hausdorff dimension subshift angle doubling

In the common case of a real algebraic curve, where k is the field of real numbers, an algebraic curve is a finite union of topological curves. If the coefficients of the polynomials belong to a field k, the curve is said to be defined over k. More generally, an algebraic curve is the zero set of a finite set of polynomials, which satisfies the further condition of being an algebraic variety of dimension one. For ensuring more regularity, the function that defines a curve is often supposed to be differentiable, and the curve is then said to be a differentiable curve.Ī plane algebraic curve is the zero set of a polynomial in two indeterminates. This is the case of space-filling curves and fractal curves. Nevertheless, the class of topological curves is very broad, and contains some curves that do not look as one may expect for a curve, or even cannot be drawn. Level curves and algebraic curves are sometimes called implicit curves, since they are generally defined by implicit equations. This definition encompasses most curves that are studied in mathematics notable exceptions are level curves (which are unions of curves and isolated points), and algebraic curves (see below). In this article, these curves are sometimes called topological curves to distinguish them from more constrained curves such as differentiable curves. In some contexts, the function that defines the curve is called a parametrization, and the curve is a parametric curve. This definition of a curve has been formalized in modern mathematics as: A curve is the image of an interval to a topological space by a continuous function. This is the definition that appeared more than 2000 years ago in Euclid's Elements: 'The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which will leave from its imaginary moving some vestige in length, exempt of any width.' Intuitively, a curve may be thought of as the trace left by a moving point.

In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. A parabola, one of the simplest curves, after (straight) lines

Hausdorff dimension subshift angle doubling