Expert subject Mathematics date November 2008 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 ... Henrik Abel , Henri Poincar , Max Noether , 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 ... 2 sup 1 has degree 2. An important classical result states that every non singular planecurve of degree 2 in a projective plane is isomorphic to the projection mathematics projection of the circle nowrap 1 x sup 2 sup y sup 2 sup 1. However, the theory of plane curves of degree 3 is already very ... id PlaneCurve title PlaneCurve Algebraic curves navbox Category Geometry geometry stub es Curva plana fr Courbe plane io Plana kurvo it Curva piana hu S kg rbe pt Curva plana ... curves , Weierstrass P function . There are many questions in the theory of plane algebraic curves ... 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 ... 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 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 math 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. math This equation has fifteen constants. However, it can be multiplied by any non zero constant without changing the curve. Therefore, the space of quartic curves can be identified with the real projective space math mathbb RP 14 math . It also follows that there is exactly one quartic curve that passes through a set of fourteen 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 Twenty eight bitangent bi tangents Three ordinary double point s. Examples Bicorn curve Bullet nose curve Deltoid curve Lemniscate of Gerono Klein quartic Toric section Kampyle of Eudoxus gallery Image Ampersandcurve.svg Ampersand curve Image Bean curve.svg Bean curve Image Bicuspid curve.svg Bicuspid curve Image Bowcurve.svg Bow curve Image Cruciform1.png Cruciform curve s Image Spiric section.svg Spiric section s Image Three leaved clover.svg Three leaved clover Image Trott.png Trott curve gallery A spiric section is a special case of a toric section Category Algebraic curves geometry stub ar es Curva cu rtica it Curva quartica ... more details
divide the plane into an interior and an exterior . A planecurve is a curve for which X ... 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 apply ... square in the plane space filling curve . The image of simple planecurve can have Hausdorff dimension ... considered in algebraic geometry . A plane algebraic curve is the locus of points f x , y 0, where .... By eliminating variables by means of the resultant , these can be reduced to plane algebraic curve ...other uses File Parabola.svg right thumb A parabola , a simple example of a curve In mathematics , a curve ... case of curve, namely a curve with null curvature . ref In current language, a line is typically ... 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 . A 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. Depending ... term curve . Hence the phrases straight line and right line were used to distinguish what are today ... geometry in the seventeenth century. This enabled a curve to be described using an equation rather ... equation s, algebraic curve s, and those that cannot, transcendental curve s. Previously, curves ... . In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general ... more details
The Curve may refer to The Curve shopping mall , a shopping mall in Malaysia The Curve film , a 1998 thriller neo noir film The Curve, an art gallery within Barbican Centre , London 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
pp move indef 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 Dungeons & Dragons , a plane of existence within the role playing game Plane Magic The Gathering , a plane of existence within the trading card game 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 , a range of Unicode code points See also Plain disambiguation Plano disambiguation disambig cs Rovina rozcestn k de Plane fr Plane homonymie hr Plane 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 time function is translated into the s plane by taking the integral of the function, multiplied by math e st math from math infty math to math infty math , where s is a complex number . math int infty infty f t e st ,dt 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. External links http dspcan.homestead.com files Ztran zlap.htm Illustration of how the s plane maps to the z plane Math stub Category Fourier analysis ru S ... more details
image Bowcurve.svg thumb A bow curve In mathematics , the bow curve is a quartic planecurve with the equation math x 4 x 2y y 3. , math The bow curve has a single triple point at x 0, y 0, and consequently is a rational curve, with geometric genus genus zero. References MathWorld title Bow urlname Bow DEFAULTSORT Bow Curve Category Algebraic curves ca Corba de llacet ... more details
Image Bicuspid curve.svg thumb 120px right Biscuspid curve where a 1 The biscuspid 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 the size of the curve. The bicuspid has only the two nodes as singularities, and hence is a curve of genus one. References MathWorld title Bicuspid Curve urlname BicuspidCurve DEFAULTSORT Bicuspid Curve Category Curves Category Algebraic curves ca Corba bic spide ro Curb bicuspid ... 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
Image Ampersandcurve.svg thumb 130px right The ampersand curve. In mathematics, 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 genus zero, with three ordinary double points, all in the real plane. References MathWorld title Ampersand Curve urlname AmpersandCurve DEFAULTSORT Ampersand Curve Category Curves Category Algebraic curves ar ca Corba signe i comercial eo Kaj signa kurbo nl Ampersandcurve zh ... more details
