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List Of Mathematical Shapes
Following is a list of some mathematically well-defined shapes. Algebraic curves *Cubic plane curve *Quartic plane curve Rational curves Degree 2 *Conic sections *Unit circle *Unit hyperbola Degree 3 *Folium of Descartes *Cissoid of Diocles *Conchoid of de Sluze *Right strophoid *Semicubical parabola *Serpentine curve *Trident curve *Trisectrix of Maclaurin *Tschirnhausen cubic *Witch of Agnesi Degree 4 *Ampersand curve *Bean curve * Bicorn *Bow curve *Bullet-nose curve *Cruciform curve *Deltoid curve * Devil's curve *Hippopede *Kampyle of Eudoxus *Kappa curve *Lemniscate of Booth *Lemniscate of Gerono *Lemniscate of Bernoulli *Limaçon **Cardioid **Limaçon trisectrix * Trifolium curve Degree 5 * Quintic of l'Hospital Degree 6 *Astroid *Atriphtaloid *Nephroid *Quadrifolium Families of variable degree *Epicycloid * Epispiral *Epitrochoid *Hypocycloid *Lissajous curve * Poinsot's spirals *Rational normal curve *Rose curve Curves of genus one * Bicuspid curve *Cassini o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tschirnhausen Cubic
In algebraic geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equation :r = a\sec^3 \left(\frac\right) where is the secant function. History The curve was studied by Ehrenfried Walther von Tschirnhaus, von Tschirnhaus, Guillaume de l'Hôpital, de L'Hôpital, and Eugène Charles Catalan, Catalan. It was given the name Tschirnhausen cubic in a 1900 paper by R C Archibald, though it is sometimes known as de L'Hôpital's cubic or the trisectrix of Catalan. Other equations Put t=\tan(\theta/3). Then applying De Moivre's formula, triple-angle formulas gives :x=a\cos \theta \sec^3 \frac = a \left(\cos^3 \frac - 3 \cos \frac \sin^2 \frac \right) \sec^3 \frac= a\left(1 - 3 \tan^2 \frac\right) ::= a(1 - 3t^2) :y=a\sin \theta \sec^3 \frac = a \left(3 \cos^2 \frac\sin \frac - \sin^3 \frac \right) \sec^3 \frac= a \left(3 \tan \frac - \tan^3 \frac \right) ::= at(3-t^2) giving a parametric equation, parametric form for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemniscate Of Booth
In geometry, a hippopede () is a plane curve determined by an equation of the form :(x^2+y^2)^2=cx^2+dy^2, where it is assumed that and since the remaining cases either reduce to a single point or can be put into the given form with a rotation. Hippopedes are bicircular, rational, algebraic curves of degree 4 and symmetric with respect to both the and axes. Special cases When ''d'' > 0 the curve has an oval form and is often known as an oval of Booth, and when the curve resembles a sideways figure eight, or lemniscate, and is often known as a lemniscate of Booth, after 19th-century mathematician James Booth who studied them. Hippopedes were also investigated by Proclus (for whom they are sometimes called Hippopedes of Proclus) and Eudoxus. For , the hippopede corresponds to the lemniscate of Bernoulli. Definition as spiric sections Hippopedes can be defined as the curve formed by the intersection of a torus and a plane, where the plane is parallel to the axis of the to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kappa Curve
In geometry, the kappa curve or Gutschoven's curve is a two-dimensional algebraic curve resembling the Greek letter . The kappa curve was first studied by Gérard van Gutschoven around 1662. In the history of mathematics, it is remembered as one of the first examples of Isaac Barrow's application of rudimentary calculus methods to determine the tangent of a curve. Isaac Newton and Johann Bernoulli continued the studies of this curve subsequently. Using the Cartesian coordinate system it can be expressed as :x^2\left(x^2 + y^2\right) = a^2y^2 or, using parametric equations, :\begin x &= a\sin t,\\ y &= a\sin t\tan t. \end In polar coordinates its equation is even simpler: :r = a\tan\theta. It has two vertical asymptotes at , shown as dashed blue lines in the figure at right. The kappa curve's curvature: :\kappa(\theta) = \frac. Tangential angle: :\phi(\theta) = -\arctan\left(\tfrac12 \sin(2\theta)\right). Tangents via infinitesimals The tangent lines of the kappa curve can al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kampyle Of Eudoxus
The Kampyle of Eudoxus (Greek: καμπύλη ραμμή meaning simply "curved ine curve") is a curve with a Cartesian equation of :x^4 = a^2(x^2+y^2), from which the solution ''x'' = ''y'' = 0 is excluded. Alternative parameterizations In polar coordinates, the Kampyle has the equation :r = a\sec^2\theta. Equivalently, it has a parametric representation as :x=a\sec(t), \quad y=a\tan(t)\sec(t). History This quartic curve was studied by the Greek astronomer and mathematician Eudoxus of Cnidus (c. 408 BC – c.347 BC) in relation to the classical problem of doubling the cube. Properties The Kampyle is symmetric about both the ''x''- and ''y''-axes. It crosses the ''x''-axis at (±''a'',0). It has inflection points at :\left(\pm a\frac,\pm a\frac\right) (four inflections, one in each quadrant). The top half of the curve is asymptotic to x^2/a-a/2 as x \to \infty, and in fact can be written as :y = \frac\sqrt = \frac - \frac \sum_^\infty C_n\left(\frac\right)^, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hippopede
In geometry, a hippopede () is a plane curve determined by an equation of the form :(x^2+y^2)^2=cx^2+dy^2, where it is assumed that and since the remaining cases either reduce to a single point or can be put into the given form with a rotation. Hippopedes are bicircular, rational, algebraic curves of degree 4 and symmetric with respect to both the and axes. Special cases When ''d'' > 0 the curve has an oval form and is often known as an oval of Booth, and when the curve resembles a sideways figure eight, or lemniscate, and is often known as a lemniscate of Booth, after 19th-century mathematician James Booth who studied them. Hippopedes were also investigated by Proclus (for whom they are sometimes called Hippopedes of Proclus) and Eudoxus. For , the hippopede corresponds to the lemniscate of Bernoulli. Definition as spiric sections Hippopedes can be defined as the curve formed by the intersection of a torus and a plane, where the plane is parallel to the axis of the to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Devil's Curve
