Classroom Tips and Techniques: The Astroid and Its Tangent Lines
Robert J. Lopez
Emeritus Professor of Mathematics and Maple Fellow
Maplesoft
Introduction
Tangents to the astroid have fixed length if restricted to a single quadrant. In fact, the envelope
of a line of fixed length that has its endpoints on the coordinate axes in the first quadrant is just
the astroid. In this month's article, we explore these two aspects of the astroid, a curve also
known as the four-cusped hypocycloid, as well as the tetracuspid, cubocycloid, and paracycle.
The first part of our investigation starts with Problem 11.1 in the 1988 Maple Calculus
Workbook, by K.O. Geddes, et. al. This collection of non-standard calculus problems was
compiled, with an Alfred P. Sloan Foundation grant, by faculty at the University of Waterloo
where Maple had its origins. I've had this book almost since it was published, and have used it as
a source of interesting calculus problems during my years in the classroom. I've recently
resurrected it in a project that culls some of the best problems for inclusion in my list of
Clickable Calculus offerings.
Problem Statement
For the purposes of my Clickable Calculus file, I've stated the original problem as follows.
Show that for the four-cusped hypocycloid (aka astroid, tetracuspid, cubocycloid, paracycle)
, the length of any tangent lying completely in the first quadrant is .
The flip side of this problem is the observation that the envelope of a line segment of fixed
length moving so that its endpoints remain in contact with the coordinate axes has to be the very
same astroid.
Some Graphics
The figure on the left is the complete astroid with
. The animation on the right shows the
tangent line along the first-quadrant portion of this astroid.
Astroid
Animation
The graph of the astroid requires some care. A first approach might be to use the implicitplot
command from the plots package. Figure 1 shows the result.
Figure 1 Trying to draw the astroid with the
implicitplot command
Just the first-quadrant portion is drawn because Maple's cube-root function produces the
principal root, which is necessarily complex. To get the real cube root, either use the surd
command or work in the confines of the RealDomain package. (Maple computes
as
, not as
. However,
=
.) Figure 2 shows the graph that results from the
first of these two options.
Figure 2
Use of the surd command and the gridrefine option
The gridrefine option in the implicitplot command is needed for the cusps on the axes to be
drawn sharply. It does a better job than simply increasing the overall number of points used.
The animation of the tangent line along the astroid requires an expression for the tangent line, a
calculation we present below.
The Tangent Line and Its Length
It's useful to write an explicit function for the first-quadrant portion of the astroid. To this end,
solve
for
and write
At the point
on the astroid, the equation of the tangent line is
The -intercept is obtained by setting
The -intercept is the -coordinate when
in this expression. The result is
. Thus, we have
, which we compute
The length of this portion of the tangent line is then
and simplify in Maple with
As predicted, the length of any tangent line lying strictly in the first quadrant is constant, and that
constant is .
Envelope of a Family of Curves
If a family of plane curves is described by
, then the equations
determine
any envelope parametrically as
. To envelope the family of fixed-length lines
whose endpoints are on the coordinate axes, we need the function
that describes the
family. To this end, consider the right triangle formed by the red line segment in Figure 3. This
segment has fixed length , and forms, with the coordinate axes, a right triangle whose sides are
and .
Figure 3
Line segment of fixed length
From the right triangle in Figure 3, we immediately have
and
as the slope of
the red hypotenuse. Hence, the slope-intercept form of the equation of the red line segment is
The function
is then
The envelope is given parametrically by
To get an explicit representation of the envelope, eliminate the parameter
via
The RootOf structures can be resolved with
The easiest thing to do here is to use the mouse to select the equation that gives the envelope
explicitly. This equation is
Clearly, the appropriate algebra is to isolate the radical, then raise both sides to the power
The first step is easy, and results in
.
The second step is also easy, and results in
A slight simplification is obtained with
A further simplification is obtained with
The two previous calls to simplify have not resulted in the obvious simplification on the left.
To obtain the final simplification, we must use
It is now just a matter of isolating the constant term, and we have
as the equation of the envelope. Clearly, this is the equation of the astroid.
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