2d curves A collection of 631 named two−dimensional curves algebraic curves are a polynomial in x other curves and y kid curve line (1st degree) 3d curve conic (2nd) cubic (3rd) derived curves quartic (4th) sextic (6th) barycentric octic (8th) caustic other (otherth) cissoid conchoid transcendental curves curvature discrete cyclic exponential derivative fractal envelope gamma & related hyperbolism isochronous inverse power isoptic power exponential parallel spiral pedal trigonometric pursuit reflecting wall roulette strophoid tractrix Some programs to draw your own For isoptic cubics: curves: • Isoptikon (866 kB) from the • GraphEq (2.2 MB) from University of Crete Pedagoguery Software • GraphSight (696 kB) from Cradlefields ($19 to register) 2dcurves in .pdf format (1882 kB) curve literature last update: 2003−06−14 higher last updated: Lennard−Jones 2002−03−25 potential Atoms of an inert gas are in equilibrium with each other by means of an attracting van der Waals force and a repelling forces as result of overlapping electron orbits. These two forces together give the so−called Lennard−Jones potential, which is in fact a curve of thirteenth degree. backgrounds main last updated: 2002−11−16 history I collected curves when I was a young boy. Then, the papers rested in a box for decades. But when I found them, I picked the collection up again, some years spending much work on it, some years less. questions I have been thinking a long time about two questions: 1. what is the unity of curve? Stated differently as: when is a curve different from another one? 2. which equation belongs to a curve? 1. unity of curve I decided to aim for simplicity: it does not matter when a curve has been reformatted in a linear way (by ways of translation, rotation or multiplication). This means that I omit constants in the equations of a curve, as been found by other authors. Example: for me the equation of the super ellipse is not but: Only the parameter 'a' affects the form of the curve. And all linear transformations of this curve do belong to this same curve 'family'. 2. which formula I don't want to swim in an ocean of formulae. Therefore I look for a formula that is as simple as possible, for covering a given curve. Trying to confine myself to Cartesian, polar, bipolar and parametric equations. Examples: • Cartesian equation: y = f(x) 2 ( Example y= x parabola) • polar equation: r = f(ϕ) Example r = ϕ (spiral of Archimedes) • bipolar equation: f(r1, r2) = a Example r1 r2 = a (Cassinian oval) • parametric equation: x = f(t), y = g(t) Example: x = t − a sin t and y = 1 − a cos t (cycloid) Sometimes the definition of a curve can not fit in one of these forms: • textual definition: let there be etc. Example: apply the following rule to a grid of black squares: when you get on a black square, make it white and turn to the right; when you get on a white square, make it black and turn to the left (ant of Langton) Sometimes a much shorter or much more elegant formula can be found, using another way of defining a curve: • complex equation: z = f(z1, z2) Example: z = f (C1(t), C2(t)) for a curve C2(t) that rolls over curve C1(t) (roulette) • differential equation: f(x, y, y') = 0 2 2 Example: x + y' =1 (quadratrix of Abdank−Abakanovicz) literature I made up a list of the literature I used. curve main last updated: 2003−05−05 definition 1) A curve can be defined as a (finite) number of arcs, combined together. Lawrence defines an arc as a valid one when its parametric equation (x,y) = (f(t), g(t)) − with t in an open interval I − obeys the following conditions: • f and g are twice continuously differentiable • for all t in I, at least one of df/dt and dg/dt is unequal zero • for all points s and t in I, (f(s), g(s)) = (f(t), g(t)) if and only if s = t However, not all curves I describe are differentiable: see e.g. the blancmange curve. A Jordan curve is defined as a closed, non−intersecting line, consisting of a finite number of regular segments. Such a curve with a direction is a contour. notes 1) In French: courbe. oval main last updated: 2002−12−17 An oval is a curve resembling a squashed circle, but without a clear mathematical definition. So an oval has not necessarily two axes of symmetry. We find oval−like curves on the following pages: • Cartesian oval • Cassinian oval • egg curve • ellipse • hippopede • oval of Booth • spiric section Sometimes an oval−like curve is constructed as connected arcs, of circles with different radii. round curve main last updated: 2001−11−17 Is an oval round? Or are only circles round? As for the Latin word 'circum', 'round' has two meanings: • specific denoting the circle form • more general, for