In geometry, a Devil's curve, also known as the Devil on Two Sticks, is a curve defined in the Cartesian plane by an equation of the form : y^2(y^2 - b^2) = x^2(x^2 - a^2). The polar equation of this curve is of the form :r = \sqrt = \sqrt. Devil's curves were discovered in 1750 by Gabriel Cramer, who studied them extensively. The name comes from the shape its central lemniscate takes when graphed. The shape is named after the juggling game diabolo The diabolo ( ; commonly misspelled ''diablo'') is a juggling or circus prop consisting of an axle () and two cups (hourglass/egg timer shaped) or discs derived from the Chinese yo-yo. This object is spun using a string attached to two hand ..., which was named after the Devil and which involves two sticks, a string, and a spinning prop in the likeness of the lemniscate. For , b, , a, it is vertical. Is , b, = , a, , the shape becomes a circle. The vertical hourglass intersects the y-axis at b,-b, 0 . The horizonta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deltoid Curve
In geometry, a deltoid curve, also known as a tricuspoid curve or Steiner curve, is a hypocycloid of three cusps. In other words, it is the roulette created by a point on the circumference of a circle as it rolls without slipping along the inside of a circle with three or one-and-a-half times its radius. It is named after the capital Greek letter delta (Δ) which it resembles. More broadly, a ''deltoid'' can refer to any closed figure with three vertices connected by curves that are concave to the exterior, making the interior points a non-convex set. Equations A hypocycoid can be represented (up to rotation and translation) by the following parametric equations :x=(b-a)\cos(t)+a\cos\left(\fracat\right) \, :y=(b-a)\sin(t)-a\sin\left(\fracat\right) \, , where ''a'' is the radius of the rolling circle, ''b'' is the radius of the circle within which the aforementioned circle is rolling. (In the illustration above ''b = 3a'' tracing the deltoid.) In complex coordinates this bec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cruciform Curve
In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree of a polynomial, degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one of not equal to zero. This equation has 15 constants. However, it can be multiplied by any non-zero constant without changing the curve; thus by the choice of an appropriate constant of multiplication, any one of the coefficients can be set to 1, leaving only 14 constants. Therefore, the space of quartic curves can be identified with the real projective space It also follows, from Cramer's theorem (algebraic curves), Cramer's theorem on algebraic curves, that there is exactly one quartic curve that passes through a set of 14 distinct points in general position, since a quartic has 14 Degrees of freedom (physics and chemistry), degrees of freedom. A quartic curve can have a maximum of: * Four connected components * Tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bullet-nose Curve
In mathematics, a bullet-nose curve is a unicursal quartic curve with three inflection points, given by the equation :a^2y^2-b^2x^2=x^2y^2 \, The bullet curve has three double points in the real projective plane, at and , and , and and , and is therefore a unicursal (rational) curve of genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ... zero. If :f(z) = \sum_^ z^ = z+2z^3+6z^5+20z^7+\cdots then :y = f\left(\frac\right)\pm 2b\ are the two branches of the bullet curve at the origin. References * Plane curves Algebraic curves {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bow Curve
In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one of not equal to zero. This equation has 15 constants. However, it can be multiplied by any non-zero constant without changing the curve; thus by the choice of an appropriate constant of multiplication, any one of the coefficients can be set to 1, leaving only 14 constants. Therefore, the space of quartic curves can be identified with the real projective space It also follows, from Cramer's theorem on algebraic curves, that there is exactly one quartic curve that passes through a set of 14 distinct points in general position, since a quartic has 14 degrees of freedom. A quartic curve can have a maximum of: * Four connected components * Twenty-eight bi-tangents * Three ordinary double points. One may also consider quartic curves over other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicorn
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 y^2 \left(a^2 - x^2\right) = \left(x^2 + 2ay - a^2\right)^2. It has two cusps and is symmetric about the y-axis. History In 1864, James Joseph Sylvester studied the curve y^4 - xy^3 - 8xy^2 + 36x^2y+ 16x^2 -27x^3 = 0 in connection with the classification of quintic equations; he named the curve a bicorn because it has two cusps. This curve was further studied by Arthur Cayley in 1867. Properties The bicorn is a plane algebraic curve of degree four and genus zero. It has two cusp singularities in the real plane, and a double point in the complex projective plane at x=0, z=0. If we move x=0 and z=0 to the origin substituting and perform an imaginary rotation on x bu substituting ix/z for x and 1/z for y in the bicorn curve, we obtain \left(x^2 - 2az + a^2 z^2\right)^2 = x^2 + a^2 z^2. This curve, a limaçon, has an ordinary double point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
