- Geometric continuity
**Geometrical**or**geometric continuity**, was a concept of geometry primarily applied to theconic sections and related shapes by mathematicians such as Leibniz, Kepler, and Poncelet. The concept was an early attempt at describing, through geometry rather than algebra, the concept ofcontinuity as expressed through a parametric function.The basic idea behind geometric continuity was that the five conic sections were really five different versions of the same shape. An

ellipse tends to acircle as the eccentricity approaches zero, or to aparabola as it approaches one; and ahyperbola tends to aparabola as the eccentricity drops toward one; it can also tend to intersectingline s. Thus, there was "continuity" between the conic sections. These ideas led to other concepts of continuity. For instance, if a circle and a straight line were two expressions of the same shape, perhaps a line could be thought of as a circle ofinfinite radius . For such to be the case, one would have to make the line "continuous" by allowing the point "x" = ∞ to be a point on the circle, and for "x" = ∞ and "x" = −∞ to be identical. Such ideas were useful in crafting the modern, algebraically defined, idea of thecontinuity of a function and ofinfinity .**Smoothness of curves and surfaces**In

CAD and other computer graphics applications, the smoothness of a curve or surface is defined by its level of geometric continuity. Acurve orsurface can be described as having "G"^{"n"}continuity, "n" being the measure of smoothness.One way to describe different levels of geometric continuity is to consider the junction of two curves and state the properties required of the curves at the join point.

*"G"

^{0}: The curves touch at the join point.

*"G"^{1}: The curves also share a commontangent direction at the join point.

*"G"^{2}: The curves also share a common center of curvature at the join point.In general, "G"

^{"n"}continuity exists if the curves can be reparameterized to have "C"^{"n"}(parametric) continuity Harv|Farin|1997|loc=Ch. 12. A reparametrization of the curve is geometrically identical to the original. The speed at which the parameter traces the curve will change, but the shape of the curve does not.Equivalently, two vector functions $f(s)$ and $g(t)$ have "G"

^{n}continuity if $f^\{(n)\}(t)$ ≠ 0 and $f^\{(n)\}(t)\; =\; kg^\{(n)\}(t)$, for a scalar $k\; >\; 0$ (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 "G"

^{1}continuity to appear smooth, for goodaesthetics , such as those aspired to inarchitecture andsports car design, higher levels of geometric continuity are required. For example, reflections in a car body will not appear smooth unless the body has "G"^{2}continuity.Fact|date=May 2008A "rounded rectangle" (with ninety degree circular arcs at the four corners) has "G"

^{1}continuity, but does not have "G"^{2}continuity. The same is true for a "rounded cube", with octants of a sphere at its corners. If an editable curve with "G"^{2}continuity is required, thencubic splines are typically chosen; these curves are frequently used inindustrial design .In some cases,

parametric continuity is also important. Consider a surface swept out by a line segment, where the endpoints of the line segment are defined by two parametric curves. In this case, the geometry of the resulting surface depends not only on the geometry of the defining curves, but also on the relative speeds.**References***

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