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Five curves people search for, each with its real, correct equation, explained and graphed. These are illustrative examples we built by hand — not a live user's drawing or a Graphiti solve result.
Examples for reference — not live app output
Circle
x² + y² = r², shown here as x² + y² = 9
A circle is the set of all points a fixed distance r (the radius) from a center point. When the center sits at the origin, the equation reduces to x² + y² = r²; for a center at (h, k) it's (x − h)² + (y − k)² = r². This example uses r = 3, so r² = 9.
Also written parametrically as x = r cos(t), y = r sin(t) for t from 0 to 2π.
Heart curve
x = 16sin³(t)
y = 13cos(t) − 5cos(2t) − 2cos(3t) − cos(4t)
There's no single tidy y = f(x) for a heart — the outline crosses itself in x, so it needs a parametric equation: separate formulas for x and y driven by a shared parameter t running from 0 to 2π. This classic combination of sine and cosine terms is one of the most common ways to trace a heart shape exactly.
A rougher approximation, |x|^(2/3) + |y|^(2/3) = 1-style curves, gets you a heart-adjacent shape without parametrization — but loses the sharp bottom point.
Parabola
y = ax² + bx + c, shown here as y = 0.15x²
A parabola is the graph of any quadratic function. In vertex form, y = a(x − h)² + k, the point (h, k) is the vertex — here it's the origin, (0, 0). The sign of a decides which way it opens: positive opens upward (a cup), negative opens downward (a dome). Making a smaller — like 0.15 here — widens the curve; making it larger narrows it.
Formally, a parabola is also the set of points equidistant from a fixed point (the focus) and a line (the directrix). For this example, that focus sits at (0, 1.67).
Spiral
r = aθ, shown here as r = 0.4θ
The simplest spiral, called the Archimedean spiral, is easiest to write in polar coordinates: the radius r grows in direct proportion to the angle θ. Each full turn (θ increasing by 2π) pushes the radius out by the same fixed amount, 2πa — which is why the coils here are evenly spaced rather than tightening or loosening.
In Cartesian form: x = aθ·cos(θ), y = aθ·sin(θ).
Sine wave
y = A sin(kx + φ) + d, shown here as y = 2sin(x)
Four numbers describe any sine wave: A, the amplitude (how tall the wave is — 2 here); k, which sets the period at 2π/k (1 here, so the period is 2π); φ (phi), a phase shift that slides the wave left or right (0 here); and d, a vertical offset that shifts the whole wave up or down (0 here).
This same shape underlies sound waves, AC electrical current, and pendulum motion — it's the building block of a huge share of periodic phenomena.
These five are illustrations. Graphiti searches across all of these families (and combinations of them) to find the closest real match to whatever you actually draw.
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