What is Circle: Definition and 1000 Discussions

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.
Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.
A circle may also be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.

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  1. D

    Coordinate chart for a circle

    Hi. I have been looking at the coordinate charts for the unit circle x^2 + y^2 = 1. In the notes I have the circle is split into 4 coordinate charts the first being - ##U_1## : x>0 , ##A_1## = y (PS without the symbols tab I have used A for the letter phi ) There are 3...
  2. M

    Segment of a circle (Exact form answer)

    Homework Statement Find the circle segment area that has the boundaries of line segment AB and the minor arc ACB. Give the area in an exact form in terms of surds and Pi. (see attachments for annotated picture & original question). Homework Equations Equation 1: Area of a segment = Sector...
  3. A

    Can a perfect circle exist in the real world?

    The (perfect) circle, defined in the Cartesian coordinates as the set of (x,y) pairs that fit the equation x^2 + y^2 = r^2 can "exist" as a mathematical abstraction, I have no problem with that. But can we have a perfect circle in the physical world? Particularly, can an object move in a...
  4. R

    What Are the Dynamics of a Particle in Vertical Circle Motion?

    Homework Statement Suppose a non-uniform circular motion where a particle of mass "m" is attached to a string, which rotates on a vertical plane. Once an initial velocity is provided to the particle at the lowest point of the trajectory, no further forces act on the particle. (Air drag is...
  5. M

    What would be general formula of a circle in form of variables ?

    General formula of a line is ax+b=0 Similarly can we have a general formula of a circle ?
  6. L

    Integral on Circle: Showing $\frac{1}{1-|z|^2}$

    How I can show the following \int _{\mathbb{T}} \frac{1}{|1-e^{-i\theta}z|^2}dm(e^{i\theta})= \frac{1}{1-|z|^2} , where z is in the unit disc dm is the normalized Lebesgue measure and T is the unite circle.
  7. Fantini

    MHB Kinematics, particle on half circle

    Here's the problem. A point traversed half a circle of radius $R = 160 \text{ cm}$ during a time interval of $\tau = 10.0 \text{ s}$. Calculate the following quantities averaged over that time: (a) the mean velocity $\langle v \rangle$; (b) the modulus of the mean velocity $ |\langle {\mathbf...
  8. Albert1

    MHB Radius Small Circle: Measurement & More

  9. T

    MATLAB Model a circle using finite difference equation in matlab

    hello. I have a MATLAB skeleton provided because i want to model a distribution with a circular geometry. all in all, i want the 3d graph of the code to be some type of cylinder. This is the code: % flat step condition for ii=1:nHi, for jj=1:nHj, if (X(ii)/R_P)<1 &...
  10. kaliprasad

    MHB Crazy Circle Illusion: Amaze Your Friends!

    Crazy Circle Illusion! | Showyou pNe6fsaCVtI
  11. Nathanael

    Regions; "Each point of the set is the center of a circle "

    "A set in the plane is called a region if it satisfies the following two conditions: 1. Each point of the set is the center of a circle whose entire enterior consists of points of the set. 2. Every two points of the set can be joined by a curve which consists entirely of points of the set."...
  12. Dethrone

    MHB Minimizing Isosceles triangle with a circle inscribed

    Find the smallest possible area of an isosceles triangle that has a circle of radius $r$ inside it. I cannot seem to find the relationship between the circle and triangle. Any hints? I'm thinking similar triangles, but I want to know if they're any other approaches before I try that.
  13. A

    Right biased LSD for circle track

    I know that some AWD vehicles used the gear ratio of a planetary or other transfer gear combined with a gear type torque biasing diff that would allow a natural mechanical advantage to apply more of the torque to the front or rear axle. Why not apply this to a rear diff for circle track...
  14. S

    MHB Find the Radius and Center of a Circle

    Can someone show me how to resolve this question? A particular circle in the standard (x,y) coordinate plane has an equation of (x − 5)2 + y2 = 38. What are the radius of the circle, in coordinate units, and the coordinates of the center of the circle? radius center F. √38 ( 5,0) G. 19 ( 5,0)...
  15. P

    Exploring the Concept of Tangent on a Large Circle

    Theoretically it is said that, tangent touches to a single point on a circle. But If my circle is very big, and large enough, then i think, it should not be a just single point where my tangent is touching, though is will be a very small portion depending on how large is the circle. If i have...
  16. A

    MHB How are Power Transmission and Reception Equations Derived?

