# Naming and Classification of Curves

## The Naming of Curves

Here are some examples of how curves got named. The list is not exhaustive.

**By Person's names**: quadratrix of Hippias, conchoid of Nicomedes, Kappa curve, limacon of Pascal, lemniscate of Bernoulli, witch of Agnesi, Folium of Descartes, trisectrix of Maclaurin, trident of Newton (aka parabola of Descartes), Tschirnhausen's cubic (aka trisectrix of Catalan, L'Hospital's cubic), Cayley's sextic, nephroid of Freeth,
Euler Spiral,
Cornu Spiral,
Euler's curve (x^y==y^x), Durer's conchoid,
Bowditch curve.

Note that in general, the name attached to a curve or a math theorem is not necessary the person who invented or worked on it.

**By method of construction**: caustics, derivative, integral, isoptic, orthoptic, parallel, pedal, radial, envelope, evolute, involute, cissoid, conchoid, roulette, glissette, strophoid, pursuit curve.

**By property**:
Conic Sections, trisectrix, quadratrix, brachistochrone (aka cycloid), tautochrone (aka cycloid), isochrone (aka semi-cubic parabola, semi-cubical parabola), anallagmatic curve.

**By shape**: astroid (star), deltoid (greek letter Delta), cardioid (hear-shaped), conchoid of Nicomedes (mussel-shaped), nephroid (kidney-shaped), cycloid (circle, wheel), folia (leaf), trident of Newton, serpentine (snake), cissoid of Diocles (Ivy-shaped), rose (aka rhodonea), catenary (chain-shaped) (aka chainette, alysoid), tractrix, spiral (coils), lemniscate of Bernoulli (ribbon-shaped), lituus, figure eight curve, bullet nose, cross curve.

**Historical reasons**:
circle, line, spiral, ellipse, parabola, hyperbola, witch of Agnesi, limacon of Pascal (snail of Pascal, originated from Roberval).

**By the form of the formula**: semi-cubic parabola, Parabola of Descartes.

Etymology to search: Right strophoid (Barrow, 1670), Kampyle of Eudoxus, Kappa curve (Gutschoven's curve), Hippopede (horse fetter) (Proclus, ~75 BC), bicorn (Sylvester, 1864), piriform (pear-shaped quartic) (De Longchamps, 1886), Clothoid, cochleoid (bernoulli, 1726), Cayley's sextic, Devil's curve (Cramer, 1750).

### Other Constructions and Names

**Brachistochrone** (from Greek, brakhus:short, chrone:time) means a curve that connects to two given points such that a particle sliding from the higher point to the lower point under ideal physical law (ideal gravitational force, no friction, no air-resistance, particle has no volume …etc.) will descent with the fastest time, among all possible curves. The unique solution is the cycloid, not straight line.

**Tautochrone** (Greek, tauto:same, chrone:time) is similar to brachistochrone. It's a curve that, connects two given points such that it takes the same amount of time for a particle to slide from any point on the curve to the lower point, under ideal physical law. This may sound paradoxical. Intuitively, the longer path the particle needs to glide, the longer time it will have to take. However, consider that acceleration rate is high when the slope is sharp, therefore it is possible for a curve whose curvature balances out so that no matter where the particle starts, it always takes the same time to reach the end. The unique solution is the cycloid.

The word **Isochrone** (Greek, iso:equal, chrone:time) is sometimes used to mean tautochrone, but other authors use it to mean the semi-cubic parabola, which is a curve which a particle will descend equal vertical distances in equal time intervals, under ideal physical law. The latter meaning will be used here.

**Anallagmatic curve** is any curve that has inversion that's itself. Circle and Cassinian oval are examples.

**Isoptic** of a given curve C and a given angle α is the locus of a point P such that P is the intersection of tangents of C that meets in angle α. If α:=π/2, then the derived curve is called a **orthoptic**. For example, the orthoptic of a parabola is its directrix, and the orthoptic of a ellipse, hyperbola, or deltoid is a circle.

