In mathematics, a planecurve is a curve in a Plane mathematics Euclidean plane cf. space curve . The most frequently studied cases are smooth plane curves including piecewise smooth plane curves , and Algebraic curvePlane algebraic curves algebraic plane curves . A smooth planecurve is a curve in a real number real Euclidean plane R sup 2 sup and is a one dimensional smooth manifold . Equivalently, a smooth planecurve can be given locally by an equation nowrap 1 &fnof x , y 0, where nowrap 1 &fnof R sup 2 sup &rarr R is a smooth function , and the partial derivative s nowrap 1 &part &fnof &part x and nowrap 1 &part &fnof &part y are never both 0. In other words, a smooth planecurve is a planecurve which locally looks like a line geometry line with respect to a smooth change of coordinates. An algebraic planecurve is a curve in an affine or projective plane given by one polynomial equation nowrap 1 &fnof x , y 0 or nowrap 1 &fnof x , y , z 0, where is a homogeneous polynomial , in the projective case. Algebraic curves were studied extensively in the 18th to 20th centuries, leading ... , among others. Every algebraic planecurve has a degree, which can be defined, in case of an algebraically closed field , as number of intersections of the curve with a generic line. For example, the circle ... result states that every non singular planecurve of degree 2 in a projective plane is isomorphic ... PlaneCurve title PlaneCurve Algebraic curves navbox Category Euclidean geometry Category Curves geometry stub es Curva plana fr Courbe plane io Plana kurvo it Curva piana hu S kg rbe pt Curva plana ... of plane curves of degree 3 is already very deep, and connected with the Weierstrass s theory ... questions in the theory of plane algebraic curves for which the answer is not known as of the beginning ... Differential geometry Algebraic curve Algebraic geometry Projective varieties References Citation first J. L. last Coolidge title A Treatise on Algebraic Plane Curves publisher Dover Publications year ... more details
In mathematics, a real planecurve is usually a real algebraic curve defined in the real projective plane . Ovals Since the real number field is not algebraically closed , the geometry of even a planecurve C in the real projective plane is not a very easy topic. Assuming no Mathematical singularity singular points , the real points of C form a number of ovals , in other words submanifolds that are topologically circle s. The real projective plane has a fundamental group that is a cyclic group with two elements. Such an oval may represent either group element in other words we may or may not be able to contract it down in the plane. Taking out the line at infinity L , any oval that stays in the finite part of the Euclidean plane affine plane will be contractible, and so represent the identity element of the fundamental group the other type of oval must therefore intersect L . There is still the question of how the various ovals are nested. This was the topic of Hilbert s sixteenth problem . See Harnack s curve theorem for a classical result. See also Real algebraic geometry Hilbert s sixteenth problem Harnack s curve theorem Ragsdale conjecture References Springer id P p072800 title Plane real algebraic curve Category Real algebraic geometry Category Algebraic curves ... more details
A quartic planecurve is a planecurve of the fourth degree. It can be defined by a quartic equation ... Curve urlname AmpersandCurve ref Bean curve The bean curve is a quartic planecurve with the equation math x 4 x 2y 2 y 4 x x 2 y 2 , math The bean curve is a algebraic curveplane algebraic curve of geometric ... is a quartic planecurve with the equation math x 2 a 2 x a 2 y 2 a 2 2 0 , math where a determines ... curve is a quartic planecurve with the equation math x 4 x 2y y 3. , math The bow curve has a single ... planecurve given by the equation math x 2y 2 b 2x 2 a 2y 2 0 , math where a and b are two parameter ... a algebraic curve rational plane algebraic curve of geometric genus genus zero. The cruciform curve has three double points in the real projective plane , at x 0 and y 0, x 0 and z 0, and y ... Greek. Three leaved clover The three leaved clover is a quartic planecurve math x 4 2x 2y 2 y 4 x 3 ... has fifteen constants. However, it can be multiplied by any non zero constant without changing the curve ... mathbb RP 14 math . It also follows that there is exactly one quartic curve that passes