Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere.... , an ellipse (Greek
Greek language
Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European.... ???e???? (elleipsis), a 'falling short') is the apparent shape of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center.... viewed obliquely from outside it, as distinct from a hyperbola
Hyperbola
In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other.... which is the shape seen from inside. It is the finite or bounded
Bounded set
In mathematical analysis and related areas of mathematics, a Set is called bounded, if it is, in a certain sense, of finite size. Conversely a set which is not bounded is called unbounded.... case of a conic section
Conic section
File:Conic sections with plane.svgIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane . A conic section is therefore a restriction of a quadric surface to the plane .... as a shape cut in a cone
Cone
A cone is a basic geometrical shape; see cone .Cone may also refer to:*Conifer cone, a seed-bearing organ on conifer plants*Cone cell, in anatomy, a type of light-sensitive cell found along with rods in the retina of the eye... by a plane
Plane (mathematics)
In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.... , the unbounded cases being the parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.... , which like the ellipse remains connected
Connected component
Connected components are part of topology and graph theory, two related branches of mathematics.* For the graph-theoretic concept, see connected component .... , and the hyperbola, which separates into two connected component
Connected component
Connected components are part of topology and graph theory, two related branches of mathematics.* For the graph-theoretic concept, see connected component .... s or branches.
Equivalently an ellipse can be defined as the locus of points
Locus (mathematics)
In mathematics, a locus is a collection of point which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve.... , or path traced out, in a plane such that the sum of the distances from the moving point to two fixed points remains constant.
Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere.... , an ellipse (Greek
Greek language
Greek is an Indo-European languages native to the southern Balkan peninsula, the language of the Greek people. It forms an independent branch within Indo-European.... ???e???? (elleipsis), a 'falling short') is the apparent shape of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center.... viewed obliquely from outside it, as distinct from a hyperbola
Hyperbola
In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other.... which is the shape seen from inside. It is the finite or bounded
Bounded set
In mathematical analysis and related areas of mathematics, a Set is called bounded, if it is, in a certain sense, of finite size. Conversely a set which is not bounded is called unbounded.... case of a conic section
Conic section
File:Conic sections with plane.svgIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane . A conic section is therefore a restriction of a quadric surface to the plane .... as a shape cut in a cone
Cone
A cone is a basic geometrical shape; see cone .Cone may also refer to:*Conifer cone, a seed-bearing organ on conifer plants*Cone cell, in anatomy, a type of light-sensitive cell found along with rods in the retina of the eye... by a plane
Plane (mathematics)
In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.... , the unbounded cases being the parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.... , which like the ellipse remains connected
Connected component
Connected components are part of topology and graph theory, two related branches of mathematics.* For the graph-theoretic concept, see connected component .... , and the hyperbola, which separates into two connected component
Connected component
Connected components are part of topology and graph theory, two related branches of mathematics.* For the graph-theoretic concept, see connected component .... s or branches.
Equivalently an ellipse can be defined as the locus of points
Locus (mathematics)
In mathematics, a locus is a collection of point which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve.... , or path traced out, in a plane such that the sum of the distances from the moving point to two fixed points remains constant. The two fixed points are then called foci
Focus (geometry)
In geometry, the foci, , are a pair of special points used in describing conic sections. The four types of conic sections are the circle, parabola, ellipse, and hyperbola.... . When the foci coincide the ellipse becomes a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center.... and the two distances then coincide as its radius
RADIUS
Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service.... . A variant of this replaces one of the foci with a straight line not passing through the remaining focus, called the directrix; in this case the locus is of a point whose distance from the remaining focus maintains a constant ratio less than one with its distance from the directrix.
Yet another definition of an ellipse, the algebraic
Algebraic
Algebraic may refer to:* Algebraic chess notation — a method used to record and describe the play of chess games.* Algebraic data types.... or implicit definition, is, up to rotation
Rotation
A rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a centerof rotation. A Three-dimensional space object rotates around a line called an axis.... and translation (geometry)
Translation (geometry)
In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the Euclidean groups . A translation can also be interpreted as the addition of a constant vector space to every point, or as shifting the Origin of the coordinate system.... , any set of points in the Cartesian plane satisfying an equation of the form
where a and b are any positive real numbers.
From this last definition it is seen that an ellipse is obtained from the unit circle
of radius 1 and hence width and height 2, by scaling the x and y coordinates with the respective factors a and b; thus determines an ellipse centered on the origin of width 14 and height 6.
