Implicit Heart

An implicit curve is the zero set of an equation f(x, y) = 0 — the set of points where the equation holds, without solving for y explicitly as a function of x. This formulation captures shapes that would require multiple branches or piecewise definitions in explicit form, including self-intersections, loops, and isolated cusps. Implicit plotting is the workhorse behind level-set methods, computational geometry, and many curve-fitting and shape-recognition algorithms.

The featured equation is (x^2 + y^2 - 1)^3 - x^2 y^3 = 0, a compact sixth-degree polynomial whose zero contour traces the familiar heart silhouette. Plotted in equal aspect on a slightly taller-than-wide domain, the curve resolves cleanly without any y = \pm\sqrt{\dots} branch handling. The downward cusp at the bottom and the two rounded lobes at the top fall out directly from the algebra; no piecewise stitching is needed.