The Toric Sections: a Simple Introduction
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Combination of Cubic and Quartic Plane Curve
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53 www.iosrjournals.org Combination of Cubic and Quartic Plane Curve C.Dayanithi Research Scholar, Cmj University, Megalaya Abstract The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid. A cross-section of the set of unistochastic matrices of order three forms a deltoid. The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid. The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six. The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856. The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides. I. Introduction Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. 1. Bicorn curve 2. Klein quartic 3. Bullet-nose curve 4. Lemniscate of Bernoulli 5. Cartesian oval 6. Lemniscate of Gerono 7. Cassini oval 8. Lüroth quartic 9. Deltoid curve 10. Spiric section 11. Hippopede 12. Toric section 13. Kampyle of Eudoxus 14. Trott curve II. Bicorn curve In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis. -
Apollonius Representation and Complex Geometry of Entangled Qubit States
APOLLONIUS REPRESENTATION AND COMPLEX GEOMETRY OF ENTANGLED QUBIT STATES A Thesis Submitted to the Graduate School of Engineering and Sciences of Izmir˙ Institute of Technology in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE in Mathematics by Tugçe˜ PARLAKGÖRÜR July 2018 Izmir˙ We approve the thesis of Tugçe˜ PARLAKGÖRÜR Examining Committee Members: Prof. Dr. Oktay PASHAEV Department of Mathematics, Izmir˙ Institute of Technology Prof. Dr. Zafer GEDIK˙ Faculty of Engineering and Natural Sciences, Sabancı University Asst. Prof. Dr. Fatih ERMAN Department of Mathematics, Izmir˙ Institute of Technology 02 July 2018 Prof. Dr. Oktay PASHAEV Supervisor, Department of Mathematics Izmir˙ Institute of Technology Prof. Dr. Engin BÜYÜKA¸SIK Prof. Dr. Aysun SOFUOGLU˜ Head of the Department of Dean of the Graduate School of Mathematics Engineering and Sciences ACKNOWLEDGMENTS It is a great pleasure to acknowledge my deepest thanks and gratitude to my su- pervisor Prof. Dr. Oktay PASHAEV for his unwavering support, help, recommendations and guidance throughout this master thesis. I started this journey many years ago and he was supported me with his immense knowledge as a being teacher, friend, colleague and father. Under his supervisory, I discovered scientific world which is limitless, even the small thing creates the big problem to solve and it causes a discovery. Every discov- ery contains difficulties, but thanks to him i always tried to find good side of science. In particular, i would like to thank him for constructing a different life for me. I sincerely thank to Prof. Dr. Zafer GEDIK˙ and Asst. Prof. Dr. Fatih ERMAN for being a member of my thesis defence committee, their comments and supports. -
Triple Conformal Geometric Algebra for Cubic Plane Curves
Received 13 February 2017; Revised 3 July 2018 (revised preprint with corrections) ; Accepted 22 August 2017 DOI: 10.1002/mma.4597 is revision 18 Sep 2017, published in Mathematical Methods in the Applied Sciences, 41(11)4088–4105, 30 July 2018, Special Issue: ENGAGE SPECIAL ISSUE PAPER Triple Conformal Geometric Algebra for Cubic Plane Curves Robert Benjamin Easter1 | Eckhard Hitzer2 1Bangkok, Thailand. Email: [email protected] Summary 2College of Liberal Arts, International The Triple Conformal Geometric Algebra (TCGA) for the Euclidean R2-plane ex- Christian University, 3-10-2 Osawa, 181-8585 Mitaka, Tokyo, Japan. Email: tends CGA as the product of three orthogonal CGAs, and thereby the representation [email protected] of geometric entities to general cubic plane curves and certain cyclidic (or roulette) Communicated by: Dietmar Hildenbrand MSC Primary: 15A66; quartic, quintic, and sextic plane curves. The plane curve entities are 3-vectors that MSC Secondary: 14H50; 53A30; linearize the representation of non-linear curves, and the entities are inner product Correspondence null spaces (IPNS) with respect to all points on the represented curves. Each IPNS Eckhard Hitzer, College of Liberal Arts, entity also has a dual geometric outer product null space (OPNS) form. Orthogonal or International Christian University, 3-10-2 conformal (angle-preserving) operations (as versors) are valid on all TCGA entities Osawa, 181-8585 Mitaka, Tokyo, Japan. Email: [email protected] for inversions in circles, reflections in lines, and, by compositions thereof, isotropic dilations from a given center point, translations, and rotations around arbitrary points in the plane. A further dimensional extension of TCGA, also provides a method for anisotropic dilations. -
Entry Curves