The informal "apparent shape" definition is formalized by defining an ellipse to be the result of projecting
Projective transformation
A projective transformation is a Transformation used in projective geometry: it is the composition of a pair of perspective projections. It describes what happens to the perceived positions of observed objects when the point of view of the observer changes.... a circle onto any plane in three dimensions that does not intersect the circle. (Cutting the circle at two points yields a hyperbola, one point a parabola.) More generally an ellipse results when any conic section, whether ellipse, parabola, or hyperbola, is projected onto a plane, provided its connected components lie properly within distinct halves of the space cut by the plane (i.e. not even touching the plane, and with the two branches of a hyperbola placed on opposite sides), and that no axis of symmetry of the conic section be parallel to the plane (to force the diverging arms of parabolas to close up).
In defining an ellipse, the vertical diameter (or "minor axis") passing through its center is known as the conjugate diameter or axis, and the horizontal diameter (or major axis)——perpendicular, or "transverse", to the conjugate——is the transverse diameter or axis passing through its center.
The distance between antipodal points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximal where the major axis cuts the ellipse and minimal where the minor axis cuts it. The semimajor axis (denoted by a in the figure) is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis
Semi-minor axis
In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis.... (denoted by b in the figure) is one half the minor axis: the line segment from the center to a nearest point of the ellipse. In the example of the preceding paragraph a = 7 and b = 3.
Common to all of these definitions is that they specify an ellipse in a plane with five real numbers, the degrees of freedom of any general conic section in a plane. In the case of the algebraic definition the translation and rotation needed to put the ellipse in general position accounts for three of the numbers. Circles as a special case of the ellipse have only three degrees of freedom, while parabolas as the limiting case of a projection of a circle where the circle touches the plane of projection have four.
In mathematics, the eccentricity, denoted e or , is a parameter associated with every Conic section#Eccentricity. It can be thought of as a measure of how much the conic section deviates from being circular.... of an ellipse, e or just e, is the ratio of the distance between the foci to the length of the major axis; this is necessarily between 0 and 1.
It can also be expressed as the sine of the angular eccentricity
Angular eccentricity
In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and Eccentricity .... , :
If the ellipse has the Cartesian equation
its eccentricity is
The eccentricity is zero if and only if in which case the ellipse is a circle.
The coordinates of the foci are and
The distance from a focal point to the centre is called the linear eccentricity of the ellipse. The distance between the foci is .
Eliminating b from the above equations gives the alternative equation for the ellipse
The distance from a point (x,y) on the ellipse to the left focal point is
The distance from the same point to the right focal point is in the same way
Adding these equations one gets
This property of the ellipse, that the sum of the distances to two given points is taking a given value, is often used as the definition of an ellipse. The method to draw an ellipse described below is an application of this.
True anomaly
The polar angle of a point on an ellipse relative the focal point is called the true anomaly of the point. Relative to the "canonical coordinate system" with origin, , at the mid-point between the focii and in which the equation of the ellipse is
one has that
As
one has that
This is the standard representation of an ellipse in polar coordinates.
Eccentric anomaly
For the point (x,y) on an ellipse with the equation
the eccentric anomaly is defined through
The direct relation between the eccentric anomaly and the true anomaly is:
With this relation the eccentric anomaly can be computed from the true anomaly. To compute the true anomaly from the eccentric anomaly a more convenient relation can be derived using using the trigonometric identity
One gets that
and as
it follows that
As and always have the same sign it follows that and are in the same quadrant
Cartesian coordinate system
In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point.... .
One therefore has that
The relation written in this form has singularities for and .
But it can also be written in the non-singular form
where is the polar argument of the vector .
For the numerical computation of the standard function atan2(y, x)
Atan2
In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real number arguments x and y not both equal to zero, atan2 is the angle in radians between the positive x-axis of a plane and the point given by the Cartesian coordinate system on it.... available in many programming languages can be used.
Generally, in mathematics, a canonical form of an object is a standard way of presenting that object.Canonical form can also mean a differential form that is defined in a natural way; #Differential forms....
with x,y defined relative another rectangular coordinate system obtained from the original one by translation and rotation. It can also be proven that if relative to one coordinate system it is true in all coordinate systems. As in the canonical coordinate system one has that it follows that . This means that A' and C' have the same sign and the equation
which in the canonical coordinate system takes the form
has a solution if and only if F' has the opposite sign to A' and C' In this case the equation can be written
In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length .... using two pins, a loop of string, and a pencil. The resulting ellipse will have each of the four sides of the rectangle as a tangent parallel to one of its axes, with each long side being parallel to the major axis and each short to the minor.
The pins are placed at the foci and the pins and pencil are enclosed inside the loop. The pencil is placed on the paper inside the loop and the string made taut. The string will form a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments.... . If the pencil is moved around with the string kept taut, the sum of the distances from the pencil to the pins will remain constant, thus satisfying the definition of an ellipse.