    Hello. In the attachments are the equations for power transmission circle and power reception circle. Does anyone know how they are derived? Pr= power reception Ps=power transmission I think Wn is the power reference.
  17. Greg Bernhardt

    What is the equation of a circle

    Definition/Summary A circle has many definitions, a classical one being "the locus of all points on a plane that are equidistant from a given point, which is referred to as the 'center' of the circle". Equations Equation for a circle with it's center as the origin and radius 'r': x^2...
  18. G

    Earth Geodesics - Rhumb Line vs Great Circle

    I have an object (A) at some altitude above the Earth ellipsoid, and a point (B) on the surface of the Earth. Since you're not confined to the surface of the Earth as you travel from A (at altitude), to B, I'm getting confused. If I were to create a (Cartesian) vector pointing from object...
  19. C

    Mass of a rubber stopper being swung in a horizontal circle

    Homework Statement Homework Statement [/b] The purpose of this lab is to determine the mass of a rubber stopper being swung in a horizontal circle. I made an apparatus that resembles the picture attached, with a string (with a rubber stopper on one end, and a weight on the other end) put...
  20. S

    Tangent to the circle at a given point

    Homework Statement I basically have the radius of the circle and its displacement from the origin, so ##(x-p)^2+(y-q)^2=r^2## And now I need to find a tangent to the circle at a given point ##(a,b)##. Or at least the slope of the tangent. How would one do that? Homework Equations...
  21. G

    Squaring the Circle - Is There More to It?

    Is the squaring of a circle just a riddle that when solved to a great degree of accuracy then that's it... riddle solved. Or there more to it then that?
  22. D

    Probability of person who sit in a circle

    Homework Statement there are 7 seats around a round table, and is labelled form A to G. Find the number of ways a committee of 3 teachers and 4 parents can sit around the table? (i ) there is no restriction my ans is 7!=5040 (ii) all teachers must sit together the ans is 1008. can...
  23. M

    MHB Area of 'that' part of the circle ....

    Using integral find the area of that part of the circle x^2 + y^2 = 16 which is exterior to the parabola y^2 = 6x.
  24. E

    Circle in the Complex Domain where Mean is not the Centre

    Hello people of Physics Forums, In my research into transmission lines, I have come across the following function: x = ( a - i * b * tan(t) ) / ( c - i * d * tan(t) ) In the above equation x, a, b, c and d are complex and t is real. If my analysis is correct, varying t from -pi/2 to...
  25. Serious Max

    Problem involving intersection of a line and a circle

    Homework Statement Erik’s disabled sailboat is floating stationary 3 miles East and 2 miles North of Kingston. The sailboat has a radar scope that will detect any object within 3 miles of the sailboat. A ferry leaves Kingston heading toward Edmonds at 12 mph. Edmonds is 6 miles due east of...
  26. G

    Motion in a Circle: True/False/Less/Greater/Equal

    Homework Statement A small mass M attached to a string slides in a circle (x) on a frictionless horizontal table, with the force F providing the necessary tension (see figure). The force is then increased slowly and then maintained constant when M travels around in circle (y). The radius of...
  27. J

    Color of central circle in Newton's rings

    Hello guys , Can someone explain me why color of central circle in transmission case is bright whereas it is dark for the reflection case? Thanks a lot,
  28. S