Image Bean curve.svg thumb right 226px The 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 genus genus zero. It has one Mathematical singularity singularity at the origin, an ordinary triple point. References Citation last1 Cundy first1 H. Martyn last2 Rollett first2 A. P. title Mathematical models origyear 1952 publisher Clarendon Press, Oxford edition 2nd isbn 978 0 906212 20 2 id MathSciNet id 0124167 year 1961 See page 72, curve 17 MathWorld title Bean Curve urlname BeanCurve DEFAULTSORT Bean Curve Category Curves Category Algebraic curves ca Corba mongeta eo Faba kurbo pt Curva feij o ... more details
In mathematics , plane geometry may refer to Euclidean plane geometry , the geometry of plane figures, geometry of a plane geometry plane , or sometimes geometry of a projective plane , most commonly the real projective plane but possibly the complex projective plane , Fano plane or others geometry of the Hyperbolic geometry hyperbolic plane or two dimensional spherical geometry . See also planecurve . mathdab bn eo Ebena geometrio ... 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 ... 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
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
5, page 66 ref It is sometimes called the offset curve but the term offset often refers also to Translation geometry translation . The term offset curve is used, e.g., in numerically controlled machining ... 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 ... x 2 y 2 . math Geometric properties As for parallel lines, a normal line to a curve is also normal ... 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 ... Visual Dictionary of Plane Curves Xah Lee Differential transforms of plane curves DEFAULTSORT Parallel Curve Category Curves Category Differential geometry geometry stub ru uk ... more details
Unreferenced date December 2009 The cruciform curve , or cross curve is a quartic planecurve given by the equation Image Cruciform1.png thumb right Cruciform with parameters b,a being 1,1 in red 2,2 in green 3,3 in blue. Image Cruciform2.png thumb right Cruciform with parameters b,a being 1,1 in red 2,1 in green 3,1 in blue. math x 2y 2 b 2x 2 a 2y 2 0 , math where a and b are two parameter s determining the shape of the curve. The cruciform curve is related by a standard quadratic transformation, x 1 x, y 1 y to the ellipse a sup 2 sup x sup 2 sup b sup 2 sup y sup 2 sup 1, and is therefore 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 0 and z 0. Because the curve is rational, it can be parametrized by rational functions. For instance, if a 1 and b 2, then math x frac t 2 2t 5 t 2 2t 3 , y frac t 2 2t 5 2t 2 math parametrizes the points on the curve outside of the exceptional cases where the denominator is zero. DEFAULTSORT Cruciform Curve Category Curves Category Algebraic curves ca Qu rtica cruciforme es Curva cruciforme nl Kruiscurve ... 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 ... more details
In geometry , curve sketching or curve tracing includes techniques that can 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 ... 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 is approximated ... those lying on the line and the others may be ignored it produce a simple approximate equation for the curve ... 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 References reflist Category Curves ca Corba transcendental es Curva trascendente pt Curva transcendental ru uk ... more details
on the dual. Degree Smooth curves If X is a smooth plane algebraic curve of degree d 1, then the dual of X is a usually singular planecurve of degree d d 1 cf. Fulton, Ex. 3.2.21 . If d 2, then d 1 1 so d d 1 d , and thus the dual curve must be singular, by duality, otherwise the bidual would have higher degree than the original curve. For a smooth ... with the line itself . Singular curves For an arbitrary plane algebraic curve X of degree d 1, its dual is a planecurve of degree math d d 1 delta math , where math delta math is the number of singularities ... euclidian plane math R 2 math , there is a beautiful classical construction of a dual curve math ... l if we identify the plane math R 2 math and its dual. Properties of dual curve File Dual curve.svg ...Image Dual curve.svg thumb right 300px Curves, dual to each other see below for Properties of dual curve properties . In projective geometry , a dual curve of a given planecurve C is a curve in the Duality projective geometry dual projective plane consisting of the set of lines tangent to C . There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If C is algebraic then so is its dual and the degree of the dual is known as the class of the original curve ... . The construction of the dual curve is the geometrical underpinning for the Legendre transformation ... y , z 0 be the equation of a curve in homogeneous coordinates . Let Xx Yy Zz 0 be the equation ... is tangent to the curve can be expressed in the form F X , Y , Z 0 which is the tangential equation of the curve. Let p , q , r be the point on the curve, then the equation of the tangent ... y p, q, r z frac partial f partial z p, q, r 0. math So Xx Yy Zz 0 is a tangent to the curve if math ... with Xp Yq Zr 0, gives the equation in X , Y and Z of the dual curve. For example, let C be the Conic ... defined curve its dual curve is defined by the following parametric equations math X x,y ... 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 ... 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 he ... more details
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