Geometric continuity

Geometrical or geometric continuity, was a concept of geometry primarily applied to the conic 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 of continuity 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 a circle as the eccentricity approaches zero, or to a parabola as it approaches one; and a hyperbola tends to a parabola as the eccentricity drops toward one; it can also tend to intersecting lines. 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 of infinite 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 the continuity of a function and of infinity.

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. A curve or surface 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 common tangent 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 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 "G"² continuity.Fact|date=May 2008

A "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, then cubic splines are typically chosen; these curves are frequently used in industrial 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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