## Classification of Curves

There are many ways to classify curves. One way is to determine
whether a curve is the graph of some polynomial equation
p[x,y]==0. The graph of a polynomial equation are called algebraic
curves.
Algebraic curve are assigned a *order*. The order of an
algebraic curve is the degree of the polynomial.
For example,
line
(a x + b y + c == 0);
circle
((x+h)^2 + (y+k)^2 -r^2 == 0), or
the deltoid ((x^2+y^2)^2 - 8
a x (x^2 - 3 y^2) + 18 a^2 (x^2 + y^2) - 27 a^4 == 0), are algebraic
curves. Curves may be easy to trace but are not algebraic. For
example, no polynomial's graph can be any
of cycloid, equiangular
spiral,
or quadratrix
of Hippias. Algebraic curves with degree greater than 2 are called
higher plane curves. Non-algebraic curves are called transcendental
curves.

There is a special class of curves known as fractal and space-filling curves. Simply, fractal curves are curves that are not smooth. All the curves covered here are such that when you keep magnifying parts of the curve, it'll eventually looks like a line, unless you are magnifying a cusp point. Fractal curves are such that no matter how large you magnify, the curve is still corrugated. In a sense, it's all cusps! Space filling curves are a special class of fractal curves, so named because they completely fill a area. Fractal and space-filling curves are discovered in late 17th century and their existence shocked mathematicians profoundly. They are interesting as a topic but this project will not deal with fractal or space-filling curves.

### Curve Family Tree

The following is rough grouping of curve family, shown as a nested list. Curves in deeper level are special cases of their parent. The phrases in parenthesis are reminders about how the curve is defined.

Not all curves in this dictionary shows here. A curve may appear under multiple parents. For example, ellipse can be considered as a special case of hypotrochoid, conic sections, lissajous, roulette.

A curve family can be defined as a particular formula (For example, the Archimedean Spiral with r==θ^n), or as a algebraic equation (For example, Conic Sections as second degree polynomials), or as a method of generation (For example, Hypotrochoid as rolling circles), or other special definitions (For example, spirals as increasing distance around a point, Cassinian Ovals as product of distances, ellipses as sum of distances, conic sections as sections of a cone.). So, there is no one absolute way to group them into families.

- roulettes (curve rolling on curve)
- cycloidal curves (circle rolling on circle/line)
- epitrochoids (circle rolling outside another circle)
- circle
- epicycloids (tracing point on circle)
- roses (r==Cos[p/q*θ])
- circle
- trifolium (3 petals)
- quadrifolium (4 petals)

- hypotrochoids (circle inside a circle. Either circle may be the rolling circle.)
- trochoids (circle rolling on a line)

- epitrochoids (circle rolling outside another circle)

- cycloidal curves (circle rolling on circle/line)
- conic sections (r==e/(1+e*Cos[θ]))
- e>1, hyperbolas
- e==1, parabola
- e<1, ellipses

- Cassinian ovals (locus of points whoes product of distances to two fixed points is a constant)
- lemniscate of Bernoulli (two loops with a double point in center)

- conchoids (shifting the curve by a constant to a point on radial line from a pole.)
- limacon of Pascal (conchoids of a circle)
- Conchoids of Nicomedes (conchoids of a line)

- cissoids (distance of two curves by a radial line from a pole)
- cissoid of a Line and a Circle
- oblique cissoid
- strophoid

- circle
- line
- conchoid of Nicomedes
- lemniscate of Bernoulli

- cissoid of a Line and a Circle
- spirals
- Archimedean spiral (r == θ^n)
- (n:=1), Archimedes' spiral
- (n:=1/2), parabolic spiral (aka Fermat's spiral)
- (n:=-1), hyperbolic spiral (aka reciprocal spiral)
- (n:=-1/2), lituus
- (n:=-0), circle

- equiangular spiral (r == E^(θ*Cot[α]))
- (α:=π/2), circle

- Euler's Spiral (aka Clothoid, Cornu's spiral)

- Archimedean spiral (r == θ^n)
- sinusoidal “spirals” (r^n==a^n*Cos[n*θ], n rational)
- (n:=-2), rectangular hyperbola
- (n:=-1), line
- (n:=-1/2), parabola
- (n:=-1/3), Tschirnhausen's cubic
- (n:=1/3), Cayley's Sextic
- (n:=1/2), cardioid
- (n:=1), circle
- (n:=2), lemniscate of Bernoulli

- lissajous

Note: “Sinusoidal spiral” is defined to be r^n==a^n*Cos[n*θ], n rational. It is not really a spiral. It is called a spiral because the polar equation has the form of a spiral but with the radius vector increases and decreases “sinuously”.

## Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.

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