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Neile's Parabola
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Deposit #80

J. B. Thoo
Manjinder S. Bains

Evolute graph


Octave graph

Thoo and Bains created the images in our animation using Octave, a high-level langauge primarily intended for numerical computations. This free software is mostly compatible with Matlab.  Open this link to learn more about Octave.

As a young man, William Neile interacted closely with his far more famous mentor, John Wallis.
Wadham College, Oxford
Neile was educated at Wadham College, Oxford, one of the newest in the 17th century.  Recently Sir Roger Penrose retired there.

Cardan was primarily a physician but made other significant contributions in addition to  mathematics.   His "cardanic suspension," a device used to hold a compass stable while sailing on the high seas, appears on a postage stamp.

Tartaglia's truncated hexahedron
Tartaglia's stellated truncated hexahedron from his 1543 edition of Euclid's "Elements."

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Evolute of a Plane Curve - The Family of Normals

This section . . . .

Animation of Normals to a Parabola

Equations and explanation

Did you know . . .
Cardano's Formula:  Cubic
Please note the second degree term is missing.  Moreover, neither Tartaglia nor Cardan knew of negative and/or imaginary solutons to equations.  However, Cardan was the first to notice a cubic might have three roots and wrote the negative solutions were "fictitious."

A brief list of references that should be in most university libraries.
From the authors of Deposit # 80 . . .
Bains, Manjinder S. and J. B. Thoo , The Normals to a Parabola and the Real Roots of a Cubic, The College Mathematics Journal, 38 (4) September 2007, pp. 272-277.
Gray, Alfred, Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, CRC Press, 1998, p. 1027.
Lockwood, E. H., A Book of Plane Curves, Cambridge University Press, 1961, pp. 167-171.
Weisstein, E. W., CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, pp. 589 - 590.  See Evolute.

The National Curve Bank thanks J. B. Thoo and Manjinder S. Bains for Deposit #80.
Be sure to see their article in the College Mathematics Journal for more information.
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