through a set ... and chemistry degrees of freedom . A quartic curve can have a maximum of Four connected components ... vertical align right width 130 image1 Ampersandcurve.svg caption1 Ampersand curve image2 Bean curve.svg caption2 Bean curve image3 Bicuspid curve.svg caption3 Bicuspid curve image4 Bowcurve.svg caption4 Bow curve image5 Cruciform1.png caption5 Cruciform curve with parameters b,a being 1,1 in red 2,2 in green 3,3 in blue. image6 Cruciform2.png caption6 Cruciform curve with parameters b,a being 1,1 ... Bicorn curve Klein quartic Bullet nose curve Lemniscate of Bernoulli Cartesian oval Lemniscate of Gerono Cassini oval L roth quartic Deltoid curve Spiric section Hippopede Toric section Kampyle of Eudoxus Trott curve Ampersand curve The ampersand curve is a quartic planecurve given by the equation math y 2 x 2 x 1 2x 3 4 x 2 y 2 2x 2. math It is an algebraic curve of Genus mathematics Graph theory ... more details
divide the plane into an interior and an exterior . A planecurve is a curve for which ... plane . A space curve is a curve for which X is of three dimensions, usually Euclidean space a skew curve anchor skew curve is a space curve which lies in no plane. These definitions also ... square in the plane space filling curve . The image of simple planecurve can have Hausdorff dimension ... curve Algebraic curves are the curves considered in algebraic geometry . A plane algebraic ... curve may be projected onto a plane algebraic curve , which however may introduce singularities such as cusp singularity cusp s or double points. A planecurve may also may also be completed in a curve in the projective plane if a curve is defined by a polynomial f of total degree d , then w ... of the curve in the projective plane and the points of the initial curve are those such w is not zero ...other uses File Parabola.svg right thumb A parabola , a simple example of a curve In mathematics , a curve ... line is a special case of curve, namely a curve with null curvature . ref In current language, a line ... Often curves in two dimensional plane curves or three dimensional space curves Euclidean space are of interest ... instances of the definition which follows. A curve is a topological space which is locally homeomorphic to a line. In every day language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola ... mathematical fields. The term curve has several meanings in non mathematical language as well. For example, it can be almost synonymous with mathematical function as in learning curve , or graph of a function as in Phillips curve . An Arc geometry arc or segment of a curve is a part of a curve that is bounded by two distinct end points and contains every point on the curve between its end points ... of the more modern term curve . Hence the phrases straight line and right line were used to distinguish ... more details
The Curve may refer to The Curve film , a 1998 thriller neo noir film The Curve Theatre, Leicester, UK. Curve theatre The Curve, an art gallery within Barbican Centre , London, UK The Curve shopping mall , a shopping mall in Malaysia See also Curve disambiguation disambiguation ... more details
S Curve can refer to the following S Curve art , an art term for a sinuous body position, noted in ancient marble sculpture Sigmoid function , a mathematical function that produces a sigmoid curve or a curve having an S shape S Curve Records , a record label Reverse curve , a type of horizontal curve disambig fr Courbe en S ... more details
wiktionarypar planePlane or planes may refer to Physical objects Aeroplane or airplane, a fixed wing aircraft Plane tool , a woodworking tool to smooth surfaces Platanus , a genus of trees with the common name plane Acer pseudoplatanus , a tree species sometimes called plane Planes genus Planes genus , a genus of crabs in the family Grapsidae called weed crabs Plane river , a river in eastern Germany Plane wherry , a Norfolk wherry in service 1931&ndash 49 Concepts Plane geometry , abstract surface which has infinite width and length, zero thickness, and zero curvature. Lattice plane , a plane in a crystal structure Clipping plane In 3D graphics Clipping plane , in computer graphics Plane esotericism , an emergent state, level or region of reality Physical plane , the physical universe in emanationist metaphysics Plane of immanence , a philosophical concept Planing sailing , a method of travelling quickly across water by using speed to lift the hull out of the water Plane sailing , an approximation used in navigation Plane Unicode , in Unicode, a big range of 65,536 2 sup 16 sup code points Mythology and Religion Plan mythology See also Planes film Planes film , an upcoming film Plain disambiguation Plano disambiguation Planar disambiguation disambig cs Rovina rozcestn k de Plane fr Plane homonymie hr Plane sr tl Plane zh yue Plane ... more details
