The Deltoid
by Dustin Qualls

Abstract

This article seeks to develop a parametric equation for a particular curve by using the geometric definition of the curve. The deltoid will be observed in this manner and the parametric equation will be found to be,
 

x(t) = a(2cos(t) + cos(2t))
y(t) = a(2sin(t) - sin(2t))

where "a" is related to the radius of the circles that generate the curve.

Description

The deltoid (aka tricuspid, or Steiner's hypocycloid) is the trace of a point on a circle rolling inside another circle three times as large in radius. This definition also serves the three-cusped hypocycloid, to which the deltoid owes its origin.

Generation of a Parametric Equation
 

The equation can best be derived by placing one cusp on the x-axis. Consider the figure of the deltoid.
 

P is the tracing point

<TOX=t

OA=2AT

AT=AP=a=radius of small circle

AT=1/3(OT)

arcTX=arcTP

<TAP=3t

OX is on the x-axis

O is the origin

inclination of AP to OX is 2t by regeneration of x-axis through A

x-axis

A line perpendicular to OX is to be dropped from A. The length from the origin O to P along the x-axis is found from the sum of horizontal lengths, O to the line and the line to P. The horizontal length from O to the line is OB = 2acos(t). The horizontal length from the generation of P on the line to P is equal to acos(2t). The parametric equation of the x-axis is equal to 2acos(t) + acos(2t).
 

x(t) = a(2cos(t) + cos(2t))

y-axis

The same perpendicular line as used in the development of the equation for x(t) is used for that of y(t). The length of the line from x-axis to A is equal to AB = 2asin(t) but the height on the y-axis of P is less than this height. P can be generated on to the perpendicular line by moving the point horizontally toward the line. This movement between points creates a line that constructs a right triangle with one leg equalling the difference in the height of AB and the y position of P. This difference is equal to asin(2t) and must be subtracted from the entire length of AB to find the height of P on the y-axis, 2asin(t)-asin(2t).
 

y(t) = a(2sin(t) - sin(2t))

History

The deltoid, named for its resemblance to the greek letter delta, has no real discoverer because of its relationship to the cycloid. Ordinary cycloids were studied by Galileo and Mersenne as early as 1599 but cycloidal curves were first conceived by Roemer (a Dane) in 1674 while studying the best form for gear teeth. The property of double generation entails the tracing of the same curve by a circle twice the size of the original rolling circle and is a property of the deltoid (see figure). Double generations of cycloidal curves were first noticed by Daniel Bernoulli in 1725.

Many have contributed to the study of Roulettes, the path of a point attached to the plane of a curve which rolls upon a fixed curve. Ignoring these others, Euler claims first consideration of the actual deltoid in 1745 in connection with an optical problem and later (1856) Steiner investigated the curve. After Steiner's contributions the curve was dubbed, "Steiner's hypocycloid."

Wheels are often supported by deltoid-shaped spokes and the sharp cusps of the curve make for a good thowing star but its anatomical relative, the Deltoid, is its most famous namesake. The Deltoid is a muscle in the human shoulder and is named for its deltoidal shape.

   

Sources

http://www.best.com/~xah/SpecialPlaneCurves_dir/specialPlaneCurves.html

http://www-groups.des.st-and.ac.uk/~history/Curves/

http://www.astro.virginia.edu/~eww6n/math/

http://www-groups.des.st-and.ac.uk/~history/Mathematicians

Lockwood, E.H. A Book of Curves. Cambridge University Press.