Visualize implicit equations f(x,y)=0 using marching squares. Supports circles, conics, and arbitrary implicit relations.
An implicit curve is traced as the set of locations where an entered expression in x and y becomes zero. The plotter samples that scalar field across the displayed coordinate region and uses marching squares to join zero-crossing cells into visible curve segments. This lets it show closed loops, vertical tangents, and several branches without rewriting the relation as y = g(x). The main interpretive trap is treating the picture as a surface plot: it displays only the zero level, not the value or height of f elsewhere.
A concrete input and expected output from the current implementation.
Input
x*x + y*y - 4
Expected output
A numerical zero-contour approximating a circle centered at (0, 0) with radius 2 and axis crossings at (-2, 0), (2, 0), (0, -2), and (0, 2).
Setting the expression to zero gives x*x + y*y = 4, so every point on the curve is distance sqrt(4) = 2 from the origin. At each listed crossing, the squared coordinates sum to 4.
Marching squares examines each rectangular grid cell using the signs of f at its four corners. A cell whose corner classifications differ can contain the zero contour. The method selects a local segment pattern for that sign configuration, and segments from adjacent cells combine into the displayed trace. The resulting line is a numerical contour, not a symbolic solution of the equation.
The entered expression represents f(x,y), with the requested curve defined by f(x,y) = 0. Writing all terms on one side turns a relation L = R into the zero expression L - R. Because x and y remain peers, one x-coordinate may correspond to several y-coordinates. Closed curves, vertical portions, and disconnected conic branches can therefore appear in one plot.
Yes. That is the purpose of an implicit plot: x and y do not need to be separated. Enter an expression whose zero set is the relation, such as x*x + y*y - 4 for a radius-2 circle. The result is a sampled contour, not an algebraic parametrization or proof that every component was found.
Marching squares makes local decisions from the samples in the current grid. A narrow loop, tangency, isolated zero, or pair of nearby branches may fall between samples. Changing the displayed region can change those samples and the apparent connections. The exact equation has not changed, and a blank plot does not prove that its real zero set is empty.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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