In mathematics and engineering , the S plane is the name for the complex plane on which Laplace transform s are graphed. It is a mathematical domain where, instead of viewing processes in the time domain modelled with time based functions, they are viewed as equations in the frequency domain . It is used as a graphical analysis tool in engineering and physics. A real function f in time t is translated into the s plane by taking the integral of the function, multiplied by math e st math from math 0 math to math infty math , where s is a complex number . math int 0 infty f t e st ,dt s in mathbb C math One way to understand what this equation is doing is to remember how Fourier analysis works. In Fourier analysis , harmonic sine and cosine waves are multiplied into the signal, and the resultant integration provides indication of a signal present at that frequency i.e. the signal s energy at a point in the frequency domain . The s transform does the same thing, but more generally. The e sup st sup not only catches frequencies, but also the real e sup t sup effects as well. s transforms therefore cater not only for frequency response, but decay effects as well. For instance, a damped sine wave can be modeled correctly using s transforms. s transforms are commonly known as Laplace transform s. In the s plane, multiplying by s has the effect of differentiating in the corresponding real time domain. Dividing by s integrates. Analysing the complex number complex roots of an s plane equation and plotting them on an Argand diagram , can reveal information about the frequency response and stability of a real time system. See also Root locus State space controls External links http dspcan.homestead.com files Ztran zlap.htm Illustration of how the s plane maps to the z plane Category Fourier analysis Mathanalysis stub de Erweiterte symbolische Methode der Wechselstromtechnik pl P aszczyzna S ru S ... more details
A Moore curve after E. H. Moore is a Geometric continuity continuous fractal space filling curve space filling curve which is a variant of the Hilbert curve . Precisely, it is the curve definitions loop version of the Hilbert curve, and it may be thought as the union of four copies of the Hilbert curves combined in such a way to make the endpoints coincide. Because the Moore curve is plane filling, its Hausdorff dimension is 2. The following figure shows the initial stages of the Moore curve. Image Moore curve stages 0 through 5.png Representation as Lindenmayer system The Moore curve can be expressed by a rewriting rewrite system L system . Alphabet L, R Constants F, , &minus Axiom LFL F LFL Production rules L &rarr &minus RF LFL FR&minus R &rarr LF&minus RFR&minus FL Here, F means draw forward , means turn left 90 , and &minus means turn right 90 see turtle graphics . Like the Hilbert curve, the Moore curve can be extended to three dimensions Image Moore3d step3.png 200 px External links A. Bogomolny, Plane Filling Curves from Interactive Mathematics Miscellany and Puzzles http www.cut the knot.org do you know hilbert.shtml, Accessed 07 May 2008. See also Hilbert curve Sierpi ski curve z order curve List of fractals by Hausdorff dimension Category Fractal curves ... more details
A normal plane may refer to The plane perpendicular to the tangent vector of a space curve see Frenet Serret formulas . A term involving gears see list of gear nomenclature . See also Normal bundle Disambig ... more details
