line could be thought of as a circle of infinite radius. For such to be the case, one would have to make the line closed by allowing the point to be a point on the circle, and for and to be identical. Such ideas were useful in crafting the modern, algebraically defined, idea of the continuity of a function and of (see projectively extended real line for more). Order of geometric continuity A curve or surface can be described as having continuity, with being the increasing measure of smoothness. Consider the segments either side of a point on a curve: : The curves touch at the join point. : The curves also share a common tangent direction at the join point. : The curves also share a common center of curvature at the join point. In general, continuity exists if the curves can be reparameterized to have (parametric) continuity. A reparametrization of the curve is geometrically identical to the original; only the parameter is affected. Equivalently, two vector functions and have continuity if and , for a scalar (i.e., if the direction, but not necessarily the magnitude, of the two vectors is equal). While it may be obvious that a curve would require continuity to appear smooth, for good aesthetics, such as those aspired to in architecture and sports car design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless the body has continuity. A (with ninety degree circular arcs at the four corners) has continuity, but does not have continuity. The same is true for a , with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with continuity is required, then cubic splines are typically chosen; these curves are frequently used in industrial design. Other concepts Relation to analyticity While all analytic functions are "smooth" (i.e. have all derivatives continuous) on the set on which they are analytic, examples such as bump functions (mentioned above) show that the converse is not true for functions on the reals: there exist smooth real functions that are not analytic. Simple examples of functions that are smooth but not analytic at any point can be made by means of Fourier series; another example is the Fabius function. Although it might seem that such functions are the exception rather than the rule, it turns out that