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Topic: Derivation of the cartesian formula for an ellipse

	Ellipse - Biocrawler (Site not responding. Last check: 2007-10-21)
		An ellipse is a type of conic section: if a cone is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse.
		The shape of an ellipse is usually expressed by a number called the eccentricity of the ellipse, conventionally denoted e (not to be confused with the mathematical constant e).
		The semi-latus rectum of an ellipse, usually denoted l (lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis.
	www.biocrawler.com /encyclopedia/Ellipse (1037 words)

	Britain.tv Wikipedia - Cartesian coordinate system
		Cartesian coordinate systems are also used in space (where three coordinates are used) and in higher dimensions.
		Using the Cartesian coordinate system geometric shapes (such as curves) can be described by algebraic equations, namely equations satisfied by the coordinates of the points lying on the shape.
		An example of a point P on the system is indicated in Figure 3, using the coordinate (3,5).
	www.britain.tv /wikipedia.php?title=Cartesian_coordinate_system (1674 words)

	Ellipse
		In mathematics, an ellipse (from the Greek for absence) is the locus of points on a plane where the sum of the distances from any point on the curve to two fixed points is constant.
		An ellipse is a type of conic section: if a conical surface is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse.
		An ellipse centered at the origin of an x-y coordinate system with its major axis along the x-axis is defined by the equation of the elliptical object.
	www.danceage.com /biography/sdmc_Ellipse (1554 words)

	Final Gallery
		Specified by the lengths of its semi-major (a) and semi-minor (b) axes, an ellipse has two foci, each located on the major axis at distance (c) from the center, such that for any point on the perimeter of the ellipse the sum of the distances to the foci is equal to the constant (2a).
		Deriving from the Cartesian equation, (1/a2) * (equation for x)2 + (1/b2) * (equation for y)2 = 1, the parametric equations x2 = a2cos2(T) and y2 = b2sin2(T) are found to satisfy the first equation, since the a squared's and b squared's cancel and sin2(T) + cos2(T) = 1.
		Returning to the polar equation of the ellipse, however, the graph of the ellipse is repositioned with one focus at the origin and its equation is derived from the (focus to point)/(point to directrix) definition of eccentricity.
	aerospacescholars.jsc.nasa.gov /has/Students/finalGall.cfm?id=682 (1356 words)

	The Ultimate Conic section Dog Breeds Information Guide and Reference (Site not responding. Last check: 2007-10-21)
		The degenerate cases, where the plane passes through the apex of the cone, resulting in an intersection figure of a point, a straight line or a pair of lines, are often excluded from the list of conic sections.
		An alternative definition of conic sections starts with a point F (the focus), a line L (the directrix) not containing F and a positive number e (the eccentricity).
		This would clearly describe an ellipse were it not for the second term under the radical, the 2 m b x: it would be the equation of a circle which has been stretched proportionally along the directions of the x-axis and the y-axis.
	www.dogluvers.com /dog_breeds/Conics (1356 words)

	ELLIPSE FACTS AND INFORMATION (Site not responding. Last check: 2007-10-21)
		In mathematics, an ellipse (from the Greek for ''absence'') is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant.
		An ellipse centred at the origin can be viewed as the image of the unit circle under a linear map associated with a symmetric matrix
		Drawing an ellipse is a common graphics primitive in standard display libraries, such as the QuickDraw and GDI interfaces on the Macintosh and Windows systems.
	www.feefriend.com /ellipse (1321 words)

	Phil 221
		Note: Newton then generalizes this result for any regular N-sided polygon inscribed in a circle, arguing that the same formula for the centripetal acceleration must hold as the number of sides of the polygon increases and the path of the ball becomes ever closer to the circumference of the circle.
		An alternative approach would be to regard mass as the constant, m, that appears in (the modern version) of Newton’s second law, N2, F =ma.
		An inertial frame of reference is either at rest in absolute space or moving with constant velocity in a straight line in absolute space.
	web.ics.purdue.edu /~curd/QQ9.html (1317 words)

	Tangents To Circles And Ellipses - GPWiki
		We may prove the correctness of this transformation by showing that applying it to the standard equation for an ellipse yields a geometric entity for which all points are equidistant from the origin (i.e.
		Assume for simplicity that the paddle is a rectangle and the ball is a sphere.
		If we cleverly treat the paddle, however, as if it were an ellipse (even though it is not and is not so rendered), we can obtain a curved upper-side which contributes itself to the perturbation applied to the ball's reflected path.
	gpwiki.org /index.php/Tangents_To_Circles_And_Ellipses (1643 words)