Image Devils curve a 0.8 b 1.svg right thumb Devil s curve for nowrap a 0.8 and nowrap b 1 . Image Devils curve a 0.0 1.0 b 1.svg right thumb Devil s curve with math a math ranging from 0 to 1 and nowrap b 1 with the curve color going from blue to red . In geometry , a Devil s curve is a curve defined in the Cartesian plane by an equation of the form math y 2 y 2 a 2 x 2 x 2 b 2 . math Devil s curves were studied heavily by Gabriel Cramer . The name comes from the shape it takes when graphed. It seems that the devil in the name of the curve is from a juggling game called diabolo , which involves two sticks, a string, and a spinning prop in the likeness of the form of this curve. The confusion is the result of the Italian word diabolo meaning devil . ref http www.2dcurves.com quartic quarticd.html ref References reflist External links http mathworld.wolfram.com DevilsCurve.html MathWorld Devil s Curve http www history.mcs.st andrews.ac.uk Curves Devils.html The MacTutor History of Mathematics University of St. Andrews Devil s curve DEFAULTSORT Devil S Curve Category Curves geometry stub ca Corba del diable es Curva del diablo fr Courbe du diable nl Duivelscurve sl Hudi eva krivulja ... more details
Refimprove date April 2007 In Anatomy , the Curve of Spee called also von Spee s curve or Spee s curvature is defined as the curvature of the mandibular occlusal plane beginning at the tip of the lower cuspid and following the buccal cusp dentistry cusp s of the posterior teeth , continuing to the terminal Molar tooth molar . According to another definition Curve of Spee is an anatomic curvature of the occlusal alignment of teeth, beginning at the tip of the lower canine, following the buccal cusps of the natural premolars and molars, and continuing to the anterior border of the ramus. Ferdinand Graf von Spee, German embryologist, 1855 1937 was first to describe anatomic relations of human teeth in the sagittal plane. The pull of the main muscle of mastication, the masseter , is at a perpendicular angle with the curve of Spee to adapt for favorable loading of force on the teeth. The Curve of Spee is, essentially, a series of slipped contact points. It is of importance to orthodontists as it may contribute to an increased overbite. Larry Andrews, in his important paper Six Keys to Normal Occlusion 1972 , stated that a flat or mild curve of Spee was essential to an ideal occlusion. The curve of Spee should not be confused with the curve of Wilson, which is the upward i.e. U shaped curvature of the maxillary and mandibular occlusal planes in the coronal plane. Category Teeth dentistry stub musculoskeletal stub de Spee Kurve ... more details
In mathematics, a trident curve also trident of Newton or parabola of Descartes is any member of the family of curve s that have the formula math xy ax 3 bx 2 cx d , math Image Trid1111.jpg thumb trident curve with a b c d 1 Trident curves are cubic plane curves with an ordinary double point in the real projective plane at x 0, y 1, z 0 if we substitute x x z and y 1 z into the equation of the trident curve, we get math ax 3 bx 2z cxz 2 xz dz 3, , math Image Trif1111.jpg thumb trident curve at y &infin with a b c d 1 which has an ordinary double point at the origin. Trident curves are therefore algebraic curve rational plane algebraic curves of geometric genus genus zero. References cite book author J. Dennis Lawrence title A catalog of special plane curves publisher Dover Publications year 1972 isbn 0 486 60288 5 page 110 geometry stub Category Algebraic curves ca Trident de Newton fr Trident de Newton sl Trizob krivulja ... more details
of Plane Curves Xah Lee Differential transforms of plane curves DEFAULTSORT Parallel Curve Category ... normal to the cutter trajectory at every point. A curve that is a parallel of itself ... can fix a circle and a point on the curve and take the envelope of the translations taking that point to the circle. Tracing the center of a circle rolled along the curve see roulette curve roulette would give one branch of a parallel. Parametric curve For a parametrically defined curve, the following equations define one branch of its parallel curve with distance math a , math the other branch is obtained ... 2 . math Geometric properties As for parallel lines, a normal line to a curve is also normal to its ... from the curve matches the radius of curvature. These are the points where the curve touches the evolute . If the initial curve is a boundary of a planar set and its parallel curve is without self ... more details
Image frenet.png thumb 300px right A space curve, Frenet E2 80 93Serret formulas Frenet Serret frame , and the osculating plane spanned by T and N . In mathematics , particularly in differential geometry , an osculating plane is a plane mathematics plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact mathematics contact at the point. The word osculate is from the Latin language Latin osculatus which is a past participle of osculari , meaning to kiss . An osculating plane is thus a plane which kisses a submanifold. The osculating plane in the geometry of Euclidean space curves can be described in terms of the Frenet Serret formulas as the linear span of the tangent and normal vectors. See also Differential geometry of curves Special Frenet vectors and generalized curvatures . Category Differential geometry differential geometry stub ja nn Oskulerande plan sl Pritisnjena ravnina zh ... more details
