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. S

    Equilibrium of bicycle going in a circle

    I have a few statements/questions, as stated below. Would appreciate your comments on them. 1) The system is not in translational equilibrium because of a net leftwards force. 2) The system should be in rotational equilibrium based on logic; But based on the FBD, if I take moments about the...
  2. D

    Linear Fractional Transformation (Mobius Transformation) from circle to line

    Homework Statement find linear fractional transformation that carries circle |z|=1 onto the line Re((1+i)w)=0 Homework Equations linear fractional transformation is of the form az+b/cz+b where ad-bc≠0 The Attempt at a Solution Re((1+i)w)=0 means that the line is just the y axis, but then I...
  3. N

    Image of Circle |z| = 3 under Mapping w = 6/z

    Homework Statement Find the image of the circle |z| = 3 in the complex plane under the mapping a) w = \frac{6}{z} b) w = \frac{6}{z} + 2i The Attempt at a Solution a) w = \frac{6}{3} = 2 So this is a circle in the w-plane of radius 2, centered on the origin? b) w =...
  4. C

    Calc-Physics 212: Charge Distribution Geometry of a Circle

    Homework Statement Hey guys. As you can see, there are 5 questions to answer regarding this question. I'm working through it and need some help regarding a few of the questions. I have A.) The electric potential at the center of the circle is Zero. B.) The value of the electric...
  5. D

    Moving a body into a circle of uniform motion

    I am trying to make a game where a drone approaches a ship, that may be moving, and then orbits it at a set distance. I know that if the drone is already moving in a circle of uniform motion then I use a = \frac{v^{r}}{r} I then take that magnitude and an angle derived from atan2 with cos...
  6. U

    Complex integral over a circle

    1. let C be the circle |z| = 2 traveled once in the positive sense. Computer the following integrals... a.∫c zez/(2z-3) dz Homework Equations I am confused as to a step in my solution, but i believe a relevant equation is if i am integrating over a circle and the function is analytic...
  7. J

    Solving a 2D Physics Problem with Unknown Circle Equation

    Hello everyone! I've found a physics problem that i don't know the solution of(maybe because of my limited knowledge). The problem is something like this: Let's say an object travels in a circular path from P to Q and Q to R in which P, Q and Rare not the center of the circle(because P, Q, R...
  8. J

    COnstant centripetal force to move in a circle?

    An interesting thought just struck me and I wanted to confirm if it is correct. Do all objects (of fixed mass) need a particular magnitude of force to keep them moving in a circle, e.g: a ball will ALWAYS need a force of 10N to keep it moving in a circle If the speed increases then the radius...
  9. O

    Gershgorin Circle Theorem, mathematical derivation of eigenvalue estimates

    I intend to use the Gershgorin Circle Theorem for estimating the eigenvalues of a real symmetric (n x n) matrix. Unfortunately, I'm a bit confused with the examples one might find on the internet; What would be the mathematical formula for deriving estimates on eigenvalues? I understand that...
  10. M

    Accelaration in a vertical circle

    A ball is attached to an inextensible string of negligable mass and hangs vertically under gravity. At time t=0, it is given a horizontal velocity u and begins to move in a vertical circle of radius r. At any time t>0, the ball is at an angle θ to the vertical and has a velocity v tangential to...
  11. W

    Find the length of this line associated with a circle

    I am struck in a place where i have to find length of a line(a in fig i.e between P1 and P2) in the form of r and Angle A. Refer to the figure: All information available are r,O,P2,and A.
  12. S

    Circle Inscribed in Triangle: Area Ratios with Inscribed Circle Tangents

    Homework Statement Consider a triangle ABC, where angle A = 60o. Let O be the inscribed circle of triangle ABC, as shown in the figure. Let D, E and F be the points at which circle O is tangent to the sides AB, BC and CA. And let G be the point of intersection of the line segment AE and the...
  13. Z

    Circle becomes a pair of parallel lines

    Does anyone know if there is a non-Euclidean geometry(or something like that) where the circle becomes a line or a pair of parallel lines? ...or even where the circle becomes a set of parallel lines? ...i'm doing some work in Guitar Theory and this situation appears... Thanks The...
  14. T