    MHB Help find eqn of circle given another circle that is tangent

    Please help me find the standard equation of the circle passing through the point (−3,1) and containing the points of intersection of the circles x^2 + y^2 + 5x = 1 and x^2 + y^2 + y = 7 I don't know how to begin, I am used to tangent lines or other points, but I don't know what is visually...
  29. S

    MHB Help find eqn of circle given another circle that is tangent

    please help me find the standard equation of the circles that have radius 10 and are tangent to the circle X^2 + y^2 = 25 at the point (3,4). the soln: (x-9)^2 + (y-12)^2 = 100, (x+3)^2 + (y+4)^2 = 100, i found the eqn that intersects the centre of the small circle and the larger one to be...
  30. W

    How to estimate 2 points on the unit circle

    I have some noisy data (x-y coordinates) containing two distinct clusters of data. Each cluster is centred at an unknown point on the unit circle. How can I estimate these two points (green points in the diagram)? We can assume the noise is Gaussian and noise power is equal in x and y...
  31. U

    What is the equation of the circumcircle of triangle PAB?

    Homework Statement Tangents drawn from a point P(2,3) to the circle $$x^2+y^2-8x+6y+1=0$$ touch the circle at the points A and B. Find equation of circumcircle of the ΔPAB. The Attempt at a Solution The chord of contact is equal to -x+3y+1=0. This is also the radical axis of the given...
  32. S

    Point of intersection between a parabola and a circle

    Homework Statement Sketch the curve C defined parametrically by ##x=t^{2} -2, y=t## Write down the Cartesian equation of the circle with center as the origin and radius ##r##. Show that this circle meets the curve C at points whose parameter ##t## satisfies the equation ##t^{4} -3t^{2}...
  33. A

    Considering a circle to be an infinite sided n-gon

    as a regular polygon increases in sides, it becomes rounder. As you increase the number of sides, the polygon will tend towards a perfect circle but never quite make it. you can only make the circle with an infinite number of sides - stopping at any other number but infinity you will only get a...
  34. O

    Is every smooth simple closed curve a smooth embedding of the circle?

    Suppose I have a smooth curve \gamma:[0,1] \to M, where M is a smooth m-dimensional manifold such that \gamma(0) = \gamma(1), and \hat{\gamma}:=\gamma|_{[0,1)} is an injection. Suppose further that \gamma is an immersion; i.e., the pushforward \gamma_* is injective at every t\in [0,1]. Claim...
  35. K

    Double integral with a circle connecting the two

    I'm trying to figure out what this one symbol was I saw. I also have a guess that I would like to see if is correct. I saw a double integral with a circle connecting the two. What does this mean? Here is my guess. Is it used when dealing with Stoke's Theorem? Since ∫F°dS =∫∫ curl(F)°dS (Both...
  36. J

    Find the equation of the circle

    Homework Statement The line \{z: y=t+x\} is mapped to a circle by the function f(z)=\frac{z-1}{1-zi} Find the equation of this circle. The Attempt at a Solution One method is to find mappings of three points on the line. These points will be mapped to the circles boundary. Then find...
  37. C

    Circle is a set of a discontinuities?

    Why is the characteristic function* of a ball in Rn continuous everywhere except on its surface?My lecturer said that a circle is a 'set of discontinuities' - what exactly does that mean? (some context: we're looking at how we can integrate over a ball. Previously we've only looked at Riemann...
  38. B

    Curvature of a circle approaches zero as radius goes to infinity

    Hello, this isn't a homework problem, so I'm hoping it's okay to post here. I would like to know the correct way to mathematically express the idea in my title. It is intuitively obvious that as the radius of a circle increases, it's curvature decreases. I looked it up and found that...
  39. bsmithysmith

    MHB Finding Coordinates on a circle with time/speed

    Marla is running clockwise around a circular track. She runs at a constant speed of 3 meters per second. She takes 46 seconds to complete one lap of the track. From her starting point, it takes her 12 seconds to reach the northernmost point of the track. Impose a coordinate system with the...
  40. P

