The HP 38G Curves Collection

By James Donnelly

Introduction

This HP 38G Curves Collection 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 curves on the MacTutor page which do not lend themselves well to the calculator, since the graphics display is quite limited.

Each curve in this collection is available for downloading as a complete HP 38G aplet. Each aplet is pre-configured with the equations and scaling parameters set to begin your exploration of the curve. Rather than attach a user program to each aplet to initialize coefficient variables, the table below suggests starting values for the coefficients.

Acknowledgements

Thanks to Colin Croft for encouragement and suggestions, and thanks to the MacTutor History of Mathematics Archive for providing a splendid web-based resource.

The Collection

Note: In Polar aplets, the independent variable is theta, which will be represented in the table by Ø until a solution is found to work around a representation problem.

HP 38G Curves Collection
Parent Aplet
and Zip File Equation Initial
Coefficient
Values Scaling Parameters Plot

Archimedes Spiral
Archimedes's name has long been associated with this classic sprial.
Polar
aspiral.zip 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
astroid.zip 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
cardioid.zip 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
catenary.zip 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
cissoid.zip 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
cochloid.zip 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
conchoid.zip 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
cycloid.zip 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
folium.zip 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
epicycl.zip 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
epitroch.zip 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
lisajous.zip 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
rhodonea.zip R=ABS(A*SIN(K*Ø)) A=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
talbot.zip 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
tricusp.zip 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
trisectr.zip 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

**HP 38G Curves Collection**
Parent Aplet and Zip File	Equation	Initial Coefficient Values	Scaling Parameters
Archimedes Spiral Archimedes's name has long been associated with this classic sprial.
Polar aspiral.zip	`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 astroid.zip	`X=ASIN(T)^3 Y=ACOS(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 cardioid.zip	`R=2A(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 catenary.zip	`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 cissoid.zip	`R=2ATAN(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 cochloid.zip	`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 conchoid.zip	`R=A+BSEC(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 cycloid.zip	`X=AT-HSIN(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=4A), the double folium* (B=0), and the trifolium (B=A).
Polar folium.zip	`R=-BCOS(Ø)+4ACOS(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 epicycl.zip	`X=(A+B)COS(T)-BCOS((A/B+1)T) Y=(A+B)SIN(T)-BSIN((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 epitroch.zip	`X=(A+B)COS(T)-CCOS((A/B+1)T) Y=(A+B)SIN(T)-CSIN((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 lisajous.zip	`X=ASIN(MT) Y=BSIN(NT)`	`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 rhodonea.zip	`R=ABS(ASIN(KØ))`	`A=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 talbot.zip	`X=(A²+F²SIN(T)²)COS(T)/A Y=(A²-2F²+F²SIN(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 tricusp.zip	`X=A(2COS(T)+COS(2T)) Y=A(2SIN(T)-SIN(2T))`	`A=1`	`T = 0 to 2*PI step .1 X = -6.5 to 6.5 Y = -3.3 to 3`
Trisectrix of Maclaurin
Polar trisectr.zip	`R=2ASIN(3Ø)/SIN(2Ø)`	`A=1`	`Ø = 0 to PI step .05 X = -6.5 to 6.5 Y = -3.1 to 3.2`

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