A pretzel curve
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 Exercitationes Geometricae, it turned out that a geometric construction problem has an extra solution, namely a formerly unknown "pretzel" curve.
Ioannis Holfeld's original text contains only a description of a parabola which is algebraically a quadratic equation in variables x and y. By contrast, this pretzel curve has a quartic equation.
Let M be a free point. Its distance from the origin will be the radius of a circle with its center in the origin. Let B be an arbitrary point on the circle and let E be the intersection of the y-axis and the line MB. Let C be the intersection point of the ray from the origin through B, and a perpendicular line to the y-axis in E. Finally, let C' be the mirror of C about point B. Now the locus of the tracer point C' (while B is moving on the circle) will be the so-called pretzel curve. (Ioannis Holfeld's parabola is defined as the locus of C.)
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 Giac computer algebra system which effectively computes Gröbner bases in the background, and the emscripten compiler which translates the C++ code into JavaScript. In the figure, however, an extra component is computed, namely the x-axis as an extra algebraic factor y(=0).
Note: There are different kinds of pretzels. This one is similar to Alton Brown's homemade soft pretzel.