    Point of tangency to a circle from a point not on the circle

    Homework Statement Find the points of tangency to a circle given by x^2+y^2=9 from point (12,9). Homework Equations dy/dx=-x/y (what I've been able to come up so far) The Attempt at a Solution Taking the derivative I got dy/dx=-x/y Let the unknown point of tangency be (a,b)...
  41. S

    Integrate complex function over unit circle

    Homework Statement Calculate ##\int _Kz^2exp(\frac{2}{z})dz## where ##K## is unit circle.Homework Equations The Attempt at a Solution Hmmm, I am having some troubles here. Here is how I tried: In general ##\int _\gamma f(z)dz=2\pi i\sum_{k=1}^{n}I(\gamma,a_k)Res(f,a_k)## where in my case...
  42. P

    MHB Rose Petal - Circle - Area problem - Can someone check my work please?

    I'd love it if someone could verify whether or not I did this problem correctly. A stained-glass window is a disc of radius 2 (graph r=2) with a rose inside (graph of r=2sin(2theta) ). The rose is red glass, and the rest is blue glass. Find the total area of the blue glass. So I set...
  43. D

    Complex Circle Equation with random variable attached to Z.

    Homework Statement |zi - 3| = Pi Homework Equations Well, it clearly has to do with a circle but I do not believe there is a general equation for what I am asking about. The Attempt at a Solution There is no general solution not trying to solve anything. I want to know exactly...
  44. samjohnny

    Relating tensions in a vertical circle help

    Homework Statement Question: A mass m attached to a light string is spinning in a vertical circle, keeping its total energy constant. Find the difference in the magnitude of the tension between the top most and bottom most points. The Attempt at a Solution So for this one I've worked...
  45. S

    New way to derive sectors of a circle (easy)

    So for starters the area of an entire circle has 360º,right? So we can say that: ##1∏r^2## is ##\equiv## to ##360º## So by that logic ##0.5∏r^2## is ##\equiv## to ##180º## And finally ##0.25∏r^2## is ##\equiv## to ##90º## Divide both sides by 9, and you get : ##0.25∏r^2/9## is...
  46. A

    Single bead on a vertical circle.

    Imagine that a single bead has a hole in it. It is passed inside a " vertical " circle with no friction. Imagine that the "vertical"circle moves by spinning around its vertical axis and that the bead is, AT THE BEGINNING, on the bottom of the vertical circle. We had another problem related to...
  47. L

    MHB Finding the point on a circle closest to the given point

    Find the curve coordinates of the point nearest to P in the circle x2 + y2 = 16 P(0,6) as the former ( see a gift ) x2 + (y-6)2 = 16 (1) solving for y = y2 = 16- x2 introducing en 1 x2 +(16-x-6)2 = 16 x2 +100-20x + x2 = 16 derivating 4x -20 and x = 5 y = sqrt ( 16-25) and i...
  48. D

    Circle Inscribed in a Parabola?

    Homework Statement Find the largest circle centered on the positive y-axis which touches the origin and which is above y=x^2 Homework Equations equation of a circle: r^2=(x-a)^2+(y-b)^2 equation of a circle centered on the y-axis: x^2+(y-b)^2=r^2 equation of a parabola: y=x^2 The...
  49. E

    Is it possible to create a metal ring with a variable diameter using segments?

    I am interested to see if there is a way of creating a metal ring which has a variable diameter, the only way I imagine that this can be done is using segments which come together to form a circle, once pulled away the diameter changes however, the problem I face is that when the segments are...
  50. S

    Tangent to a Circle Homework Solution

    Homework Statement Given that the line ##y=mx+c## is a tangent to the circle ##(x-a)^{2} +(y-b)^{2} =r^{2}##, show that ##(1+m^{2}) r^{2}=(c-b+ma)^{2}##Homework Equations Quadratic discriminant, sum and product of rootsThe Attempt at a Solution I substituted y=mx+c into the equation of the...
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