    Average Acceleration of a Fly on a Rotating Hand

    Homework Statement A fly sitting on the end of the second hand of a clock is traveling in a circle. The second hand has a length of 25 cm. Calculate the average acceleration of the fly. Homework Equations The Attempt at a Solution\ since only one measurment is given who would i calculate for...
  15. N

    Complex Analysis - Finding the equation of a circle

    Homework Statement If \frac{z}{z + 3} is purely imaginary, show that z lies on a certain circle and find the equation of that circle.The Attempt at a Solution So, \frac{z}{z + 3} = \frac{x + iy}{x + iy + 3} Multiplying by the complex conjugate (and simplifying), we get, \frac{x^{2} + y^{2}...
  16. Z

    Prove Analytically: Inversion of a Circle is Also a Circle

    Homework Statement Given the unit circle (in the Euclidean plane) centered at the origin x^2+y^2=1, and a general circle D with equation (x-a)^2+(y-b)^2=c^2 that does not pass through the origin (ie the center of inversion, ie a^2+b^2≠c^2, prove analytically that the inversion of D in the...
  17. G

    Bucket of water being swung in a circle.

    I am studying for a test coming up in a few weeks, so what I generally do is be familiar with every aspect of the question. Anyways, going back to the old swinging a pale of water in circular motion example. The force of tension and gravity would be the forces keeping the bucket in circular...
  18. N

    Finding the radius of a circle in a graph

    Homework Statement A circle of maximal area is inscribed in the region bounded by the graph of y = -x^2-7x+12 and the x axis. The radius of this circle is of the form (sqrt(p) + q)/r where, p, q and r are integers and are relatively prime.What is p+q+r Homework Equations Vertex form...
  19. B

    Block moving down circle, a straight path, then into a spring

    Homework Statement Consider the track shown in the figure. The section AB is one quadrant of a circle of radius 2.0 and is frictionless. B to C is a horizontal span 3.5 long with a coefficient of kinetic friction = 0.27. The section CD under the spring is frictionless. A block of mass 1.0...
  20. B

    Squaring the Circle: Why is Finding an Exact Solution so Difficult?

    The ancient problem of squaring the circle is to find a square with the exact area of a given circle. I don't see why this problem is so hard. Say a circle has radius 5. All you have to do is √(∏25) Then that's the length of the side of the square. Help me out here. Why is that not a...
  21. P

    Why does the Unit Circle work?

    In my Trig class, we learned about the unit circle and its relationship to the various trig functions (sin, cos, etc.). I am curious to know why the unit circle works the way it does, and the how it was "derived" so to speak. More specifically, why does radius of the circle have to be 1 for...
  22. D

    Osculating Circle for vector-valued function?

    Hi! This is my first time on Physics Forum. (which shows how desperate I am on figuring out this question). Homework Statement r(t) = <3sin(t),4cos(t)> There is a unique circle with the following properties: 1. It passes through the point r(∏/2) 2. At the point r(∏/2), the tangent...
  23. I

    What is the Derivation for a Point on a Circle?

    Hi All, I am working on the problem of bead on wire and got stuck on some basic derivation detail. I took the same approach as Andrew Witkin used in his slide (page 13): http://www.cs.cmu.edu/~baraff/sigcourse/slidesf.pdf Here is the screenshot of the slide page 13...
  24. X

    Work required to move charged particle traveling in a circle

    Homework Statement Hi everyone and thank you in advance for your time. I just had this problem on a physics exam (that everyone in the class bombed, and I mean everyone, including the best students). I honestly couldn't care less about the grade, but I really want to understand where I went...
  25. B

    Vector motion in a circle

    Homework Statement The Attempt at a Solution I understand how to get the derivatives for velocity and acceleration but when the book plugs in the numbers I disagree. velocity at pi 2 sin pi/2 = 0 2 cos pi/2 = -1j therefore, 0 + - 1j = -1j (the books says -2i) velocity at (3pi)/2 2 sin...
  26. P

    Why is integral of 1/z over unit circle not zero?

