Curve encyclopedia
Math curve encyclopedia
30 classic curves, each with an interactive parameter preview, its equation in LaTeX, key properties, and a one-click path to a Manim animation. Drag the sliders to feel how each parameter reshapes the curve.
Parametric curves
Lissajous Curve
The figure drawn by two perpendicular oscillations x = sin(at + δ), y = sin(bt) — the oscilloscope’s signature art.
Superellipse
The Lamé curve |x/a|ⁿ + |y/b|ⁿ = 1 that interpolates between ellipse and rectangle — the geometry of the squircle.
Butterfly Curve
Temple Fay’s 1989 transcendental curve whose sine and exponential terms flutter into a butterfly.
Heart Curve
The parametric valentine x = 16sin³t with a four-term cosine partner — mathematics’ most shared curve.
Polar curves
Rose Curve
The polar curve r = a·cos(kθ) that blooms into k or 2k petals depending on whether k is odd or even.
Cardioid
The heart-shaped curve r = a(1 + cos θ) traced by a point on a circle rolling around an equal circle.
Limaçon
The family r = b + a·cos θ that morphs from an inner-loop snail through the cardioid to a convex oval as b grows.
Lemniscate of Bernoulli
The figure-eight curve r² = a²·cos 2θ — the set of points whose distances to two foci multiply to a constant.
Cissoid of Diocles
The cusped curve r = 2a·sin θ·tan θ invented around 180 BC to solve the ancient problem of doubling the cube.
Rolling curves (roulettes)
Cycloid
The arch traced by a point on a rolling wheel — solution of both the brachistochrone and tautochrone problems.
Epicycloid
The flower of cusps traced by a circle rolling outside a fixed circle; k = R/r sets the cusp count.
Hypocycloid
The star traced by a circle rolling inside a fixed circle; k = 3 gives the deltoid, k = 4 the astroid.
Hypotrochoid
The spirograph curve: a pen fixed at distance d from the center of a circle rolling inside another.
Epitrochoid
The outer-rolling trochoid family whose two-lobed member shapes the Wankel rotary engine housing.
Astroid
The four-cusped star x^{2/3} + y^{2/3} = a^{2/3}, also the envelope of a sliding ladder.
Deltoid
The three-cusped hypocycloid, home of Steiner’s theorem: all Simson lines of a triangle envelope a deltoid.
Nephroid
The two-cusped epicycloid — the bright caustic you see when sunlight reflects inside a cup.
Spirals
Archimedean Spiral
The constant-pitch spiral r = a + bθ, whose successive turns stay exactly 2πb apart.
Logarithmic Spiral
The self-similar spiral r = a·e^{bθ} that crosses every radius at the same angle — nature’s growth curve.
Fermat Spiral
The two-armed spiral r² = a²θ whose coils pack tighter as they grow — the pattern behind sunflower seed heads.
Golden Spiral
The logarithmic spiral that widens by the golden ratio φ every quarter turn, approximated by the Fibonacci arc construction.
Involute of a Circle
The path of a string end unwinding from a circle — the profile that makes modern gear teeth run smoothly.
Conic sections
Parabola
The set of points equidistant from a focus and a directrix — the shape of projectile paths and satellite dishes.
Ellipse
The locus with a constant sum of distances to two foci — the true shape of planetary orbits.
Hyperbola
The two-branched conic with a constant difference of focal distances, hugging its asymptotes forever.
Classic Cartesian curves
Sine Wave
The graph y = A·sin(ωx) of pure oscillation — the atom every periodic signal is built from.
Catenary
The hyperbolic-cosine curve y = a·cosh(x/a) of a hanging chain — subtly different from a parabola.
Tractrix
The “drag curve” with a constant-length tangent — rotate it and you get the pseudosphere of hyperbolic geometry.
Witch of Agnesi
The bell-shaped cubic y = 8a³/(x² + 4a²), misnamed “witch” by a translation slip, and the Cauchy distribution’s profile.
Folium of Descartes
The looped cubic x³ + y³ = 3axy that sparked the Descartes–Fermat duel over tangent lines.