Implicit Bouquet

Implicit curves — defined as zero sets f(x, y) = 0 — are the natural language for algebraic geometry over the real plane. Many classical shapes are most cleanly described this way: folia, lemniscates, astroids, cardioids, and conics all have compact polynomial defining equations but resist neat expression as y = f(x). Plotting them via marching-squares or similar methods avoids the branch-handling that explicit form would require.

The bouquet combines three classical implicit curves on one equal-aspect plane: x^3 + y^3 - 3xy = 0 (the folium of Descartes, first studied by Descartes in 1638), (x^2 + y^2)^2 - 2.5(x^2 - y^2) = 0 (a Bernoulli lemniscate), and |x|^{2/3} + |y|^{2/3} - 1 = 0 (the astroid traced by a small circle rolling inside a four-times-larger one). Overlaying them shows how loops, cusps, and symmetric petals coexist in the same coordinate system while each defining equation remains in its natural algebraic form.