	Cartesian_coordinates (Site not responding. Last check: 2007-10-21)
		Cartesian coordinate systems are also used inside space, where three coordinates are used, & inside higher dimensions.
		The modern Cartesian coordinate system inside two dimensions, also called a rectangular coordinate system, is commonly defined by two axes - at right angles to each other - forming a plane, an xy-plane,.
		An example 0f a point P on teh system is indicated inside Figure 3 - using teh coordinate, 3,5,.
	www.demandtwinother.info /Cartesian_coordinates (2393 words)

	The Ellipse ........................................
		The more stretched out an ellipse is, the larger the value of the eccentricity...
		The eccentricity of an ellipse is always a value between 0 and 1.
		On a Cartesian plane, an ellipse has an equation based on the values of a and b.
	www.worsleyschool.net /science/files/ellipse/facts.html (553 words)

	Final Answers - Science - NUMERICANA
		The derived subgroup of a group is generated by its commutators.
		Euler's criterion: Modulo an odd prime p, a square to the power of (p-1)/2 is 1.
		Heron's Formula (for the area of a triangle) is related to the Law of Cosines.
	home.att.net /~numericana/answer (5892 words)

	Hyperbolic Trigonometric Functions
		Hence, an object on a constant-speed circular orbit, centered on one focal-point will be seen from the other focal-point as moving at the correct speed, as it passes each vertex of the ellipse.
		Acceleration is the second derivative of the radius vector from the fixed central point body of mass M. By Newton's law of gravitation, the force is proportional to the inverse-square of the distance r between the two objects.
		Carl Friedrich Gauss (1727-1855) derived the Gauss theorem and from it the flux-divergence theorem.
	www.rism.com /Trig/hyperbol.htm (9850 words)

	Unit of Study #154 - DDN Curriculum
		An explanation of each step by the teacher or the student is important at this stage.
		During the discussion it is crucial that the teacher explain the derivation of the circle formula from the distance formula.
		The fourth part of the lesson involves the review of lines, circles, ellipses and hyperbolas using the outline format and having the students again fill in the information as the outline is completed on the board by the teacher.
	ddncurriculum.k12.sd.us /view.asp?ID=154 (2726 words)

	Kepler's Third Law Text - Physics Forums Library
		Kepler spent most of his life trying to derive his relationship between the period of orbit and the radius of orbit along with the rest of his laws.
		But ofcrouse the ellipse equations that describe the ellipse in Cartesian Coordinates will have to be modified for its fit to be derived into Keplers 3rd law.
		It should be possible to derive Kepler's third law from his first two, since those two plus centripetal force are sufficient to prove the inverse square central force law, and the third law follows from inverse square and centripetal force.
	www.physicsforums.com /archive/index.php/t-8977.html (1243 words)

	Mathematical Chronology
		Arbuthnot publishes an important statistics paper in the Royal Society which discusses the slight excess of male births over female births.
		There is an investigation of a curve that becomes known as "the witch of Agnesi".
		It summarises all the work done in the field of mechanics since the time of Newton and is notable for its use of the theory of differential equations.
	www-groups.dcs.st-and.ac.uk /~history/Chronology/full.html (6672 words)

	ellipse (Site not responding. Last check: 2007-10-21)
		Algebraically, an ellipse is a curve in the Cartesian plane defined byan equation of the form
		The eccentricity is a positive number less than 1, or 0 in the case of a circle.The greater the eccentricity is, the larger the ratio of a to b,and therefore the more elongated is the ellipse.
		The ellipse shown in the image below has an eccentricity of approximately 0.8733.The distance between the foci is 2ae.
	www.holdemseriespoker.com /wiki/?title=Ellipse (1520 words)

	Circle (Site not responding. Last check: 2007-10-21)
		A circle is the degenerate case of an
		Ellipse with equal semimajor and semiminor axes (i.e., with
		In Cartesian Coordinates, the equation of a circle of
	www.math.sdu.edu.cn /mathency/math/c/c305.htm (379 words)

