Circle As Limiting Case of Other Figures
The circle can be viewed as a limiting case of each of various other figures:
- A Cartesian oval is a set of points such that a weighted sum of the distances from any of its points to two fixed points (foci) is a constant. An ellipse is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle. A circle is also a different special case of a Cartesian oval in which one of the weights is zero.
- A superellipse has an equation of the form for positive a, b, and n. A supercircle has b = a. A circle is the special case of a supercircle in which n = 2.
- A Cassini oval is a set of points such that the product of the distances from any of its points to two fixed points is a constant. When the two fixed points coincide, a circle results.
- A curve of constant width is a figure whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines. The circle is the simplest example of this type of figure.
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Famous quotes containing the words circle, limiting, case and/or figures:
“A beauty is not suddenly in a circle. It comes with rapture. A great deal of beauty is rapture. A circle is a necessity. Otherwise you would see no one. We each have our circle.”
—Gertrude Stein (1874–1946)
“I am truly free only when all human beings, men and women, are equally free. The freedom of other men, far from negating or limiting my freedom, is, on the contrary, its necessary premise and confirmation.”
—Mikhail Bakunin (1814–1876)
“He was then in his fifty-fourth year, when even in the case of poets reason and passion begin to discuss a peace treaty and usually conclude it not very long afterwards.”
—G.C. (Georg Christoph)
“Families suffered badly under industrialization, but they survived, and the lives of men, women, and children improved. Children, once marginal and exploited figures, have moved to a position of greater protection and respect,... The historic decline in the overall death rates for children is an astonishing social fact, notwithstanding the disgraceful infant mortality figures for the poor and minorities. Like the decline in death from childbirth for women, this is a stunning achievement.”
—Joseph Featherstone (20th century)