The foci are placed as follows. First draw a circle centered on a corner of the rectangle, having radius the short side, namely 2b. The two long sides, of length 2a, are then respectively tangent to and centered on this circle. From the far end of the latter long side, draw a tangent to the circle. By the Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles.... , the length of this tangent is , which is the distance between the foci of an ellipse of these dimensions. Place the foci symmetrically on the line parallel to and midway between the two long sides, at the separation just determined.
The length of the loop is determined simply by hooking it over one focus and adjusting it so as to just reach the middle of the more remote short side.
The length of the portion of the loop from the center of the rectangle to the hooking focus and back to the center equals the separation of the foci, by symmetry of their placement. The other portion of the loop, from the center to the side and back, is of length 2a, which is therefore also the distance from one focus, to the pencil, thence to the other focus.
Equations
An ellipse with a semimajor axis a and semiminor axis b, centered at the point and having its major axis parallel to the x-axis may be specified by the equation
In mathematics, parametric equations are a method of defining a curve. A simple kinematics example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion.... as
where may be restricted to the interval .
Parametric form of an ellipse rotated counterclockwise by an angle :
The formula for the directrices is
If = 0 and = 0 (i.e., if the center is the origin (0,0)), then we can express this ellipse in polar coordinates by the
equation
With one focus at the origin, the ellipse's polar equation is
In differential geometry, the Gauss map maps a surface in Euclidean space R3 to the unit sphere S2. Namely, given a surface X lying in R3, the Gauss map is a continuous map N: X ? S2 such that N is a unit vector orthogonal to X at p, namely the normal ve... form:
has normal .
Semi-latus rectum and polar coordinates
The semi-latus rectum of an ellipse, usually denoted (lowercase L
L
L or l, described in English language as L with stroke, is a letter of the Polish alphabet, Kashubian alphabet, Sorbian alphabet, Lacinka alphabet , Wymysorys, Navajo language, Dene Suline language, Inupiaq language and Dogrib language alphabets, and of several proposed alphabets for the Venetian language.... ), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular
Perpendicular
In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective.... to the major axis. It is related to and (the ellipse's semi-axes) by the formula or, if using the eccentricity,
In mathematics, the polar coordinate system is a dimension coordinate system in which each point on a plane is determined by an angle and a distance.... , an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation
An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle f to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin f, provided f is not 90°.
Area and circumference
The area enclosed by an ellipse is pab, where (as before) a and b are one-half of the ellipse's major and minor axes respectively.
The circumference is the distance around a closed curve. Circumference is a kind of perimeter.... of an ellipse is ,
where the function is the complete elliptic integral
Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler.... of the second kind
Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler.... .
An approximation is an Accuracy and precision representation of something that is still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as Function , shapes, and physical laws.... is Ramanujan
Srinivasa Ramanujan
Srinivasa Ramanujan Ivengar Fellow of the Royal Society, better known as Srinivasa Ramanujan was an Indian mathematician, who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions.... 's:
An approximation is an Accuracy and precision representation of something that is still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as Function , shapes, and physical laws.... :
For the special case where the minor axis is half the major axis, we can use:
Determining the length of an irregular arc segment ? also called rectification of a curve ? was historically difficult. Although many methods were used for specific curves, the advent of calculus led to a general formula that provides closed-form expression in some cases.... of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral
Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler.... . The inverse function
Inverse function
In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself.... , the angle subtended as a function of the arc length, is given by the elliptic functions.
Stretching and projection
An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a line). Similarly, any oblique projection
Oblique projection
Oblique projection is a simple type of graphical projection used for producing pictorial, two-dimensional of three-dimensional objects.... onto a plane results in a conic section. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.
A mirror is an object with one surface polished, which leads to reflection and another opaque. The most familiar type of mirror is the plane mirror, which has a flat surface.... with a light source at one of the foci. Then all rays are reflected
Reflection (physics)
Reflection is the change in direction of a wavefront at an wiktionary:interface between two differentmedium so that the wavefront returns into the medium from which it originated.... to a single point — the second focus. Since no other curve has such a property, it can be used as an alternative definition of an ellipse. In a circle, all light would be reflected back to the center since all tangents are orthogonal to the radius.
