Curve encyclopedia
Math curve encyclopedia
Parametric curves
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Both coordinates follow one parameter: Lissajous figures, Bézier curves, hearts and butterflies.
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.
Bézier curve
The cubic curve behind fonts, SVG paths and CSS easing: two end points and two handles, blended by repeated linear interpolation.
Lemniscate of Gerono
The figure eight x⁴ = a²(x² − y²), drawn by x = a·cos t, y = a·sin t·cos t: two lobes of total area 4a²/3.
Polar curves
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The radius as a function of the angle: roses, cardioids, lemniscates and the classical curves of Greek geometry.
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.
Cassini oval
Points whose distances to two foci multiply to b²: one oval, a figure eight, or two separate loops.
Conchoid of Nicomedes
Slide a ruler through a fixed point and mark a fixed distance b beyond a line: the shell-shaped curve Nicomedes used to trisect angles.
Strophoid
The right strophoid y²(a − x) = x²(a + x): a loop through the origin and two arms reaching for the asymptote x = a.
Trisectrix of Maclaurin
The cubic 2x(x² + y²) = a(3x² − y²): a loop and an asymptote, studied by Maclaurin in 1742 for trisecting angles.
Cochleoid
r = a·sin θ / θ: a snail-shell spiral that passes through the pole at every multiple of π.
Superformula
Johan Gielis’s one formula for stars, flowers, squares and circles: four numbers reshape a single polar curve.
Maurer rose
Walk around the rose r = sin nθ in steps of d degrees and join the dots: 361 straight lines weave a rose.
Rolling curves (roulettes)
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Traced by a point on a rolling circle: cycloids, epicycloids, hypocycloids and spirograph patterns.
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.
Trochoid
A pen fixed at distance d from the centre of a rolling wheel: waves for d < r, the cycloid for d = r, loops for d > r.
Spirals
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Curves that wind around a point: Archimedean, logarithmic, golden, Fibonacci and Euler 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.
Hyperbolic spiral
r = a/θ: it winds in towards the pole forever while its far end runs along the line y = a.
Lituus
r²θ = a²: a shepherd’s-crook spiral named by Roger Cotes, whose sectors have equal area whenever their angles have equal ratio.
Euler spiral (clothoid)
The curve whose curvature grows in proportion to its length: the transition bend of every motorway and railway.
Spiral of Theodorus
Right triangles with legs √n and 1 stacked around a point: the hypotenuses are √2, √3, √4 … and the outer edge spirals out.
Fibonacci spiral
Quarter circles drawn in squares of side 1, 1, 2, 3, 5, 8 …: the popular approximation of the golden spiral.
Conic sections
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Sections of a cone: the parabola, the ellipse and the hyperbola.
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
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Curves given by an equation in x and y: the catenary, the witch of Agnesi, the sigmoid, and classical cubics and quartics.
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.
Devil’s curve
The quartic y²(y² − b²) = x²(x² − a²), studied by Cramer in 1750: a figure eight between two arcs, or a circle crossed by two lines.
Bicorn
The cocked-hat curve y²(a² − x²) = (x² + 2ay − a²)²: two arcs meeting in sharp cusps at (±a, 0).
Kampyle of Eudoxus
x⁴ = a²(x² + y²): two branches opening outward from (±a, 0), traditionally linked to Eudoxus and the doubling of the cube.
Piriform curve
The pear-shaped quartic a⁴y² = b²x³(2a − x): a teardrop with a cusp, enclosing exactly the area πab of an ellipse.
Serpentine curve
y = abx/(x² + a²): an S-bend that rises to b/2, falls back and fades to the x-axis on both sides.
Tschirnhausen cubic
27ay² = (a − x)(8a + x)²: a cubic with a loop, the caustic of a parabola lit at right angles to its axis.
Semicubical parabola
y² = ax³, the cusp curve: the first algebraic curve other than a straight line whose arc length was found exactly (Neile, 1657).
Quadratrix of Hippias
Where a falling line meets a turning radius: one of the oldest named curves, used to trisect angles and square the circle.
Sigmoid function
The logistic curve y = 1/(1 + e^(−kx)): growth that starts slow, speeds up, and levels off at 1.
Fractal curves
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Curves built by repeating one rule: the Koch snowflake, the dragon curve and the space-filling curves.
Koch snowflake
A triangle whose edges bend again and again: the perimeter grows without bound while the area stays finite.
Hilbert curve
A single path that visits every cell of a 2ⁿ × 2ⁿ grid; in the limit it passes through every point of the square.
Dragon curve
Fold a strip of paper in half n times and open every fold to a right angle: the edge draws the Heighway dragon.
Lévy C curve
Replace every segment by two legs of a right isosceles triangle, again and again: the segments bloom into a C.
Sierpiński arrowhead curve
One unbroken path whose limit is the Sierpiński triangle, drawn with nothing but 60° turns.
Peano curve
The first space-filling curve (1890): a snake through a 3ⁿ × 3ⁿ grid whose limit covers the whole square.