Questions tagged [circles]

For questions concerning circles. A circle is the locus of points in a plane that are at a fixed distance from a fixed point.

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472 views

How to calculate the average distance between two points in concentric annuli

I tried to solve this analytically but gave up and ran a simulation. I would still like to know if it can be done explicitly. Consider a circle radially divided into 'zones', so say zone 1 is the ...
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1answer
43 views

Degree measure of circles

One slice of a circle which has been divided into 360 slices is one degree right? If this is the case, won't bigger circles have bigger slices and therefore bigger degrees? Why is one degree of a ...
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63 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 $M$...
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404 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 ...
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78 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 ...
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394 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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1answer
77 views

Graphing Circles, Ellipses, Parabolas, and Hyperbolas

I need help plotting a curve on a graph where the distance from focus1 is always the same ratio to the distance from focus2. For instance, lets assume focus1 is -5 along the x axis, and focus2 is +5 ...
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2answers
242 views

High-School level question concerning circle and arcs

This question somehow is unsolvable to me. Any idead/hints wil be much appreciated. $AB$ is a chord which is cut ny the chords $CD$ and $EC$ in the circle. Givens: $\frown{AC} +\frown{BE}=\frown{AD}...
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1answer
253 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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252 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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1answer
857 views

From any arbitrary point $P$ on $y =\cos x$ tangents $PA$ and $PB$ are drawn to a circle which passes through

From any arbitrary point $P$ on $y =\cos x$ tangents $PA$ and $PB$ are drawn to a circle which passes through the points $(1,0)$ and $(3,0)$ and touches the circle $x^2+y^2-2x-8=0$ and have its ...
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171 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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88 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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1answer
349 views

Mapping a distorted ellipse onto a circle

I have a circular label pasted on a cylindrical object. In the image, this circle looks like a asymmetrical ellipse. I know the radius of the cylinder and that of the label. What mapping do I need to ...
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0answers
3k 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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2answers
727 views

Square covered with circles

I have a square 800x800 and i need to fully cover it with the least number of circles possible, each circle has a radius of 150. QUESTIONS: - What pattern would be the best to use? Clover, diamon or ...
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96 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 - \...
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379 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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1answer
235 views

Approximate radius of a group of n packed circles

I am looking for a formula to estimate the radius of a circle which would hold n number of circles with some radius r. I understand this is part of the packing problem which does not have a definite ...
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822 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 My ...
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1answer
160 views

How to constrain disks that intersection of them is inside unit circle

I have two disks $(x-a_1)^2+(y-b_1)^2\leq r_1^2$ and $(x-a_2)^2+(y-b_2)^2\leq r_2^2$, where $a_1$, $b_1$, $r_1$, $a_2$, $b_2$, $r_2$ are all known. What kind of constraint can I put on $a_i$, $b_i$ ...
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2answers
150 views

Circles tangent to a parabola

For the past two weeks I was struggling with solving the following problem. Description of variables: $(x_n,y_n)$ - center point of the circle $C_n$ $r_n$ - radius of the circle $C_n$ Given the ...
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1answer
543 views

Interval and Circle

Can anyone give me a proof of why the circle $S^1$ and the closed interval $[0,1]$ are not homotopically equivalent? (Using the basic definition and not the fundamental group!)
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1answer
37 views

What are the conditions required for the perpendicular bisectors of all sides of a quadrilateral to intersect?

What are the conditions required for the perpendicular bisectors of all sides of a quadrilateral to intersect? Actually, this question came in my mind while I was thinking about how a circle can ...
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71 views

Expected overlap of n circles of equal area randomly placed inside a circle of larger area

Say I have an outer circle of area $\Omega$. If I randomly position n circles of area $\omega$ ($0 \le \omega \le \Omega$) completely inside the outer circle, then what is the expected overlap area of ...
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16 views

Semi-Conjugacy and the preservation of wandering intervals

This is mostly a checking of my understanding. Currently I am working through the proof of Denjoy's Theorem in Katok and Hasselblatt's Introduction to the Modern Theory of Dynamical Systems, and ...
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2answers
31 views

Formulae for calculating line and circle intersection

Very simple problem, yet I can't find a solution online and can't seem to keep track of the algebra. Given $ax+by+c=0$ and $x^2+y^2=r^2$ , Solve for x constraints: ($a,b,c =$ any real number) ($x,...
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1answer
64 views

Real life applications of a circle? (Conics)

for my Math 2U assignment, we have to discuss real life applications of different conic sections. However, apart from the wheel, I cannot find or think of any other real life applications of the ...
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1answer
23 views

reflections of a point around the sides of a triangle: the area stays constant

Given a triangle $\Delta ABC$ and a point $P$, we define $P_A, P_B, P_C$ as the reflections of $P$ around $BC, AC, AB$ respectively. Now, $P_A, P_B, P_C$ are collinear if and only if $P\in(ABC)$. (1) ...
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54 views

Probability of two circles colliding in intersection area of two bigger circles

I have a small circle with area $A_s$ that is bound to be in a bigger circle with area $A_b$. The probability of the small circle being at a specific place in the bigger circle is: $$P = A_s/A_b$$ ...
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28 views

