Concepts inExact arrangements on tori and Dupin cyclides
Dupin cyclide
In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of any standard torus. In particular, the standard (or circular) tori are themselves examples of Dupin cyclides. They were discovered by (and named after) Charles Dupin in his 1803 dissertation under Gaspard Monge. The key property of a Dupin cyclide is that it is a channel surface (envelope of a one parameter family of spheres) in two different ways.
more from Wikipedia
Torus
In geometry, a torus (pl. tori) is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit. When the axis is tangent to the circle, the resulting surface is called a horn torus; when the axis is a chord of the circle, it is called a spindle torus.
more from Wikipedia
Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold. The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of dimension two).
more from Wikipedia
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.
more from Wikipedia
Asymptote
In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors. In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity.
more from Wikipedia
Parameter space
In science, a parameter space is the set of values of parameters encountered in a particular mathematical model. Often the parameters are inputs of a function, in which case the technical term for the parameter space is domain of a function. Parameter spaces are particularly useful for describing families of probability distributions that depend on parameters. More generally in science, the term parameter space is used to describe experimental variables.
more from Wikipedia
Curve
In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but which is not required to be straight. This entails that a line is a special case of curve, namely a curve with null curvature. Often curves in two-dimensional or three-dimensional (space curves) Euclidean space are of interest.
more from Wikipedia
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R ¿ for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections.
more from Wikipedia