Type: XY - Bell curve
,exp(-(x²)),7,-2,2,2,0.5,100,-2,2,4,2,2,
Type: XY - Catenary
,cosh(x),7,-2,2,2,0.5,100,-4,4,8,4,2,
Type: XY - Conic sections
,sqr((1 - a)x² + 2x + a + 1),5,0.0,2.0,2,0.5,100,-5,7,12,5,6,
Type: XY - Cubic
,ax³+bx²-c,7,-2,2,2,0.5,100,-3,3,6,3,3,
Type: XY - Exponential
,be^x + ae^-x,5,-2,2,0.5,2,100,-2,2,6,3,3,
Type: XY - Hyperbolic
,ax + b/x,5,-2,2,0.5,2,100,-2,2,6,3,3,
Type: XY - Inverse quadratic
,1/(x² + a),5,-2,2,-2,2,100,-2,2,6,3,3,
Type: XY - Neile's semi-cubical parabola
,ax²^(1/3),5,0,2,2,0.5,100,-2,2,4,2,1,
Type: XY - Quadratic
,ax²+bx+c,7,-2,2,2,0.5,100,-3,3,6,3,3,
Type: XY - Quadratrix of Hippias
,x/tan x,7,-2,2,2,0.5,100,-10,10,20,10,10,
Type: XY - Sin(ax)/x
,sin(ax)/x,6,0,5,-2,2,100,-5,5,10,5,3,
When x = 0 both the numerator and the denominator of the expression are zero. Nevertheless the expression has a perfecly respectable value at that point - namely a.
Type: XY - Trigonometric
,a sin x + b cos x,5,-2,2,1,2,100,-2,2,12,3,6,
Type: Polar - A weird family of curves
,(2cos θ - a) / (cos θ + a),5,0.0,2,1,2,100,0,2,8,3,4,
This family of curves all pass through the points (0,1) and (0,-1) with the apparent exception of the case a = 0.
The explanation of this anomaly is that when a = 0, the curve is undefined at θ = π/2 and - π/2 so it is not actually a complete circle.
Examine what happens when a = 0.01!
Type: Polar - Archimedean spiral
,θ/(2π),9,-1,3,-2,2,200,0,10,10,5,5,
In the Archimedean spiral, the radius increases by the same amount with each revolution.
Type: Polar - Cissoid of Diocles
,tan θ sin (θ+a),3,0.0,1.5,-2,2,100,0,1,4,2,2,
Type: Polar - Cochleoid
,sin(θ) / θ,1,1.1,3,-2,2,100,-3.0,3.0,4,2,2,
Type: Polar - Conchoid
,a + 1/cos θ,5,0,2.0,-2,2,100,-1,1,6,2,3,
Type: Polar - Double Folium
,4 (sin θ)² cos(aθ),5,1,2.0,-2,2,100,0,1,8,4,4,
Type: Polar - Fermat's spiral
,√(abs θ),1,1.1,3,-2,2,500,0,8,12,6,6,
Type: Polar - Folium of Descartes
,(3 sinθ cosθ)/((sinθ)³ + (cosθ)³),9,-1,3,-2,2,100,-0.5,0.5,6,3,3,
Type: Polar - hyperbolic spiral
,a/θ,5,1,5,-2,2,100,0,2,8,4,4,
Type: Polar - Kappa curve
,a/tan θ,5,1,5,-2,2,100,0,2,8,4,4,
Type: Polar - Lemniscate of Bernouilli
,√cos(aθ),1,2,5,-2,2,100,0,2,4,2,2,
Type: Polar - Limacon of Pascal
,b + 2a cos θ,4,-1,3,-2,2,100,0,2,12,6,6,
Type: Polar - Lituus (Bishop's crook curve)
,a/√θ,5,1,5,-2,2,100,0,2,8,4,4,
Type: Polar - Logarithmic (equiangular) spiral
,b^θ,9,-1,3,1.1,2,200,0,10.00,20,10,10,
This spiral stays the same shape as it grows. It is often found in nature. It also gives the impression that you are looking down a long tunnel
Type: Polar - Propellers
,(cos(aθ))²,4,0.5,2,-2,2,100,0,2,4,2,2,
Type: Polar - Quartic hyperbolae
,(a + 1) / (a - (cos θ)²),5,0.0,2.0,-2,2,100,0,2,12,6,6,
Type: Polar - Rhodonea curves
,cos(aθ),3,1.0,3,-2,2,100,0,2,4,2,2,
Type: Polar - Right strophoid
,cos(aθ) / cos θ,3,1.0,3,-2,2,100,0,2,4,2,2,
Type: Polar - Trisectrix
,1 + a cosθ,5,1.0,2.5,-2,2,100,0.0,2.0,6,2,3,
Type: XY parametric - Archimedean spiral (parametric)
at cos(bt),at sin(bt),1,0.5,2,10,9,1000,0,5,6,3,3,
Type: XY parametric - Astroids
sgn cos t ×(abs cos t)^a,sgn sin t ×(abs sin t )^a,11,0,5,10,9,100,-3.3,3.3,4,2,2,
The sgn and abs convolutions are only necessary to complete all the quadrants.
