The following curve was recently found by Roman Hašek and Jan Zahradník at the University of South Bohemia. When they read a 18th century book, Ioannis Holfeld's [url=http://home.pf.jcu.cz/~hasek/Holfeld/]Exercitationes Geometricae[/url], it turned out that a geometric construction problem has an extra solution, namely a formerly unknown "pretzel" curve.[br]Ioannis Holfeld's original text contains only a description of a parabola which is algebraically a quadratic equation in variables [i]x[/i] and [i]y[/i]. By contrast, this pretzel curve has a quartic equation.[br]

Let [i][color=#0000ff]M[/color][/i] be a free point. Its distance from the origin will be the radius of a circle with its center in the origin. Let [i][color=#1155cc]B[/color][/i] be an arbitrary point on the circle and let [i]E[/i] be the intersection of the [i]y[/i]-axis and the line [i][color=#0000ff]M[/color][color=#1155cc]B[/color][/i]. Let [i][color=#ff7700]C[/color][/i] be the intersection point of the ray from the origin through [i][color=#1155cc]B[/color][/i], and a perpendicular line to the [i]y[/i]-axis in [i]E[/i]. Finally, let [i][color=#ff0000]C'[/color][/i] be the mirror of [i][color=#ff7700]C[/color][/i] about point [i][color=#1155cc]B[/color][/i]. Now the locus of the tracer point [i][color=#ff0000]C'[/color][/i] (while [i][color=#1155cc]B[/color][/i] is moving on the circle) will be the so-called pretzel curve. (Ioannis Holfeld's parabola is defined as the locus of [i][color=#ff7700]C[/color][/i].)[br]GeoGebra has already support to compute the algebraic equation of a locus curve. This is also possible in its web version, thanks to Bernard Parisse's [url=https://www-fourier.ujf-grenoble.fr/~parisse/giac.html]Giac[/url] computer algebra system which effectively computes [url=http://www.scholarpedia.org/article/Groebner_basis]Gröbner bases[/url] in the background, and the [url=https://en.wikipedia.org/wiki/Emscripten]emscripten[/url] compiler which translates the C++ code into JavaScript. In the figure, however, an extra component is computed, namely the [i]x[/i]-axis as an extra algebraic factor [i]y[/i](=0).[br]Note: There are different kinds of pretzels. This one is similar to [url=http://altonbrown.com/homemade-soft-pretzels-recipe/]Alton Brown's homemade soft pretzel[/url].