Parent Aplet 
Equation 
Initial
Values 
Scaling Parameters 
Plot 
Archimedes SpiralArchimedes's name
has long been associated with this classic sprial. 
Polar 
R=A*Ø 
A=.05 
Ø = 0 to 14*PI step .1309
X = 6.5 to 6.5
Y = 3.1 to 3.2 

Astroid 
Parametric 
X=A*SIN(T)^3
Y=A*COS(T)^3 
A=1 
T = 0 to 2*PI step .1
X = 2.6 to 2.6
Y = 1.24 to 1.28 

CardioidCardioid (heart shaped)
curves come in many varieties. One common instance of the use of this
curve is to describe the shape of the area covered by a "cardioid
microphone". 
Polar 
R=2*A*(1+COS(Ø)) 
A=1 
Ø = 0 to 2*PI step .1309
X = 6.5 to 6.5
Y = 3.4 to 2.9 

Catenary
The catenary function describes the shape of a rope or chain as it
hangs freely by both ends. 
Function 
Y=A*COSH(X/A)) 
A=1 
X = 6.5 to 6.5
Y = 1.3 to 6 

Cissoid of Diocles
What happens at PI/2? 
Polar 
R=2*A*TAN(Ø)*SIN(Ø) 
A=1 
Ø = 0 to PI step .1309
X = 6.5 to 6.5
Y = 3.1 to 3.2 

Chochleoid
A member of the "snail form" curve family. 
Polar 
R=A*SIN(Ø)/Ø 
A=1.5 
Ø = 0 to 4*PI step .1309
X = 1.625 to 1.625
Y = .475 to 1.1 

Conchoid
A "shell form" curve. 
Polar 
R=A*+B*SEC(Ø) 
A=3
B=1 
Ø = 0 to 2*PI step .065
X = 6.5 to 6.5
Y = 5.21 to 5.38 

Cycloid
The pattern that a point H on a disk makes as the disk of
diameter A rolls along the Xaxis. Notice what happens when A<H
or A>H. 
Parametric 
X=A*TH*SIN(T)
Y=AH*COS(T) 
A=2
H=2 
T = 0 to 24 step .185
X = 0 to 50
Y = 3.5 to 20.7
Xtick = 5, Ytick = 5 

Folium
Folium means "leaf shaped". There are three cases of this
curve  the folium (B=4*A), the double folium (B=0), and the
trifolium (B=A). 
Polar 
R=B*COS(Ø)+4A*COS(Ø)*SIN(Ø)^2 
A=1, B=4
A=1, B=0
A=2, B=2 
Ø = 0 to PI step .13
X = 6.5 to 6.5
Y = 3.4 to 2.9 

Epicycloid
The pattern that a point on the circumference of a disk makes as the
disk of diameter b rolls around a disk of diameter a. 
Parametric 
X=(A+B)*COS(T)B*COS((A/B+1)*T)
Y=(A+B)*SIN(T)B*SIN((A/B+1)*T) 
A=4
B=2.5 
T = 0 to 31.5 step .2
X = 21.66 to 21.66
Y = 12 to 9
Axes not drawn 

Epitrochoid
The pattern that a point a distance c from the center of a disk
of diameter b makes as it rolls around a disk of diameter a. 
Parametric 
X=(A+B)*COS(T)C*COS((A/B+1)*T)
Y=(A+B)*SIN(T)C*SIN((A/B+1)*T) 
A=5
B=3
C=5 
T = 0 to 18.85 step .05
X = 31.2 to 31.2
Y = 16.54 to 13.69
Axes not drawn 

Lissajous Curves 
Parametric 
X=A*SIN(M*T)
Y=B*SIN(N*T) 
A=3
B=2
M=3
N=4 
T = 0 to 2*PI step .1
X = 6.5 to 6.5
Y = 3.1 to 3.2 

Rhodonea
A visual "cousin" of the folium family, named for its flower
shape. Note that k controls the number of petals  what happens
when k is odd or even? 
Polar 
R=A+ABS(B*SIN(K*Ø)) 
A=0
B=2.5
K=3 
Ø = 0 to 2*PI step .1309
X = 7.8 to 7.8
Y = 4.02 to 3.54 

Talbot's Curve 
Parametric 
X=(A²+F²*SIN(T)²)*COS(T)/A
Y=(A²2*F²+F²*SIN(T)²)*SIN(T)/B 
A=1.1
B=.666
F=1 
T = 0 to 2*PI step .05
X = 1.567 to 1.567
Y = .7475 to .7716
Xtick = .25, Ytick = .25 

Tricuspoid 
Parametric 
X=A*(2*COS(T)+COS(2*T))
Y=A*(2*SIN(T)SIN(2*T)) 
A=1 
T = 0 to 2*PI step .1
X = 6.5 to 6.5
Y = 3.3 to 3 

Trisectrix of Maclaurin 
Polar 
R=2*A*SIN(3*Ø)/SIN(2*Ø) 
A=1 
Ø = 0 to PI step .05
X = 6.5 to 6.5
Y = 3.1 to 3.2 