    Ok I can do the integral and see that it is equal to 2∏i, but thinking about it in terms of 'adding up' all the points along the curve I feel like every every point gets canceled out by its antipode, e.g. 1/i and -1/i.
  27. D

    Circumference of a circle in Poincare Half Plane

    i am trying to figure out how to calculate the circumference of a circle in the Poincare Half Plane. I know that vertical lines are geodesics so using the arclength formula, the distance between 2 points (x_0, y_0) and (x_1, y_1) on a vertical line is ln(y_1/y_0) . Thus, if i have a circle...
  28. I

    Impulse for a car when it drives in a circle

    Homework Statement A 1400kg car is driving at a constant velocity, 5.3 m/s. It turns 90 degrees in 4.6 seconds. And then it slams into a tree and it takes 350 ms to stop the car. What is the impulse on the car (a) due to the turn? (b)Due to the collision with the tree? What is the...
  29. I

    Impulse, momentum, and force during motion around a circle

    Hello, How would you calculate the impulse during the time which a car drives in a circle. You are given the car's mass and constant velocity and the time that the car was turning and how many degrees the car turned. Thanks in advance!
  30. S

    Magnetic Field at center of circle on rectangular circuit

    Homework Statement A circuit consists of 7 sections of wire. The figure looks like a rectangle of length 9cm and width 5cm with a circle of diameter 3 cm cutting right through the middle of one of the 9cm sides so that the two sides of the circle are in parallel. Each of the sections of wire...
  31. D

    MHB Circle radius 2 complex integration

    Gamma is a circle of radius 2 oriented counterclockwise. $$ \int_{\gamma}\frac{dz}{z^2+1} = \int_{\gamma} = \frac{i}{2}\left[\int_{\gamma}\frac{1}{z+i}dz-\int_{\gamma}\frac{1}{z-i}dz\right] $$ $\gamma(t) = 2e^{it}, \ \ \gamma'(t) = 2ie^{it}$ $$ \int_{\gamma}\frac{2ie^{it}}{2e^{it}+i}dz $$...
  32. B

    Electric Field of a quarter circle segment

    Homework Statement A quarter circle segment has a uniform linear charge density of λ. Starting with the E-field due to point charges, show that the magnitude of the E-field at the center of curvature(which is distance R away from all points on the quarter circle) is E= (kλ√(2))/R Homework...
  33. B

    Center of circle from two points and a tangent angle

    So my problem is this: I need to figure out the center of a circle given two points. At one of the points, I know the tangent angle. So I know (x1, y1, θ1) and (x2, y2) and need to find (xc, yc). I also need to do this on a computer so I need some sort of closed-form solution. The way I...
  34. S

    Find Radius of Circle for 2kg Mass and 4kg Mass

    Homework Statement A block of mass m1=2kg is attached to a cord. The cord goes down through a hole in the table and is attached to mass m2=4 kg hanging below the table. The 2 kg mass moves on the table in a circle at a speed of 3.5 m/s the table top is friction less and there is no friction...
  35. S

    Calculating the Radius of a Circle for Masses Attached by a Cord

    Homework Statement A block of mass m1=2kg is attached to a cord. The cord goes down through a hole in the table and is attached to mass m2=4 kg hanging below the table. The 2 kg mass moves on the table in a circle at a speed of 3.5 m/s the table top is friction less and there is no friction...
  36. G

    Work of kinetic friction on block in vertical circle

    Homework Statement 1. Homework Statement A block of mass 0.015kg enters the bottom of a circular, vertical track with a radius R = 0.3m at an initial velocity of 4/ms. If the block loses contact with the track at an angle of 130 degrees, what is Wk, the work done by kinetic friction...
  37. G

    Block in a verticle circle with friction

    Homework Statement A block of mass 15g enters the bottom of a circular, vertical track with a radius R = 0.5m at an initial velocity of 4/ms. If the block loses contact with the track at an angle of  = 130, what is Wk, the work done by kinetic friction...
  38. D