Image Swastika curve.svg right thumb 349px The swastika curve. The swastika curve is the name given by Cundy and Rollett ref Mathematical Models by Martyn Cundy H. Martyn Cundy and A.P. Rollett, second edition, 1961 Oxford University Press , p. 71. ref to the quartic curve quartic planecurve with the Cartesian coordinates Cartesian equation math y 4 x 4 xy, , math or, equivalently, the polar coordinates polar equation math r 2 tan 2 theta 2. , math The curve looks similar to the right handed swastika , but can be inverted with respect to a unit circle to resemble a left handed swastika. The Cartesian equation then becomes math x 4 y 4 xy. , math references External links http mathworld.wolfram.com SwastikaCurve.html Mathworld Article http www.jstor.org view 00255572 ap060315 06a00150 0 Mathematical Notes Category Curves nl Swastika kromme ta ... more details
File Radiodrome simple y bw.png thumb 100px right Simple pursuit curve Image Radiodrome params colour.png thumb 250px right Curves of pursuit with different parameters A curve of pursuit is a curve constructed by analogy to having a point geometry point or points which represents pursuers and pursuees, and the curve of pursuit is the curve traced by the pursuers. With the paths of the pursuer and pursuee parameterized in time, the pursuee is always on the pursuer s tangent . That is, given F t the pursuer follower and L t the pursuee leader , there is for every t with F &prime t 0 an x such that math L t F t xF prime t . , math Multiple pursuers Image Four point pursuit curve.gif thumb left baseline 150px animation Curve of pursuit of vertex geometry vertices of a square Typical drawings of curves of pursuit have each point acting as both pursuer and pursuee, inside a polygon , and having each pursuer pursue the adjacent point on the polygon. See also Radiodrome Logarithmic spiral Tractrix External links Commons Curve of pursuit http mathworld.wolfram.com PursuitCurve.html Mathworld , with a slightly narrower definition that L &prime t and F &prime t are constant Differential transforms of plane curves Category Curves geometry stub de Radiodrome es Curva de persecuci n pl Krzywa pogoni ru uk ... more details
In mathematics , a transcendental curve is a curve that is not an algebraic curve . ref name newman Newman, JA, The Universal Encyclopedia of Mathematics , Pan Reference Books, 1976, ISBN 0330243969, Transcendental curves . ref Here for a curve, C , what matters is the point set typically in the plane mathematics plane underlying C , not a given parametrisation. For example, the unit circle is an algebraic curve pedantically, the real points of such a curve the usual parametrisation by trigonometric function s may involve those transcendental function s, but certainly the unit circle is defined by a polynomial equation. The same remark applies to elliptic curve s and elliptic function s and in fact to curves of genus mathematics genus 1 and automorphic function s. The properties of algebraic curves, such as B zout s theorem , give rise to criteria for showing curves actually are transcendental. For example an algebraic curve C either meets a given line L in a finite number of points, or possibly contains all of L . Thus a curve intersecting any line in an infinite number of points, while not containing it, must be transcendental. This applies not just to sinusoidal curves, therefore but to large classes of curves showing oscillations. The term is originally attributed to Gottfried Wilhelm von Leibniz Leibniz . Further examples Cycloid Trigonometric function s Logarithm ic and exponential function exponential functions Archimedes spiral Logarithmic spiral Catenary tricomplex number Cosexponential functions Tricomplex cosexponential References reflist Category Curves ca Corba transcendental es Curva trascendente pt Curva transcendental ru uk ... more details
