The butterfly curve is a
transcendental plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
discovered by Temple H. Fay of
University of Southern Mississippi
The University of Southern Mississippi (Southern Miss or USM) is a public research university with its main campus located in Hattiesburg, Mississippi. It is accredited by the Southern Association of Colleges and Schools to award bachelor's, ma ...
in 1989.
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Equation
The curve is given by the following
parametric equations:
:
:
:
or by the following
polar equation
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
:
:
The term has been added for purely aesthetic reasons, to make the butterfly appear fuller and more pleasing to the eye.
Developments
In 2006, two mathematicians using
Mathematica analyzed the function, and found variants where leaves, flowers or other insects became apparent.
See also
https://books.google.com/books?id=AsYaCgAAQBAJ&dq=OSCAR+RAMIREZ+POLAR+EQUATION&pg=PA732
*
Butterfly curve (algebraic)
In mathematics, the algebraic butterfly curve is a plane algebraic curve of degree six, given by the equation
:x^6 + y^6 = x^2.
The butterfly curve has a single singularity with delta invariant three, which means it is a curve of genus se ...
Oscar’s Butterfly Polar Equation''r'' = (cos 5θ)
2 + sin 3θ + 0.3 for 0 ≤ θ ≤ 6π
(A polar equation discovered by Oscar Ramirez, a UCLA student, in the fall of 1991.)
References
External links
Butterfly Curve plotted in WolframAlpha
Plane curves
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