List of curves
Table of contents
- Functions
- Sigmoid functions
- Bell shaped curves
- Curves
- Generalized spirography
- Remainder
- Bibliography
Functions
- Rectangle wave
- y = sgn(mod(x-1,2)-1)
- Triangle wave
- y = |mod(x,2)-1|
- Sawtooth wave
- y = mod(x,1)
Sigmoid functions
y = x/sqrt(x^2+1)
y = x/(|x|+1)
y = x/(|x|^n+1)^(1/n)
y = tanh(x)
y = erf(sqrt(pi)/2*x)
y = 2/pi*arctan(pi/2*x)
y = 2/pi*gd(pi/2*x)
y = x/ln(exp(|x|)+1)
Bell curves
y = exp(-x^2)
y = 1/cosh(x)
Curves
- Circle
- x^2+y^2 = r^2
x = r*cos(t)
y = r*sin(t)
t in [0,2pi)
r: radius
- Ellipse
- (x/a)^2+(y/b)^2 = 1
x = a*cos(t)
y = b*sin(t)
t in [0,2pi)
- Hyperbola
- x^2-y^2 = 1
x = cosh(t)
y = sinh(t)
t in (-∞,∞)
- Square
- |x|+|y| = 1
x = sgn(cos(t))*cos(t)^2
y = sgn(sin(t))*sin(t)^2
t in [0,2pi)
- Spiral
- x = t*cos(2pi*t)
y = t*sin(2pi*t)
t in [0,n)
- Logarithmic spiral
- x = a*exp(b*t)*cos(t)
y = a*exp(b*t)*sin(t)
t in (-∞,∞)
a>0, b≠0
- Cardioid
- x = (1-cos(t))*cos(t)
y = (1-cos(t))*sin(t)
t in [0,2pi)
- Astroid
- x = cos(t)^3
y = sin(t)^3
t in [0,2pi)
- Quadrifolium
- x = sin(2t)*cos(t)
y = sin(2t)*sin(t)
t in [0,2pi)
- Figure-of-eight loop
- y^2 = x^2-1/4*x^4
x = 2cos(t)
y = sin(2t)
t in [0,2pi)
- Curves with loops
- y^2 = x^2-x^3
- True lemniscate
- (x^2+y^2)^2 = 2a^2*(x^2-y^2)
- Cassinian ovals
- (x^2+y^2)^2 - 2c^2*(x^2-y^2) = a^4-c^4
- Strophoid
- (a-x)*y^2 = (a+x)*x^2
x = a*(t^2-1)/(1+t^2)
y = a*t*(t^2-1)/(1+t^2)
t in (-∞,∞)
a: diameter of the loop
Generalized spirography
- Quadrifolium
- (x^2+y^2)^3=(x^2-y^2)^2
x = cos(2t)*cos(t)
y = cos(2t)*sin(t)
t in [0,2pi)
- Ten leafs
- x = cos(5t)*cos(4t)
y = cos(5t)*sin(4t)
t in [0,2pi)
- Starfish
- x = (4+sin(5*t))*cos(t)
y = (4+sin(5*t))*sin(t)
t in [0,2pi)
Remainder
- Lattice
- sin(x) = sin(y)
- Rings
- sin(x)+sin(y)=1
10*sin(x)*sin(y)=1
- Difficult to draw
- gamma(x*y) = gamma(x+y)
- Infinitely quick oscillation
- f(x) = sin(1/x)
x in (0,∞)
Bibliography
- »List of curves«. In: Wikipedia.
- John J. O'Connor, Edmund F. Robertson:
»Famous Curves Index«. In: MacTutor History of Mathematics archive, April 2015.
- J. Dennis Lawrence: »A catalog of special plane curves«. Dover Publications, 1972.
- Jürgen Köller: »Einfach geschlossene Kurven«.
In: Mathematische Bastelleien, 2010.
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