a closed loopless curve The second meaning is found in expressions as: a circular road, a circular saw, a circular tour. They all don't match the precise circle form. Conclusion: there are to meanings of 'round' a strict one (circle) and a more general one. main last updated: algebraic 2003−05−03 curve A curve is algebraic when its defining Cartesian equation is algebraic, that is a polynomial in x and y. The order or degree of the curve is the maximum degree of each of its terms x y . An algebraic curve with degree greater than 2 is called a higher plane curve. An algebraic curve is called a circular algebraic curve, when the points (±1, ± i) are on the curve. In that case the highest degree of the Cartesian equation is divisible by 2 2 (x + y ). The circle is the only circular conic section. The curve got its name from the fact that it contains the two imaginary circular points: replace x by x/w and y by y/w, and let the variable w go to zero, we obtain the circular poins. A bicircular algebraic curve passes twice through the points (±1, ± i). In this case the highest degree of the 2 2 2 Cartesian equation is divisible by (x + y ) . Every algebraic polynomial is a Bézier curve. Given a set of points Pi the Bézier curve of this control polygon is the convex envelope of these points. When a curve is not algebraic, we call the curve (and its function) transcendental. In the case a function is sufficiently sophisticated it is said to be a special function. main last updated: polynomial 2002−12−29 1) A polynomial is literally an expression of some terms, distinct powers in one ore more variables. On this page we talk about a polynomial in one variable, x. When talking about two variables (x and y), we call it an algebraic curve. A curve for which f(x,y) is a constant, is called a equipotential curve. Other names for the curve are: isarithm, isopleth. As said, here the polynomial in x, each term is an entire power of x, and the function is also called an entire (rational) function. It is also called an algebraic function. The highest power in x is the degree (or: order) of the polynomial. An entire function of 2nd degree is called a quadratic function; an entire function of 4th degree is called an biquadratic function. Some specific polynomials can be found as: • line y is linear in x. • parabola y is quadratic in x. • cubic • quartic The root to the nth power is the inverse function of the (simple) polynomial with equation y = x . A broken function is a function that is the quotient of two polynomials.Together with the polynomials do they form the group of rational functions. The following polynomials have a special significance: • cubic parabola • generated polynomial • orthogonal polynomial ♦ Hermite polynomial ♦ Laguerre polynomial ♦ Legendre polynomial ♦ Tchebyscheff polynomial • Pochhammer symbol • power series • recurrent polynomial notes 1) polus (Gr.) = many, nomen (Lat.) = name. cubic polynomial last updated: 2002−03−25 parabola A polynomial of the third degree who has its own name is the cubic parabola. Its name has been derived from the parabola, the cubic part refers to the third degree. The curve has to be distinguished from the cubic parabola that is a divergent parabola. polynomial last updated: generated 2003−05−18 polynomial Some polynomials are defined inside an expression, e.g. inside an series with infinite coefficients: Bernoulli polynomial The Bernoulli polynomials (Bi) are used in some formulae that relate a series and a integral. Some slowly converging series can be written as an (easy to solve) integral. Euler polynomial The Euler polynomials (Ei) are used to clarify phenomena of 1) some alternating series . notes 1) Temme 1990 p.13. polynomial last updated: orthogonal 2002−11−10 polynomial According to algebraic theory systems of perpendicular (orthogonal) polynomials fi can be constructed, that span a so called function space. Orthogonality has the meaning of being zero of the inproducts <fi, fj> for i j. This function inproduct is defined as above mentioned integral, where w(x) is called the weight function. [−1 T ?? n Always true is that fi is a polynomial of degree i with the , 1] (x) remarkable quality that all points of intersection with the x−axis are real and different, and are in the interval [a, b]. Some orthogonal polynomials have been given special attention, and were named to their explorer: − polynomial interval w(x) fn(x) abbreviation Hermite [− , ] Hn (x) Laguerre [0, ] Ln (x) Legendre [−1 , 1] 1 Pn (x) The polynomials are also solutions of differential equations with the same name (equation of Hermite, Laguerre and so on), with whom I don't want you to torture.