Plotting implicit equations
An implicit equation relates x and y without putting one alone on the left side. The classic example is the equation of a circle: x² + y² = 25. There's no easy way to write y = … here without breaking it into two pieces (positive and negative roots). DesmosGraph plots implicit equations directly, in one go, no manual splitting required.
Why implicit?
Some curves simply cannot be expressed as a single function y = f(x). Vertical lines, circles, ellipses, hyperbolas, and most algebraic curves require the implicit form. Even when an explicit form exists, the implicit form is often more concise and reveals symmetry better.
Your first implicit plot
In the graphing calculator, type:
x^2 + y^2 = 25
You see a circle of radius 5 centred at the origin. Easy. Now try:
x^2 / 9 + y^2 / 4 = 1 (ellipse) x^2 / 4 - y^2 / 9 = 1 (hyperbola) y^2 = 4*x (sideways parabola)
How it works under the hood
To draw F(x, y) = 0, we sample F over a grid and use the marching squares algorithm to find where the function crosses zero. The result is a fast, accurate piecewise-linear approximation of the curve. The key advantage: it works for any expression, no matter how exotic.
Famous implicit curves to try
- Heart:
(x² + y² − 1)³ = x²·y³ - Lemniscate of Bernoulli:
(x² + y²)² = 2(x² − y²) - Folium of Descartes:
x³ + y³ = 3xy - Astroid:
x^(2/3) + y^(2/3) = 1 - Cassini ovals:
(x² + y² + a²)² − 4a²x² = b⁴
Equations vs. inequalities
Replacing = with <, >, ≤ or ≥ gives an inequality, which DesmosGraph fills as a region. For example:
y < sin(x) x^2 + y^2 <= 9
Inequalities are great for showing solution sets in linear programming, feasible regions, or just artistic shading.
Common gotchas
- Self-intersections like the folium are fine; the plotter handles crossings naturally.
- Tangent points (where the curve touches but doesn't cross zero) can occasionally be missed if your view zoom is far out. Zoom in and the resolution increases.
- Vertical asymptotes are best treated by re-arranging algebraically before plotting if you need extreme precision.
What's next?
Check out the sliders & animation tutorial to make your implicit curves come alive with parameters, or browse the examples gallery for inspiration.