|
|
| Line 1: |
Line 1: |
| {{refimprove|date=August 2012}}
| | I am Oscar and I completely dig that title. South Dakota is where me and my spouse live. Doing ceramics is what love performing. Bookkeeping is my occupation.<br><br>Feel free to visit my web site - [http://glskating.com/groups/stay-yeast-infection-free-with-these-helpful-suggestions/ glskating.com] |
| [[Image:Butterfly curve.png|thumb|<math>x^6 + y^6 = x^2</math>]]
| |
| [[Image:Butterlink.png|thumb|The butterfly curve singularity link]]
| |
| In [[mathematics]], the algebraic '''butterfly curve''' is a [[algebraic curve|plane algebraic curve]] of [[sextic plane curve|degree six]], given by the [[equation]]
| |
| | |
| :<math>x^6 + y^6 = x^2.\,\!</math>
| |
| | |
| The butterfly curve has a single [[Mathematical singularity|singularity]] with [[delta invariant]] three, which means it is a curve of [[geometric genus|genus]] seven. The only plane curves of genus seven are singular, since seven is not a [[triangular number]], and the minimum degree for such a curve is six, so the butterfly curve aside from its appearance is possibly interesting as an example.
| |
| | |
| The butterfly curve has branching number and multiplicity two, and hence the singularity [[link (knot theory)|link]] has two components, pictured at right.
| |
| | |
| The area of the algebraic butterfly curve is given by (with [[gamma function]] <math>\Gamma</math>)
| |
| :<math>4 \cdot \int_0^1 (x^2 - x^6)^{\frac{1}{6}} dx = \frac{ \Gamma(\frac{1}{6}) \cdot \Gamma(\frac{1}{3})}{3 \sqrt{\pi}} \approx 2.804,</math>
| |
| and its [[arc length]] ''s'' by
| |
| :<math>s \approx 9.017.</math> | |
| | |
| ==See also==
| |
| * [[Butterfly curve (transcendental)]]
| |
| | |
| == References ==
| |
| <references/>
| |
| *{{MathWorld|title=Butterfly Curve|id=ButterflyCurve}}
| |
| | |
| == External links ==
| |
| *{{OEIS2C|A118292}} Sequence for the area of algebraic butterfly curve
| |
| *{{OEIS2C|A118811}} Sequence for the arc length of algebraic butterfly curve
| |
| | |
| [[Category:Curves]]
| |
| {{geometry-stub}}
| |
I am Oscar and I completely dig that title. South Dakota is where me and my spouse live. Doing ceramics is what love performing. Bookkeeping is my occupation.
Feel free to visit my web site - glskating.com