Transcendental curve

A plane curve whose equation in rectangular Cartesian coordinates is not algebraic. In contrast to algebraic curves (cf. Algebraic curve), transcendental curves can have an infinite number of points of intersection with a straight line and an infinite number of points of inflection. On transcendental curves one encounters points of a special nature that do not exist on algebraic curves: points of termination, with the property that a circle of sufficiently small radius with centre at this point intersects the curve only at a single point; corner points (points of fracture) at which two branches of the curve come together, each of them having a tangent at this point; asymptotic points, which a branch of the curve approaches continuously, forming an infinite number of rotations around the point. Some transcendental curves have distinctive peculiarities of form (for example, they have a dotted branch consisting of an infinite set of isolated points).

One of the attempts to classify transcendental curves is based on the fact that in the overwhelming majority of known transcendental curves (as for all algebraic curves) the slope $y'$ of the tangent at each point of the curve is the root of an algebraic equation, the coefficients of which are polynomials in $x$ and $y$. In other words, the differential equation of the majority of known transcendental curves is a first-order equation of the form

$$\sum_{r=0}^nf_r(x,y)(y')^{n-r}=0,$$

where $f_0,\ldots,f_n$ are polynomials without common factors. This circumstance enables one to classify all algebraic curves, as well as almost-all transcendental curves (except, for example, the Cornu spiral), into groups of so-called pan-algebraic curves, which are distinguished by degree $n$ and rank $\nu$ — the maximal degree of the polynomials $f_0,\ldots,f_n$. For example, for curves of order three, $n=1$, $\nu=2$; for the Archimedean spiral, $n=2$, $\nu=4$. Pan-algebraic curves possess many of the properties inherent to algebraic curves. For example, the notions of the Hessian (algebraic curve) and the polar can be generalized to them. Concerning the attempts at further classifications of pan-algebraic curves, see [1].

Examples of transcendental curves are spirals, the catenary, the Dinostratus quadratrix, the cycloid, as well as the graphs of the transcendental functions: exponential, logarithmic, trigonometric, etc. (cf. Transcendental function).

References

[1]	A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[a1]	K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)