Geometric continuity

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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; it was possible to construct ellipses towards the limit of either a circle or a parabola, and likewise to construct hyperbola that approached the shapes of a parabola or a straight line. 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, could a line not 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 $x = \infty$ to be a point on the circle, and for $x = \infty$ and $x = -\infty$ to be identical. Such ideas were useful in crafting the modern, algebraically defined, idea of the continuity of a function and of infinity.

[edit] 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 Gⁿ continuity, 'n' being the measure of smoothness.

To describe different levels of geometric continuity, it is easiest to consider the join between two curves and state the properties required of the curves at the join point.

G⁰: The curves meet at a common join point.

G¹: The curves share a common tangent at the join point.

G²: The curves share a common tangent and curvature at the join point.

While it may be obvious that a curve would require G¹ continuity to appear smooth, for good aesthetics, such as those aspired to in architecture and sports car design, much higher levels of geometric continuity are required.

For curves describing motion parametric continuity is also important.