ENTRY CURVES [ENTRY CURVES] Authors: Oliver Knill, Andrew Chi, 2003 Literature: www.mathworld.com, www.2dcurves.com astroid An [astroid] is the curve t (cos3(t); a sin3(t)) with a > 0. An asteroid is a 4-cusped hypocycloid. It is sometimes also called a tetracuspid,7! cubocycloid, or paracycle. Archimedes spiral An [Archimedes spiral] is a curve described as the polar graph r(t) = at where a > 0 is a constant. In words: the distance r(t) to the origin grows linearly with the angle. bowditch curve The [bowditch curve] is a special Lissajous curve r(t) = (asin(nt + c); bsin(t)). brachistochone A [brachistochone] is a curve along which a particle will slide in the shortest time from one point to an other. It is a cycloid. Cassini ovals [Cassini ovals] are curves described by ((x + a) + y2)((x a)2 + y2) = k4, where k2 < a2 are constants. They are named after the Italian astronomer Goivanni Domenico− Cassini (1625-1712). Geometrically Cassini ovals are the set of points whose product to two fixed points P = ( a; 0); Q = (0; 0) in the plane is the constant k 2. For k2 = a2, the curve is called a Lemniscate. − cardioid The [cardioid] is a plane curve belonging to the class of epicycloids. The fact that it has the shape of a heart gave it the name. The cardioid is the locus of a fixed point P on a circle roling on a fixed circle. In polar coordinates, the curve given by r(φ) = a(1 + cos(φ)). catenary The [catenary] is the plane curve which is the graph y = c cosh(x=c). -
Geometry September 5, 2013
Outline of Geometry September 5, 2013 Contents MATH>Geometry.............................................................................................................................................................. 2 MATH>Geometry>Cartography ................................................................................................................................... 3 MATH>Geometry>Construction .................................................................................................................................. 3 MATH>Geometry>Coordinate System ........................................................................................................................ 3 MATH>Geometry>Fractal Geometry ........................................................................................................................... 4 MATH>Geometry>Fractal Geometry>Fractal Set ................................................................................................... 6 MATH>Geometry>Mapping ........................................................................................................................................ 6 MATH>Geometry>Plane .............................................................................................................................................. 7 MATH>Geometry>Plane>Angle ............................................................................................................................. 7 MATH>Geometry>Plane>Angle>Size .............................................................................................................. -
Polynomial Lemniscates and Their Fingerprints: from Geometry to Topology
POLYNOMIAL LEMNISCATES AND THEIR FINGERPRINTS: FROM GEOMETRY TO TOPOLOGY ANASTASIA FROLOVA, DMITRY KHAVINSON, AND ALEXANDER VASIL'EV Abstract. We study shapes given by polynomial lemniscates, and their fingerprints. We focus on the inflection points of fingerprints, their num- ber and geometric meaning. Furthermore, we study dynamics of zeros of lemniscate-generic polynomials and their `explosions' that occur by plant- ing additional zeros into a defining polynomial at a certain moment, and then studying the resulting deformation. We call this dynamics polynomial fireworks and show that it can be realized by a construction of a non-unitary operad. 1. Introduction A polynomial lemniscate is a plane algebraic curve of degree 2n, defined as Qn a level curve of the modulus of a polynomial p(z) = k=1(z − zk) of degree n with its roots zk called the nodes of the lemniscate. Lemniscates have been objects of interest since 1680 when they were first studied by French-Italian astronomer Giovanni Domenico Cassini see, e.g., [14], and later christened as "ovals of Cassini". 14 years later Jacob Bernoulli, unaware of Cassini's work, described a curve forming `a figure 8 on its side', which is defined as a solution of the equation (1) (x2 + y2)2 = 2c2(x2 − y2); or jz − cj2jz + cj2 = c4; where z = x+iy. (Curiously, his brother Johann independently discovered lem- niscate in 1694 in a different context.) Observe that the level curves jp(z)j = R can form a closed Jordan curve for sufficiently large R, and split into n dis- connected components when R ! 0+. -
Collected Atos