The basic formula is simply:
          X  =  (cos t) ^ a
          Y  =  (sin t) ^ a
Type: XY parametric - Binet's formula
(c^t - cos(πt)×c^-t)/sqr5,(sin(πt)×c^-t)/sqr5,1,-10,10,-2,1.618,1000,-10,10,6,3,3,
This is the complex version of Binet's formula for the nth Fibonacci number:
    (p^n - (-p)^-n) / √5
where p is the golden ratio (1 + √5)/2
Type: XY parametric - Cubic
t,t³,1,0.5,2,10,9,1000,-3,3,8,4,4,
Type: XY parametric - Cycloid
t - a sin(t),1 - a cos(t),5,0,2,1,2,100,-5,5,10,5,5,
Consider a disc of unit radius rolling along a straight line. The cycloid is the locus of a point at a distance of a from the disc
Type: XY parametric - Ellipses
a  sin t,(4-a)cos t,7,0.0,3,-2,2,100,-3.3,3.3,8,4,4,
Type: XY parametric - Ellipses
a cos(t),sin(t),5,0.5,2,1,9,100,-3.2,3.2,6,3,3,
Type: XY parametric - Epicyclic curves
cos(t) + a cos(bt),sin(t) + ac sin(bt),1,0.333,0.8,3,1,300,-10,10,4,2,2,
a is the angle between the curve and the circle through the same point.
Type: XY parametric - Folium of Desctartes
3t/(1+t³),3t²/(1+t³),1,0.5,2,10,9,200,-5,5,6,3,3,
Type: XY parametric - Hyperbolae
a  / cost,(4-a)tan t,7,0.0,3,-2,2,100,-3.3,3.3,8,4,4,
Type: XY parametric - Iterative ellipse
x - y/b,y + x/c,1,0.5,2,50,100,1000,0,5,6,3,3,
Type: XY parametric - Lissajou's figures
2cos(bt + aπ),2sin t,5,0,1,2,2,100,-3.3,3.3,6,3,3,
b determines the ratio of the frequencies
a determines the phase difference at the start
Type: XY parametric - Nephroid
cos(t) + a cos(bt),sin(t) + a sin(bt),3,0.5,1.5,3.0,9,100,-3.2,3.2,6,3,3,
Type: XY parametric - Nicomedes conchoid
a tan(t) + sin(t),cos(t),5,-1.0,0.0,1,2,100,-1.6,1.6,4,2,2,
Type: XY parametric - Piriform
1 + cos(t),a sin(t) ( 1 + cos t),5,0.5,2,1,9,100,-3.2,3.2,6,3,3,
Type: XY parametric - Semi-cubic parabola
t³,at²,5,0.5,2,10,9,200,-5,5,6,3,3,
Type: XY parametric - Spirograph figures (Epicycloids)
(1 + a)cos(t) + ab cos(t(1 + a)/a),(1 + a)sin(t) + ab sin(t(1 + a)/a),9,-0.4,0.4,0.7,9,300,-10,10,4,2,2,
Sirograph figures are formed when a circle of radius a rolls round (inside or outside) a unit circle.

b is the distance of the penpoint from the centre of the moving circle (expressed as a fraction of the radius).
Type: Polar parametric - Infinity symbol
sin(at),cos t,2,1.0,2.0,-2,2,100,-3.2,3.2,4,2,2,
Type: Functional - A family of hyperbolae
,xy = ay² + 1,5,0.5,2,2,2,100,-5,5,10,5,5,
Type: Functional - Bicorn (cocked hat)
,(x² + 2ay - a)² = (a² - x²)y²,5,0.5,2,2,2,100,-3,3,6,3,3,
Type: Functional - Cubic equations (1a - simple wiggle)
,y = bx³ + cx²  - ax,5,-2,2,1,1,100,-3,3,6,3,3,
Type 1 curves have only one infinite value.
