## Introduction

This HP 38G Curves Collection was originally created by Jim Donnelly, one of the designers of the HP38G.  It was inspired in part by requests from students and teachers and by the Famous Curves Collection on the MacTutor History of Mathematics Archive web page. Many of the curves presented here are described on the MacTutor page, along with interesting notes about the mathematician who first described the curve. There are also curves on the MacTutor page which do not lend themselves well to the calculator, since the graphics display is quite limited.

It is easiy to create aplets for these curves yourself, and the process of creating an aplet for the Astroid curve is shown below the table.  Note: In Polar aplets, the independent variable is theta, which will be represented in the table below by Ø.

Parent Aplet Equation Initial
Values
Scaling Parameters Plot
Archimedes Spiral
Archimedes'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
Cardioid
Cardioid (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 X-axis. Notice what happens when A<H or A>H.

Parametric X=A*T-H*SIN(T)
Y=A-H*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=(A²-2*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

### Creating a Curve aplet for the Astroid curve

You can, of course, just enter the equation into the appropriate aplet (Function, Parametric or Polar)  but you may want to create a copy, renamed for the curve, so that you have it stored permanently. instructions are given below on how to do this.

 In the information for the Astroid curve, the parent aplet is listed as the Parametric aplet.  Go to the LIB view and SAVE a copy of the Parametric aplet under the new name of Astroid. START the aplet and enter the equation into the SYMB view. Since the equation is partially controlled by a parameter A, change to the HOME view and store the suggested initial value in A. Go to the PLOT SETUP view and enter the values given in the table. PLOT the image. To explore the effect of A on the shape of the curve, return to HOME and change the value of A.  Then press PLOT once to see the original graph and then again to force a redraw using the new value.