Mathematical Documentation of the objects realized in the visualization program 3D-XplorMath Select the Table Of Contents (TOC) of the desired kind of objects: Table Of Contents of Planar Curves Table Of Contents of Space Curves Surface Organisation Go To Platonics Table of Contents of Conformal Maps Table Of Contents of Fractals ODEs Table Of Contents of Lattice Models Table Of Contents of Soliton Traveling Waves Shepard Tones Homepage of 3D-XPlorMath (3DXM): http://3d-xplormath.org/ Tutorial movies for using 3DXM: http://3d-xplormath.org/Movies/index.html Version November 29, 2020 The Surfaces Are Organized According To their Construction Surfaces may appear under several headings: The Catenoid is an explicitly parametrized, minimal sur- face of revolution. Go To Page 1 Curvature Properties of Surfaces Surfaces of Revolution The Unduloid, a Surface of Constant Mean Curvature Sphere, with Stereographic and Archimedes' Projections TOC of Explicitly Parametrized and Implicit Surfaces Menu of Nonorientable Surfaces in previous collection Menu of Implicit Surfaces in previous collection TOC of Spherical Surfaces (K = 1) TOC of Pseudospherical Surfaces (K = −1) TOC of Minimal Surfaces (H = 0) Ward Solitons Anand-Ward Solitons Voxel Clouds of Electron Densities of Hydrogen Go To Page 1 Planar Curves Go To Page 1 (Click the Names) Circle Ellipse Parabola Hyperbola Conic Sections Kepler Orbits, explaining 1=r-Potential Nephroid of Freeth Sine Curve Pendulum ODE Function Lissajous Plane Curve Catenary Convex Curves from Support Function Tractrix -
Focus (Geometry) from Wikipedia, the Free Encyclopedia Contents
Focus (geometry) From Wikipedia, the free encyclopedia Contents 1 Circle 1 1.1 Terminology .............................................. 1 1.2 History ................................................. 2 1.3 Analytic results ............................................ 2 1.3.1 Length of circumference ................................... 2 1.3.2 Area enclosed ......................................... 2 1.3.3 Equations ........................................... 4 1.3.4 Tangent lines ......................................... 8 1.4 Properties ............................................... 9 1.4.1 Chord ............................................. 9 1.4.2 Sagitta ............................................. 10 1.4.3 Tangent ............................................ 10 1.4.4 Theorems ........................................... 11 1.4.5 Inscribed angles ........................................ 12 1.5 Circle of Apollonius .......................................... 12 1.5.1 Cross-ratios .......................................... 13 1.5.2 Generalised circles ...................................... 13 1.6 Circles inscribed in or circumscribed about other figures ....................... 14 1.7 Circle as limiting case of other figures ................................. 14 1.8 Squaring the circle ........................................... 14 1.9 See also ................................................ 14 1.10 References ............................................... 14 1.11 Further reading ............................................ 15 1.12 External -
A Case Study on 3Dimensional Shapes of Curves and Its Application
International Journal of Latest Engineering and Management Research (IJLEMR) ISSN: 2455-4847 www.ijlemr.com || Volume 03 - Issue 02 || February 2018 || PP. 62-67 A Case Study on 3dimensional Shapes of Curves and Its Application K. Ponnalagu1, Mythili.R2, Subaranjani.G3 1Assistant Professor, Department of Mathematics, Sri Krishna Arts and Science College, Coimbatore, Tamilnadu, India. 2III B.Sc Mathematics, Department of Mathematics, Sri Krishna Arts and Science College, Coimbatore, Tamilnadu, India. 3III B.Sc Mathematics, Department of Mathematics, Sri Krishna Arts and Science College, Coimbatore, Tamilnadu, India. Abstract: In this paper, we are going to study about the 3-dimensional curves and its application. The curves come under the topic Analytical Geometry. The purpose of this paper is to show how the 3-dimensional curves are applied in real life and also what are the benefits of using these types of curves. The curves discussed are helix, asteroid, cycloid and solenoid. The curves are investigated using its parametric equations. Some applications of these curves are DNA, Roller coaster, pendulum, electro-magnetic wires, magnetic field, etc. are illustrated with example problems. A detailed mathematical analysis of 3-d curves is discussed below. Keywords: Asymptote, Axis, Cartesian co-ordinates, Curvature, Vertex I. INTRODUCTION In Analytical Geometry, a curve is generally a line, in which its curvature has not been zero. A Curve is also called as a curved line, it is similar to a line, but it need not be a straight line. Some examples of these curves are parabola, ellipse, hyperbola, etc. The curves are classified into two types. They are (i) Open curve- Open curve is defined as a curve whose end does not meet. -
Arxiv:1704.00897V1 [Math-Ph]