Type 1a has a single wiggle.
It is monotonic and has no assymptotes.
Type: Functional - Cubic equations (1b - assymptotic wiggle)
,y³ + y  = x³ - bx - a,5,-2,2,1,1,100,-3,3,6,3,3,
Type 1 curves have only one infinite value.
This one is monotonic.
The presence of a y³ term means that the curves are assymptotic to the line y = x.
Type: Functional - Cubic equations (1c - bell-shaped curve)
,x²y  + by = a,5,0,2,0.5,1,100,-3,3,6,3,1,
Type 1 curves have only one infinite value.
This one is monotonic.
It is of the form y = 1 / (x² + 1)
Since the function contains only even powers of x, it is symmetrical about the y axis.
Type: Functional - Cubic equations (1d - crossover curve)
,x²y  + by = ax,5,0,2,0.5,1,100,-2,2,4,2,2,
Type 1 curves have only one infinite value.
This one is monotonic.
It is is of the form y = x / (x² + 1)
Type: Functional - Cubic equations (1e - bipartite curve)
,y³ - ay + x² = b,5,-4,4,1,1,100,-4,4,8,4,4,
Type 1 curves have only one infinite value.
For certain values of a, this one splits into two parts.
Since the function contains only even powers of x, it is symmetrical about the y axis.
Type: Functional - Cubic equations (1e - bipartite curve2)
,y² = x(x - a)(x - b),7,-2,4,3,1,100,-2,6,8,2,4,
Type 1 curves have only one infinite value.
Since the function contains only even powers of x, it is symmetrical about the y axis.
It crosses the X axis at x = 0, x = a and x = b.
It consists of two parts - a loop and a 'hyperbola'
Type: Functional - Cubic equations (1f - single loop)
,y³ + axy - x³ = c,5,-4.0,4.0,-0.5,0.0,100,-5,5,10,5,5,
Type 1 curves hove one infinite value.
This one is multivalued and with c <> 0 it can become bi-partite
Type: Functional - Cubic equations (2a - interrupted parabola)
,xy = ax³ + by,3,0.2,1,1,2,100,-5,5,10,5,5,
Type 2 curves go off to infinity in two places.
This is due to the xy term.
It is of the form y = x² / (x - 1)
Type: Functional - Cubic equations (2b - bell and parabola)
,x²y  + y² = a,6,-2,2,0,1,100,-4,4,8,4,6,
Type 2 curves go off to infinity in two places.
Since the function contains only even powers of x, it is symmetrical about the y axis.
For a < 0 the curve consists of two 'hyperbolae'
For a > 0 the curve consists of a bell-shaped piece and a 'parabola'.
Type: Functional - Cubic equations (3a)
,x²y =a x + y,3,0.5,2.5,1,2,100,-5,5,10,5,5,
Type 3 curves go off to infinity in three places.
This is due to the presence of the x²y term
This one is of the form y = x / (x - 1)(x + 1)
Type: Functional - Cubic equations (3b)
,x²y =a x² + y,3,0.5,2.5,1,2,100,-5,5,10,5,5,
Type 3 curves go off to infinity in three places.
This is due to the presence of the x²y term
This one is of the form y = x³ / (x - 1)(x + 1)
Type: Functional - Cubic equations (3c - triple hyperbola)
,y³ - x²y = a,7,-2,2,0.2,1,100,-4,4,8,4,4,
Type 3 curves go off to infinity in three places.
Since the function contains only even powers of x, it is symmetrical about the y axis.
The curves are in three bits with assymptotes at y = x,
y = -x and y = 0.
Type: Functional - Ellipses
,ax² + by² = 4,5,0.5,2,1.00,2,100,-3,3,6,3,3,
Type: Functional - Folium of Descartes
,x³ + y³ = 3xy,1,0.5,2,2,2,100,-3,3,6,3,3,
Type: Functional - Lines
,ax + by = 1,5,0.5,2,1,2,100,-4,4,8,4,4,
Type: Functional - Pear-shaped quartic
,x³(a - x) = y²,5,1,3.0,2,2,100,0.2,3,6,2,3,
Type: Functional - Quadratic family
,y² + ax² = bx + c,6,-1.0,1.5,1.5,1.0,100,-4,4,8,4,4,
a < 0 : hyperbola
a = 0 : parabola
a > 0 : ellipse (a = 1 : circle)
Type: Functional - Squircles
,x^a + y^a = 1,5,1,5,1,2,100,-4,4,8,4,4,