In mathematics , the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates X Y Z by the Fermat equation math X n Y n Z n. math Therefore in terms of the Euclidean plane affine plane its equation is math x n y n 1. math An integer solution to the Fermat equation would correspond to a nonzero rational number solution to the affine equation, and vice versa. But by Fermat s last theorem it is now known that for n 3 there are no nontrivial integer solutions to the Fermat equation therefore, the Fermat curve has no nontrivial rational points. The Fermat curve is non singular and has genus mathematics genus math n 1 n 2 2. math This means genus 0 for the case n 2 a conic and genus 1 only for n 3 an elliptic curve . The Jacobian variety of the Fermat curve has been studied in depth. It is isogenous to a product of simple abelian varieties with complex multiplication . Fermat varieties Fermat style equations in more variables define as projective varieties the Fermat varieties . Related studies citation first1 Benedict H. last1 Gross first2 David E. last2 Rohrlich year 1978 title Some Results on the Mordell Weil Group of the Jacobian of the Fermat Curve journal Inventiones Mathematicae volume 44 issue 3 pages 201 224 url http www.kryakin.com files Invent mat 282 8 29 44 44 01.pdf doi 10.1007 BF01403161 . Category Algebraic curves Category Diophantine geometry ca Corba de Fermat he sl Fermatova krivulja ru ... more details
In the differential geometry of surfaces , an asymptotic curve is a curve always tangent to an asymptotic direction of the surface where they exist . It is sometimes called an asymptotic line , although it need not be a line mathematics line . An asymptotic direction is one in which the normal curvature is zero. Which is to say for a point on an asymptotic curve, take the plane mathematics plane which bears both the curve s tangent and the surface s surface normal normal at that point. The curve of intersection of the plane and the surface will have zero curvature at that point. Asymptotic directions can only occur when the Gaussian curvature is negative or zero . There will be two asymptotic directions through every point with negative Gaussian curvature, these directions are bisected by the principal curvature principal directions . The direction of the asymptotic direction are the same as the asymptote s of the hyperbola of the Dupin indicatrix . ref cite book title Geometry and Imagination author David Hilbert authorlink David Hilbert coauthors Stephan Cohn Vossen Cohn Vossen, S. year 1999 publisher American Mathematical Society isbn 0 8218 1998 4 ref A related notion is a Monstar curvature line , which is a curve always tangent to a principal direction. References MathWorld urlname AsymptoticCurve title Asymptotic Curve http www.seas.upenn.edu cis70005 cis700sl10pdf.pdf Lines of Curvature, Geodesic Torsion, Asymptotic Lines http www.mathcurve.com surfaces asymptotic asymptotic.shtml Asymptotic line of a surface at Encyclop die des Formes Math matiques Remarquables in French language French references Category Curves Category Differential geometry of surfaces Category Surfaces differential geometry stub ar cs Asymptotick k ivka eo Asimptota kurbo fr Branche parabolique ru ... more details
curve of degree three. Plane algebraic curves An algebraic curve defined over a field F may be considered ... homogeneous polynomial functions, which define the corresponding curve in projective space P sup n sub . For a plane algebraic curve we have a single equation f x , y , z ..., if the curve is a plane projective algebraic curve, defined by a single homogeneous polynomial ... at a curve in the real affine plane there might be singular P modulo the stalk, or alternatively as the sum ... where the sum is taken over all singular points P of the complex projective planecurve. Singularities ... with slope t through the rational point, and intersection with the plane quadratic curve this gives ... genus one with a rational point a common model is a nonsingular Cubic planecurve cubic curve ... Jacobian variety Klein quartic List of curves Hilbert s sixteenth problem Cubic planecurve ... Roch theorem for algebraic curves Quartic planecurve Rational normal curve Weber s theorem Geometry ...In algebraic geometry , an algebraic curve is an algebraic variety of dimension of an algebraic variety dimension one. The theory of these curve s in general was quite fully developed in the nineteenth ... the curve is defined by setting each g sub i sub 0. Using the resultant , we can eliminate all but two of the variables and reduce the curve to a birational geometry birationally equivalent planecurve, f x , y 0, still with coefficients in F . The degree of this planecurve the degree of a polynomial degree of f is usually the product of the degrees of the polynomials g sub i sub and also the degree of an algebraic variety degree of the initial curve. The planecurve possesses often additional singularities. For example, eliminating z between the two equations ... 3 z 1 0, which defines an intersection of a cone and a plane in three ... 1, which is the equation of a hyperelliptic curve . Projective curves It is often desirable ... more details