PEDAL COORDINATES, DARK KEPLER AND OTHER FORCE PROBLEMS PETR BLASCHKE Abstract. We will make the case that pedal coordinates (instead of polar or Cartesian coordinates) are more natural settings in which to study force problems of classical mechanics in the plane. We will show that the trajectory of a test particle under the influence of central and Lorentz-like forces can be translated into pedal coordinates at once without the need of solving any differential equation. This will allow us to generalize Newton theorem of revolving orbits to include nonlocal transforms of curves. Finally, we apply developed methods to solve the “dark Kepler problem”, i.e. central force problem where in addition to the central body, gravitational influences of dark matter and dark energy are assumed. 1. Introduction Since the time of Isaac Newton it is known that conic sections offers full description of trajectories for the so-called Kepler problem – i.e. central force problem, where the force varies inversely as a square of 1 the distance: F r2 . There is also∝ another force problem for which the trajectories are fully described, Hook’s law, where the force varies in proportion with the distance: F r. (This law is usually used in the context of material science but can be also interpreted as a problem∝ of celestial mechanics since such a force would produce gravity in a spherically symmetric, homogeneous bulk of dark matter by Newton shell theorem.) Solutions of Hook’s law are also conic sections but with the distinction that the origin is now in the center (instead of the focus) of the conic section. -
24 May 2014 the Lemniscate of Bernoulli, Without Formulas
The Lemniscate of Bernoulli, without Formulas Arseniy V. Akopyan Abstract In this paper, we give purely geometrical proofs of the well-known properties of the lemniscate of Bernoulli. What is the lemniscate? A polynomial lemniscate with foci F1, F2, ..., Fn is a locus of points X such that the product of distances from X to the foci is constant ( Y FiX = const). The n-th root of this value is called the radius of the lemniscate. | | i=1,...,n It is clear, that lemniscate is an algebraic curve of degree (at most) 2n. You can see a family of lemniscates with three foci in Figure 1. Fig. 1. The lemniscate with three foci. A lemniscate with two foci is called a Cassini oval. It is named after the astronomer Giovanni Domenico Cassini who studied them in 1680. The most well-known Cassini oval is the lemniscate of Bernoulli which was described by Jakob Bernoulli in 1694. This arXiv:1003.3078v2 [math.HO] 24 May 2014 is a curve such that for each point of the curve, the product of distances to foci equals quarter of square of the distance between the foci (Fig. 2). Bernoulli considered it as a modification of an ellipse, which has the similar definition: the locus of points with the sum of distances to foci is constant. (Bernoulli was not familiar with the work of Cassini). It is clear that the lemniscate of Bernoulli passes through the midpoint between the foci. This point is called the juncture or double point of the lemniscate. X F1 O F2 Fig. -
Seashell Interpretation in Architectural Forms
Seashell Interpretation in Architectural Forms From the study of seashell geometry, the mathematical process on shell modeling is extremely useful. With a few parameters, families of coiled shells were created with endless variations. This unique process originally created to aid the study of possible shell forms in zoology. The similar process can be developed to utilize the benefit of creating variation of the possible forms in architectural design process. The mathematical model for generating architectural forms developed from the mathematical model of gastropods shell, however, exists in different environment. Normally human architectures are built on ground some are partly underground subjected different load conditions such as gravity load, wind load and snow load. In human architectures, logarithmic spiral is not necessary become significant to their shapes as it is in seashells. Since architecture not grows as appear in shells, any mathematical curve can be used to replace this, so call, growth spiral that occur in the shells during the process of growth. The generating curve in the shell mathematical model can be replaced with different mathematical closed curve. The rate of increase in the generating curve needs not only to be increase. It can be increase, decrease, combination between the two, and constant by ignore this parameter. The freedom, when dealing with architectural models, has opened up much architectural solution. In opposite, the translation along the axis in shell model is greatly limited in architecture, since it involves the problem of gravity load and support condition. Mathematical Curves To investigate the possible architectural forms base on the idea of shell mathematical models, the mathematical curves are reviewed.