problem should be solved using only ...
2
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105 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 ...
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116 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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69 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 ...
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1k 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 ...
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174 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 ...
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88 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 - ...
2
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229 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 ...
2
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0answers
318 views

Drawing a Great Circle between two given points on Earth

I need to draw a great circle arc between two latitude and longitude points. For sake of example we will use the coordinates for LAX and JFK. JFK is 40.64°N / 73.78°W LAX is 33.94°N / 118.41°W ...
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27 views

Prove a harmonic range from a familiar picture

During solving some simple problem (10th grade), I found this interesting problem, which I got no clue to solve it clean and properly. Hope someone can give me some hint to solve it. Thanks. Given ...
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50 views

The arc length of a circle section if radius is changing?

I would like to find the angle subtended by an arc of a circle with a changing radius. The main issue is that the radius is changing by a non-linear factor as shown below: The integral on the left ...
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0answers
19 views

circle segment height by given fill fraction

At work I was facing the problem of how to calculate the height of a water column inside an horizontal cylinder given the volume of the liquid. A plot of this function and a visual explanation can be ...
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31 views

On the existence of a continuous section of $\zeta\in\mathbb{S}^1\mapsto(\zeta^m,\zeta^n)\in\mathbb{S}^1\times\mathbb{S}^1$.

Let $(m,n)\in\mathbb{Z}^2$ and let define the following map: $$f:\left\{\begin{array}{ccc} \mathbb{S}^1&\rightarrow&\mathbb{S}^1\times\mathbb{S}^1\\ \zeta&\mapsto&(\zeta^m,\zeta^n) ...
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27 views

Circle homography

I'm attending a 3d-graphics course and I want to figure out which homograpic transformations conserve a circle's equation. The circle's equation is given as: Circle = $x^2 + y^2 + Ax + By + C = 0 $ ...
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45 views

Evaluating the curve (line) integral of a complex function along three circles in $\mathbb{C}$

I want to find the the curve integral $$\int_γ \frac{1}{1 - z + z^2 - z^3 } dz$$ with $γ$ passing through the following sets counter-clockwise once. a) $\{z \in \mathbb{C}, |z - i| = 1\}$ b) ...
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48 views

Finding locus of circle passing through extremities of the two rods

Two thin rods AB and CD of length 2a and 2b moves along OX and OY where O is the origin. Find the locus of the center of the circle passing through the extremities of the two rods. My attempt:- ...
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112 views

Circle continuty principle proof

Circular continuity principle: If a circle C has one point inside and one point outside another circle C' , then the two circles intersect in two distinct points. I read this on Euclidean and ...
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0answers
39 views

Length of an arc of a circle when the angle is infinitesimally small

The task is to express the length of an arc of a circle trapped between two radii named $r$ if the angle between them is infinitesimally small, named $d\theta$. The solution to this problem is ...
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46 views

How many discs necessary to cover a big circle?

Let be a circle a radius R,and other discs of radius r,palpable. I can cover the circle,completly, with a minimum number of discs,N.I can't cut any disc. What is the value of N,according to R and r? ...
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0answers
53 views

Integrating a prob distr over the set of possible circles within an annulus

Let $z$ be the measured coordinates of a point on a circle $c$ with center $x$ and radius $r$. Assume the probability of measuring $z$ given the circle $c$ is normally distributed by the distance ...
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247 views

Calculating the Area of a Circle Occupied by a Rectangle

This is a question regarding how to calculate the area of a circle occupied by a rectangle when that rectangle is larger than the circle (see this link for a example image ...