Discover what algebraic curves are: the zero sets of polynomial equations. Explore lines, conics, and beautiful higher-degree curves.
An algebraic curve is the set of all points (x, y) that satisfy a polynomial equation f(x, y) = 0. This simple definition leads to an incredibly rich theory connecting algebra and geometry.
From straight lines to exotic curves with loops and cusps, every algebraic curve tells a story about its defining polynomial. The degree of the polynomial determines the curve's fundamental properties.
Adjust the coefficients of a general degree-2 polynomial and watch the curve change in real-time. Try the presets to see familiar shapes emerge.
Adjust the sliders to change coefficients. The curve shows all points (x, y) where the polynomial equals zero.
Explore a collection of famous algebraic curves. Click on each to see its equation and discover its unique properties.
The fundamental circle with radius 1 centered at the origin
Click on different curves to explore their shapes. Notice how higher-degree curves have more complex features.
The degree of a curve is the highest total power of x and y in any term. Higher degree means more complex possible shapes.
You've learned the essential concepts of algebraic curves:
Next: We'll dive deep into degree-2 curves (conics) and discover why circles, ellipses, parabolas, and hyperbolas are all related — they're all slices of a cone!