In geometry , curve sketching or curve tracing includes techniques that can be used to produce a rough idea of overall shape of a planecurve given its equation without computing the large numbers of points ... of a curve Determine the x and y intercepts of the curve. The x intercepts are found by setting y equal to 0 in the equation of the curve and solving for x . Similarly, the y intercepts are found by setting x equal to 0 in the equation of the curve and solving for y Determine the symmetry of the curve. If the exponent of x is always even in the equation of the curve then the y axis is an axis of Reflection symmetry symmetry for the curve. Similarly, if the exponent of y is always even in the equation of the curve then the x axis is an axis of symmetry for the curve. If the sum of the degrees of x and y in each term is always even or always odd, then the curve is Rotational symmetry symmetric about the origin and the origin is called a center of the curve. Determine any bounds on the values of x and y . If the curve passes through the origin then determine the tangent lines there. For algebraic ... gives the points where the curve meets the line at infinity . Determine the asymptote s of the curve. Also determine from which side the curve approaches the asymptotes and where the asymptotes intersect the curve. ref Hilton Chapter III 2 ref Newton s diagram Newton s diagram also known as Newton s parallelogram , after Isaac Newton is a technique for determining the shape of an algebraic curve ... sup y sup sup in the equation of the curve. The resulting diagram is then analyzed to produce information about the curve. Specifically, draw a diagonal line connecting two points on the diagram so that every other point is either on or to the right and above it. There is at least one such line if the curve passes through the origin. Let the equation of the line be q p r . Suppose the curve ... for the curve. There may be several such diagonal lines, each corresponding to one or more branches ... more details
curves in the Euclidean plane . The fixed curve is kept invariant the rolling curve is subjected to a continuous ... attached to the rolling curve, another given curve is carried along the moving plane, a family of congruent ...In the differential geometry of curves , a roulette is a kind of curve , generalizing cycloid s, epicycloid ..., a roulette is the curve described by a point called the generator or pole attached to a given curve as it rolls without slipping along a second given curve that is fixed. More precisely, given a curve attached to a plane which is moving so that the curve rolls without slipping along a given curve attached to a fixed plane occupying the same space, then a point attached to the moving plane describes a curve in the fixed plane called a roulette. In the illustration, the fixed curve blue is a parabola , the rolling curve green is an equal parabola, and the generator is the vertex of the rolling ... name 2dcurves cubicc Special cases and related concepts In the case where the rolling curve is a line ... curve. If the rolling curve is a circle and the fixed curve is a line then the roulette is a trochoid ... concept is a glissette , the curve described by a point attached to a given curve as it slides along ... curve another way to express this constraint is that the point of contact of the two curves is the instant .... Modeling the original curves as curves in the complex plane , let math r,f mathbb R to mathbb C math ... If the fixed curve is a catenary and the rolling curve is a line mathematics line , we have math f ... of roulettes class wikitable Fixed curve Rolling curve Generating point Roulette Any curve Line mathematics Line Point on the line Involute of the curve Line mathematics Line Circle Any Trochoid Line ... roulette roulette.shtml Roulette with straight fixed curve on www.mathcurve.com ref Circle Circle ... trochoid on mathcurve.com ref Parabola Equal parabola parameterized in opposite direction Vertex curve ... transforms of plane curves Category Curves ca Ruleta geometria de Rollkurve es Ruleta geometr a ... more details