Sound waves are reflected in a similar way, so in a large elliptical room a person standing at one focus can hear a person standing at the other focus remarkably well. Such a room is called a whisper chamber. Examples are the National Statuary Hall
National Statuary Hall
National Statuary Hall is a chamber in the United States Capitol devoted to sculptures of prominent United States. The hall, also known as the Old Hall of the House, is a large, two-story, semicircular room with a second story gallery along the curved perimeter.... at the U.S. Capitol (where John Quincy Adams
John Quincy Adams
John Quincy Adams was an Foreign relations of the United States and Politics of the United States who served as the List of Presidents of the United States President of the United States from March 4, 1825 to March 4, 1829.... is said to have used this property for eavesdropping on political matters), at an exhibit on sound at the Museum of Science and Industry
Museum of Science and Industry (Chicago)
The Museum of Science and Industry is located in Chicago, Illinois in Jackson Park , in the Hyde Park, Chicago neighborhood adjacent to Lake Michigan.... in Chicago
Chicago
Chicago is the largest city in the U.S. state of Illinois and the Midwestern United States, as well as the List of United States cities by population city in the United States with more than 2.8 million residents.... , in front of the University of Illinois at Urbana-Champaign
University of Illinois at Urbana-Champaign
The University of Illinois at Urbana-Champaign is a public university research university in the state of Illinois, United States. It is the oldest and largest campus in the University of Illinois system.... Foellinger Auditorium, and also at a side chamber of the Palace of Charles V, in the Alhambra
Alhambra
The Alhambra is a palace and fortress complex of the Moors rulers of Emirate of Granada in southern Spain , occupying a hilly terrace on the southeastern border of the city of Granada.... .
Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy.... explained that the orbit
ORBit
ORBit is a Common Object Request Broker Architecture 2.4 compliant Object Request Broker . It features mature C , C++ and Python bindings, and less developed bindings for Perl, Lisp , Pascal , Ruby , and Tcl.... s along which the planets travel around the Sun are ellipses in his first law of planetary motion
Kepler's laws of planetary motion
In astronomy, Kepler's three laws of planetary motion are*"The orbit of every planet is an ellipse with the sun at a Focus ."*"A line joining a planet and the sun sweeps out equal areas during equal intervals of time."... . Later, Isaac Newton
Isaac Newton
Sir Isaac Newton, Fellow of the Royal Society was an English people physicist, mathematician, Astronomy, Natural philosophy, Alchemy, and Theology and one of the the 100 in human history.... explained this as a corollary of his law of universal gravitation
Newton's law of universal gravitation
Isaac Newton's law of universal gravitation is an empirical physical law describing the gravitational attraction between bodies with mass. It is a part of classical mechanics and was first formulated in Newton's work Philosophiae Naturalis Principia Mathematica, first published on July 5 1687.... .
In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus.... , if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus.
In classical mechanics, a harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hooke's law:... in two or more dimension
Dimension
In mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists in... s is also an ellipse, but this time with the origin of the force located at the center of the ellipse.
In optics, an index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive index in a crystal.... describes the refractive index
Refractive index
The refractive index of a medium is a measure for how much the speed of light is reduced inside the medium. For example, typical soda-lime glass has a refractive index of 1.5, which means that in glass, light travels at times the speed of light in a vacuum.... of a material as a function of the direction through that material. This only applies to materials that are optically anisotropic. Also see birefringence
Birefringence
Birefringence, or double refraction, is the decomposition of a Ray of light into two rays when it passes through certain types of material, such as calcite crystals or boron nitride, depending on the polarization of the light.... .
Ellipses in computer graphics
Drawing an ellipse as a graphics primitive is common in standard display libraries, such as the Macintosh QuickDraw
QuickDraw
QuickDraw is the 2D Computer graphics library and associated Application programming interface which is a core part of the classic Apple Macintosh Mac OS.... API, the Windows Graphics Device Interface
Graphics Device Interface
The Graphics Device Interface is a Microsoft Windows application programming interface and core operating system component that is responsible for representing graphical objects and transmitting them to output devices such as computer display and computer printer.... (GDI) and the Windows Presentation Foundation
Windows Presentation Foundation
The Windows Presentation Foundation , formerly code-named Avalon, is a graphical subsystem in .NET Framework 3.0 , which uses a markup language, known as Extensible Application Markup Language, for rich user interface development.... (WPF). Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. An efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken (IEEE CG&A, Sept. 1984).
The following is example JavaScript code using the parametric formula for an ellipse to calculate a set of points. The ellipse can be then approximated by connecting the points with lines.
/*
This functions returns an array containing 36 points to draw an
ellipse.* @param x X coordinate
@param y Y coordinate
@param a Semimajor axis
@param b Semiminor axis
@param angle Angle of the ellipse
/
function calculateEllipse(x, y, a, b, angle, steps)
One beneficial consequence of using the parametric formula is that the density of points is greatest where there is the most curvature. Thus, the change in slope between each successive point is small, reducing the apparent "jaggedness" of the approximation.
See also
External links
at
- An interactive sketch showing how to trace the curves of the ellipse and hyperbola. (Requires Java.)
- Another interactive sketch, this time showing a different method of tracing the ellipse. (Requires Java.)
A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle....
