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The superellipsoid is a generalization of the ellipsoid by allowing different exponents of the variables in the algebraic representation. It is similarly a generalization of the superellipse to three dimensions. The version called the superquadratic ellipsoid is defined by the equation (|x|^(2/e)+|y|^(2/e))^(e/n)+|z|^(2/n)=1, where e and n are the east-west and north-south exponents, respectively. This superellipsoid can be rendered in POVRay® with the command superellipsoid{...

Mathematica: Export Plot Values as CSV. GitHub Gist: instantly share code, notes, and snippets.

Given two circles with one interior to the other, if small tangent circles can be inscribed around the region between the two circles such that the final circle is tangent to the first, the circles form a Steiner chain. The simplest way to construct a Steiner chain is to perform an inversion on a symmetrical arrangement on n circles packed between a central circle of radius b and an outer concentric circle of radius a (Wells 1991). In this arrangement, sin(pi/n)=(a-b)/(a+b), (1) so the...

Wolfram Language function: Plot a curve defined by its curvature. Complete documentation and usage examples. Download an example notebook or open in the cloud.

Wolfram Language function: Evaluate the Fabius function. Complete documentation and usage examples. Download an example notebook or open in the cloud.

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The Cornu spiral is a plot in the complex plane of the points B(t)=S(t)+iC(t), (1) where S(t) and C(t) are the Fresnel integrals (von Seggern 2007, p. 210; Gray 1997, p. 65). The Cornu spiral is also known as the clothoid or Euler's spiral. It was probably first studied by Johann Bernoulli around 1696 (Bernoulli 1967, pp. 1084-1086). A Cornu spiral describes diffraction from the edge of a half-plane. The quantities C(t)/S(t) and S(t)/C(t) are plotted above. The slope of the curve's...

A helix, sometimes also called a coil, is a curve for which the tangent makes a constant angle with a fixed line. The shortest path between two points on a cylinder (one not directly above the other) is a fractional turn of a helix, as can be seen by cutting the cylinder along one of its sides, flattening it out, and noting that a straight line connecting the points becomes helical upon re-wrapping (Steinhaus 1999, p. 229). It is for this reason that squirrels chasing one another up and...

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Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

But what kinds of shapes can folding produce? The pictures below show what happens when the local curvature -- which is... – from A New Kind of Science

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.