Valentine's day science

Paul Doherty (pauld@exploratorium.edu)
Sat, 13 Feb 1999 13:31:24 -0800

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Message-Id: <l03110712b2eb9c96e47a@[192.174.2.173]>

Date: Sat, 13 Feb 1999 13:31:24 -0800

To: pinhole@exploratorium.edu

From: Paul Doherty <pauld@exploratorium.edu>

Subject: Valentine's day science

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Paul D's activity was about cardioids. Here is a text version, a microsoft
word 5 version is attached. The attached version includes images and tables.

Polar Plot - cardiod

Won't you be my Valentine?

Introduction

A simple plot in polar coordinates will produce a heart shaped function
known as a cardiod.

Material

Low-tech version
Polar plotting paper (Available in the book: "Graph paper from your copier")
A scientific calculator that can compute sine functions

High-tech version
A computer with access to the Web

Assembly

Low-tech version, none.
High-tech version
Go to
http://mss.math.vanderbilt.edu/cgi-bin/MSSAgent/~pscrooke/MSS/plotpolar.def
or http://www.webmath.com/polar.html

To Do and Notice

Low-tech
Have students plot the function r =3D 1 - sin(T) where r is the radius and t
is the angle in degrees with 0<T<360

High-tech
Enter the function 1-sin(t)
and the range tmin=3D 0, tmax=3D2*pi, notice that this angle is in radians.
press the plot button.

What's Going On?

The curve you produced is known as a cardiod.

Etc.

Try variations on this function.
Predict what the graph will look like first.
r =3D 1 + sin(t)
r =3D 1 - 1.25 * sin(t)
r =3D 1
r =3D 1 -sin(t)
r =3D 1 + cos(t)
r =3D 1 - (sin(t))^3
r =3D 1 - (sin(t))^9
r =3D 1 - (sin(t))^99

Appendix:

Table of values for sin T For use with the polar plotting activity:

(*Table is not available in the text version)

Sources of information for Cardioids
http://www.best.com/~xah/SpecialPlaneCurves_dir/Cardioid_dir/cardioid.html

http://www-groups.dcs.st-andrews.ac.uk/%7Ehistory/Curves/Cardioid.html

The cardiod was studied by Roemer (1674) in an investigation for the best
form of gear teeth. The name cardioid (heart-shaped; from Greek root cardi,
meaning heart) was first used by de Castillon in the Philosophical
Transactions of the Royal Society of 1741. Its length is found by La Hire
in 1708. Cardioid is a special case of limacon of Pascal: a family of
curves studied and named after Etienne Pascal (1588-1640)

A cardioid can be defined as the trace of a point on a circle that rolls
around a fixed circle of the same size without slipping.

(* image not available in text version)
You can create a cardioid by rolling one donut shaped refrigerator magnet
around another fixed one. Make a mark with nail polish on one edge of the
magnet then plot the path of this mark as the magnet it is on rolls around
a fixed magnet. The resulting path is a cardioid.

Cartesian equation: (x+y-2ax) =3D 4a(x+y)
Polar equation: r =3D 2a(1+cos())

These formulas describe a cardioid such that the distance from the cusp to
its opposite point is 4*a, where a is a scaling factor equal to the radius
of a genrating circle.

The response of one type of microphone, sensitivity versus angle, is a
cardiod. So are many antennas.
(* Image of microphone sensitivity versus angle is not available in text
version.)

Paul Doherty =A91999

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