	Pipe Flow in Arbitrary Geometries
		With the aim of finding the average velocity of an incompressible, homogeneous fluid through a pipe of arbitrary cross-section, from which the maximum fluid velocity and average flow rate can be found, the conservation of momentum equation is chosen as a starting point.
		The Laplace Equation is an elliptic equation, and the particular case of interest is a defined as being equal to zero on the boundary and thus is also a boundary value problem.
		Whether or not the point was inside the cross-section was governed by an if statement with a series of geometrical criteria.
	www.maths.tcd.ie /~healyw/projectpsi/projectpsi.htm (2948 words)

	ekpolarmyst.html
		Carl claimed that the equation’s graph could actually be an ellipse--you can never be sure you’re seeing the whole graph until you get a whole ellipse.
		is an irrational number, no step size (which must be rational) will make the calculator use this value.
		By using the trigonometric addition formulas, one can show that the formulas give equivalent coordinates.
	www2.edc.org /mathproblems/getp.asp?name=ekPolarMyst (742 words)

	Math 1910 - Calculus I
		Topics covered will include inverse functions, techniques and applications of integration, an introduction into the modeling and techniques for solving simple first order differential equations, the study of parametric equations as well as the polar coordinate system and its use, conic sections, sequences and series to include conditions and tests for convergence.
		An understanding of the Taylor series, a generalization of the Maclaurin series, to include the ability to present a function as either a Taylor or Maclaurin series,
		To be able to recite on demand the general behavior of hyperbolic functions, their inverses, their derivatives (both for hyperbolic as well as their inverses), to write from memory the hyperbolic inverse functions in terms of logarithmic functions, their domain of definition and range,
	www.tn.regentsdegrees.org /courses/syllabi/math1920.htm (3432 words)

	The Hindu : Ecstasy of mathematics
		IN EARLIER articles we emphasised that the ecstasy of mathematics can be experienced not only by an Euler or Ramanujan but also by an aspirant to a career in creative research.
		Our country abounds in such youthful talent which has to generate its own energy and momentum to reach the destination — the solution of an unsolved problem, or at least the derivation of an unnoticed result.
		The properties of iL* and L* are connected to the striking features of the velocity transformation formula in which either all three velocities are less than unity or two of them greater than and one less than unity.
	www.hindu.com /thehindu/seta/2003/06/05/stories/2003060500040200.htm (1311 words)

	cartesian_coordinates (Site not responding. Last check: 2007-10-21)
		See coordinates for other commonly used coordinate systems such as polar coordinates and coordinate systems for usage of the term in advanced mathematics.
		(Early systems allowed "oblique" axes, that is, axes that did not meet at right angles.) All the points in a Cartesian coordinate system taken together form a so-called Cartesian plane.
		The choice of letters comes from a convention, to use the latter part of the alphabet to indicate unknown values.
	www.consumerresource.com /wiki/?title=Cartesian_coordinates (1457 words)

	[No title]
		Rutherford scattering: repulsive Coulomb potential, orbit equation, impact parameter and scattering angle and their relation, differential cross section, derivation of Rutherford scattering formula, remark on deviations from the Rutherford behavior; CHPT.
		Equation of motion for center of mass motion in the presence of external and internal forces, vanishing of the internal part for central forces, total angular momentum and its decomposition in internal (spin) and external (orbital) contributions, total kinetic energy as a sum of relative and cm motion;
		Hamilton's variational principle: minimum principle of the action to define the dynamics, example of free fall: definition of variations in position and velocity, Lagrange function, derivation of Newton's 2.
	tonic.physics.sunysb.edu /~rapp/lecture.html (1130 words)

	Super Cartesian (Site not responding. Last check: 2007-10-21)
		Though he is often credited with Shop Golf Shoes its invention, the Danish poet and scientist Piet Hein (1905-1996) did not discover the super-ellipse.
		The general Cartesian notation of the form comes from the FrenchFig.
		Four points are marked: (2,3) in Farm Toy Hyperbolic spiral; Kappa curve; Derivation of the cartesian formula for an ellipse; Spring (math) Super ellipse; Cylinder (geometry)Super Cartesian Diver Kit.
	cartesian.adult-beaver.com /super-cartesian.htm (233 words)

	PHYSICS BIBLIOGRAPHY (Site not responding. Last check: 2007-10-21)
		Pierre Duhem, To Save the Phenomena, an Essay on the Idea of Physical Theory from Plato to Galileo, translated from the French edition of 1908 by E. Doland and C. Maschler.
		Donn MacMinn, "An analysis of Ptolemy's treatment of retrograde motion," Journal for the History of Astronomy, Vol.
		Noel Swerdlow, "The Derivation and first Draft of Copernicus's Planetary Theory: A Translation of the Commentariolus with Commentary," Proceedings of the American Philosophical Society, Vol.
	www.ipst.umd.edu /Faculty/brush/physicsbibliography.htm (3091 words)

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