Implicit Equations

In mathematics, an implicit equation is a relation of the form R(x₁,...,x_n) = 0, where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x² + y² - 1 = 0.

Examples

Given an X/Y equation (using the infix library):

(use 'implicit-equations.plot)
(use 'infix.macros)

(defn quadrifolium [x y]
  (infix (x ** 2 + y ** 2) ** 3 - x ** 2 * y ** 2))

By specifying some bounds, we can now render to PNG images, with the draw command - this renders onto a BufferedImage when the equation crosses zero:

(draw quadrifolium "graph.png" {:bounds 1 :line-width 4})

to produce:

Other equations produce the following plots:

Knot Curve

(defn knot-curve [x y]
  (infix (x ** 2 - 1) ** 2 - y ** 2 . (3 + 2 . y)))

Biology

(defn biology [x y]
  (infix sin(sin x + cos y) - cos(sin(x . y) + cos x)))

Chain-mesh

(defn chain-mesh [x y]
  (infix sin((x ** 2) + (y ** 2)) - cos(x . y)))

Checkerboard

(defn checkerboard [x y]
  (infix exp(sin x + cos y) - sin(exp(x + y))))

Dizzy

(defn dizzy [x y]
  (infix abs(sin(x ** 2 - y ** 2)) - (sin(x + y) + cos(x . y))))

Glint

(defn glint [x y]
  (infix abs(sin(x ** 2 + 2 . x . y)) - sin(x - 2 . y)))

Spira

(defn spira [x y]
  (infix sin(x ** 2 + y ** 2) - sin(x ÷ y ** 2))

Bands

(defn bands [x y]
  (infix sin(1 - x ** 2) . sin(2 - y ** 2) . x . y - cos (2 - y ** 2)))

Refined from: http://mathematica.stackexchange.com/questions/19590/what-is-a-good-way-to-plot-some-difficult-implicit-equations

References

• http://www.xamuel.com/graphs-of-implicit-equations/
• http://www.peda.com/grafeq/
• http://www.padowan.dk/
• http://doc.sagemath.org/html/en/reference/plot3d/sage/plot/plot3d/implicit_plot3d.html
• http://mathematica.stackexchange.com/questions/69421/plot-of-implicit-equations
• http://wmueller.com/precalculus/newfunc/4.html
• https://en.wikipedia.org/wiki/Implicit_curve
• http://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html
• http://www.implicitcad.org/
• https://christopherolah.wordpress.com/2011/11/06/manipulation-of-implicit-functions-with-an-eye-on-cad/
• https://en.wikipedia.org/wiki/Marching_squares

License

The MIT License (MIT)

Copyright (c) 2016 Richard Hull

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.