Image Hyperelliptic curve.svg 300px right thumb A hyperelliptic curve with the equation y sqrt math x 4 x 2 1 math . The red curve f x shows the principal square root, whereas the green curve g x shows the other square root. In algebraic geometry , a hyperelliptic curve is an algebraic curve given by an equation ... field of such a curve or possibly of the Jacobian variety on the curve, these being two concepts that are the same for the elliptic function case, but different in this case. Genus of the curve The degree of the polynomial determines the genus mathematics genus of the curve a polynomial of degree 2 g 1 or 2 g 2 gives a curve of genus g . When the degree is equal to 2 g 1, the curve is called an imaginary hyperelliptic curve . Meanwhile, the curve that has degree 2 g 2 is mentioned a real hyperelliptic curve . This statement about genus remains true for g 0 or 1, but those curves are not called hyperelliptic . Rather, the case g 1 if we choose a distinguished point is an elliptic curve . Hence ... in the projective plane . This feature is specific to the case n 4. Therefore in giving such an equation to specify a non singular curve, it is almost always assumed that a non singular model also .... The singular point at infinity can be removed since this is a curve by the normalization integral closure process. It turns out that after doing this, there is a cover of the curve with two affine ..., with the curve C being defined as a ramified double cover of the projective line , the ramification ... 2 are hyperelliptic, but for genus 3 the generic curve is not hyperelliptic. This is seen heuristically ..., which is less than 3 g &minus 3, the number of moduli of a curve of genus g , unless g is 2. Much ... curves is read from the theory of canonical curve s, the canonical mapping being 2 to 1 on hyperelliptic curves but 1 to 1 otherwise for g 2. Trigonal curve s are those that correspond to taking ... to be separable. Hyperelliptic curves can be used in hyperelliptic curve cryptography for cryptosystem ... more details
For a planecurve C and a given fixed point P , the pedal curve , of C is the locus of points X so that PX is perpendicular to a tangent to the curve passing through X . The point P is called the pedal point . The pedal curve is the first in a series of curves C sub 1 sub , C sub 2 sub , C sub 3 ... at R the point corresponding to P on the moving plane is X , and so the roulette is the pedal curve ... P in case P lies on T or forms with P a line perpendicular to T . The pedal curve is the set of such points ..., so PXRY is a possibly degenerate rectangle. The locus of points Y is called the contrapedal curve. The orthotomic of a curve is its pedal magnified by a factor of 2 so that the center of similarity ... equation Take P to be the origin. For a curve given by the equation F x , y 0, if the equation ... p , and replacing p , by r , produces a polar equation for the pedal curve. ref Edwards p. 164 ref Image PedalCurve1.gif 500px right thumb Pedal curve red of an ellipse black . Here a 2 and b 1 so the equation of the pedal curve is 4 x sup 2 sup y sup 2 sup x sup 2 sup y sup 2 sup sup 2 ... R r , be a point on the curve and let X p , be the corresponding point on the pedal curve. Let ... curve. ref Edwards p. 164 5 ref For example, ref Follows Edwards p. 165 with m 1 ref let the curve be the circle given by r a cos . Then math a cos theta a sin theta tan psi math so math tan ... 2 . math From the pedal equation The pedal equation s of a curve and its pedal are closely related. If P is taken as the pedal point and the origin then it can be shown that the angle between the curve and the radius vector at a point R is equal to the corresponding angle for the pedal curve at the point X . If p is the length of the perpendicular drawn from P to the tangent of the curve i.e. PX ... equation of the curve is f p , r 0 then the pedal equation for the pedal curve is ref Williamson p. 228 ... components of math vec v math with respect to the curve. Then math vec v parallel math is the vector ... more details
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