Maximum number of nodes in a circle, with distance constraint

Given a circle $C$ with known radius $r$, I want to determine the maximum number of nodes in the circle, where there is a distance constraint between each two nodes equal to $s$, i.e. each two nodes ...
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11 views

plane edge intersections embedded in higher dimensional space

Let's say we have some D-dimensional euclidean space, and we have some circles of dimension 0 to D-1 (circle dimensionality meaning the minimum number of vectors needed to fully define it, so a ...
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46 views

A circle inscribed in a quadrilateral

A circle is inscribed inside a trapezoid $ABCD$ ($AB || CD$). $M, N, P, Q$ are the midpoints of the sides $AB, BC, CD, AD$, respectively. If $AD = d$ and $BC=c$, express the perimeter of $MNPQ$. (...
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15 views

Reflect point inside Poincarè Disk Model

I have a question: is there a formula which map points onto the Poincarè Disk Model starting from points which are outside the unit Disk? For example, I have a point p with norm = 2, so it is outside ...
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1answer
45 views

Prove that three common chords are concurrent

Three circles intersect each other as shown. Prove that the three common chords are concurrent. Now the book does this by proving that the chord out of E and through M is the same for circle (2) and ...
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27 views

Maximum number of common chords that are existent between two Conics

The maximum no. Of common chords between a circle and a parabola is 6. this is because they can have at most Four Points of intersection. However I have doubt regarding other combination of conics. ...
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18 views

Circles in the case of $Distance=\sqrt{x^2+y^2}$, with $x$ as a real number, and $y$ as a real multiple of $a+bi$, with $a$ and $b$ as constants

In a 2d space, in which $Distance=\sqrt{x^2+y^2}$, with $x$ being a real number, and $y$ being a real multiple of $a+bi$, with $a$ and $b$ both being real none zero constants, would it be possible to ...
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25 views

The center of incircle and circumcircle coincide

How to show that if the centres of the circumscribed circle and the inscribed circle in $\triangle ABC$ match (coincide), $\triangle ABC$ is equilateral? I have an idea but I'm not sure. Let $\angle ...
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42 views

Probability of obtuse triangles formed on a circle

There are 16 equally spaced points on the circumference of a circle. If 3 points out of these 16 points are selected randomly, What is the probability that they will form an obtuse angled triangle?
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23 views

What is the coordinate value after moving counterclockwise by distance $d$ from a coordinate on the ellipse?

Let me define an ellipse function as follows: Assuming $a \ge b$, $$ f(x,y) = \frac{(x-x_0)^2}{a^2} + \frac{(y-y_0)^2}{b^2} = 1,$$ where $(x_0,y_0)$ is the origin of the ellipse, and $a$ and $b$ are ...
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1answer
44 views

Calculating point on circle given angle and distance traveled without calculating radius

I want to calculate the X,Y coordinates of a point on a circle given only the distance and angle traveled, without calculating the radius as an intermediate step. My starting point (0,0) is at the ...
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62 views

Approximation for the equation of an imperfect circle

Given two random points $(x_0,y_0)$ and $(x_1,0)$ and some condition, I have an equation $C[(x-x_0)^2+(y-y_0)^2]=[(x-x_1)^2+y^2]^2$ which is a perfect circle if I ignore the power (2) on the RHS. ...
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1answer
72 views

Problem about incentre of a triangle.

Through the incentre $I$ of triangle $ABC$ a straight line is drawn intersecting $AB$ and $BC$ at points $M$ and $N$, respectively, in such a way that the triangle $BMN$ is acute- angled. On the ...
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1answer
47 views

Symbol for cyclic quadrilateral

Is there a symbol to denote, say, $ABCD$ is a cyclic quadrilateral? (I very much doubt it.)
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2answers
94 views

Solving a System of Quadratic Equations for Sound Triangulation

I am currently attempting to solve a system of quadratic (and linear) systems that I have run into while trying to triangulate sound. My hypothetical setup includes 3 sensors on a perfectly ...
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0answers
25 views

Relationship between a circle inscribed in a square and a sphere inscribed in a cylinder

The ratio of the area of a circle to the area of the square it is inscribed in is equal to ${\pi\over 4}$ and the ratio of the volume of a sphere to the volume of the cylinder it is inscribed in is $2\...
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97 views

For any orientation-preserving circle homeomorphism $f$ with rational rotation number , $\Omega(f) = Per(f)$

Here's a problem that I've been unable to solve, even though I've tried. Let $f: \mathbb{S}^{1} \to \mathbb{S}^{1}$ be an orientation-preserving circle homeomorphism with rational rotation number. ...
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29 views

Circle can always represented as $\{z:|z-c|=k|z-d|\}$

I am reading Howie's Complex Analysis. There I see this remark: The observation that $c$ and $d$ are inverse points is the key to showing that every circle can be represented as $\{z:|z-c|=k|z-d|\}$...
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polar graphing, supposed range is between $0$ to $\pi,$ but graph drawn range from $-\pi/2$ to $\pi/2$

I'm currently studying "area bound by polar curves" on khan academy, and below is one of the question I have encountered I am puzzled by part $2$ of the official solution. I can't seem to figure out ...
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72 views

Determining the value of $\frac{BC}{CE}$ in a cyclic pentagon $ABCDE$.

Let $ABCDE$ be a cyclic pentagon, where $AC=2, AD=3, BD=5, BE=1, \frac{CD}{DE} = \frac{10}{3}$. What is the value of $\frac{BC}{CE}$? I worked with the area of specific triangle with trigonometry. ...