    MHB Circle radius 2 oriented counterclockwise

    gamma is a circle of radius 2, centered at the origin, and oriented counterclockwise $\displaystyle\int_{\gamma}\frac{dz}{z^2+1} =\int_{\gamma}\frac{dz}{(z+i)(z-i)}=\frac{1}{2}\int_{\gamma}\frac{\frac{1}{z-i}}{z-(-i)}dz+\int_{\gamma}\frac{\frac{1}{z+i}}{z-i}dz = 4\pi...
  39. J

    What does it mean to find the Area (e.g. area of a circle)?

    What does it mean to find the area? I've read somewhere and the person says, it means to find the space enclosed, but I still don't know what that means. I understand what area intuitively means, but not logically.
  40. A

    The emf of a string in a circle - magnetic fields

    Homework Statement there is a cylindrical area in xy plane with a magnetic field (into the plane) that changes with time like that: (dB/dt)=δ the magnetic field outside the area is 0. the radius of the cylinder is a. we pick a string in the circle with lentgh L which is smaller than the...
  41. Rapier

    Calculating Great Circle Distance

    Homework Statement Find the distance from Atlanta to San Francisco if Atlanta is @ 33.75°N and 84.40°W and San Francisco is @ 37.78°N and 122.42°W. Explain your strategy. Homework Equations cos phi = z/rho cos theta = x/r r = rho*sin phi The Attempt at a Solution I believe that...
  42. C

    Find a center of circle given a point and radius

    Hi all, How do I find the center of a circle given a point (tangent to the circle) and the radius. Thanks, CG
  43. B

    Area of a circle within a circumscribed triangle

    If you have a triangle circumscribed around a circle, how do you find the area of that circle? Say that the triangle is an equilateral triangle with side length of 8 cm. I found the area of the triangle using Heron's formula: 16√3 cm^2. Apparently the answer is 16π/3 cm^2. I'm just confused...
  44. A

    Ellipse inside a circle - intersection

    Hi everyone. I hope I've found the right place for my first post here. I have a geometry problem which I need to solve for a piece of software I'm writing, and I'm hoping someone might be able to help me. I have a non-rotated ellipse inside a circle, as in this diagram. I know the x and y...
  45. F

    How to find all points along a great circle given two points on circle

    Hi, I've researched this problem all across the web and most answers involve finding the distance between two points on a great circle, mostly in nautical terms, using latitude and longitude, etc. but that isn't the answer I'm looking for. It's generally agreed if you have two points on a...
  46. M

    Analyzing Bolt Tension & Shear: A Mohr's Circle Approach

    How would I analize a bolt in tension and shear. Imagine an "L" bracket bolted in the configuration below "||" represents a bolt. There is a force "<--" at the top of the bracket. <--| ...| ...|_____||_ The bolt will see tension and shear. There will be tension in the y direction (Sigma...
  47. R

    Circumference C of a circle of radius R inscribed on a sphere

    Homework Statement By employing spherical polar coordinates show that the circumference C of a circle of radius R inscribed on a sphere S^{2} obeys the inequality C<2\piR The Attempt at a Solution I proved C=2\piR\sqrt{1-\frac{R^2}{4r^2}} So if r>R, then the equality is correct. Am I right...
  48. H

    Why there are 360 degress in a circle

    Mentor comment: Harmony360 original post violated our rules. The new wording is mine. D H So, why there are 360 degress in a circle?
  49. U

    IF i was a 1 dimensional being living in a 1 dimensional Universe being a circle.

    How could i mathematically proove that I am living on a circle. Almost got it last night , just need an insight to figure out an equation. Ty.
  50. P

    What Determines the Detachment Point of a Ball Rolling on a Circle?

    Homework Statement A ball on top of a circle is acted on by a FG. The ball rolls to a certain point on the circle, and detaches. Ffr Is negligable. Given info: Diameter Of the circle, Mass of ball no ffr x = the distance from the balls posistion to the top of the big circle. x is what is...
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