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 
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 infinity.
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This article incorporates text from the Encyclopædia Britannica Eleventh Edition, a publication now in the public domain.
