Article

Hypocycloid

By , 22 Dec 2009
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Introduction

Math problems have a charm of their own. Besides, they help to develop a programmer's skill. Here, we describe a student's exam task: "Develop an application that models the behaviour of a Hypocycloid".

Background

A cycloid is the curve defined by the path of a point on the edge of a circular wheel as the wheel rolls along a straight line. It was named by Galileo in 1599 (http://en.wikipedia.org/wiki/Cycloid).

A hypocycloid is a curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. It is comparable to the cycloid, but instead of the circle rolling along a line, it rolls within a circle.

Use Google to find a wonderful book of Eli Maor, Trigonometric Delights (Princeton, New Jersey). The following passage is taken from this book.

I believe that a program developer must love formulas derivation. Hence, let us find the parametric equations of the hypocycloid.

A point on a circle of radius `r` rolls on the inside of a fixed circle of radius `R`. Let `C` be the center of the rolling circle, and `P` a point on the moving circle. When the rolling circle turns through an angle in a clockwise direction, `C` traces an arc of angular width `t` in a counterclockwise direction. Assuming that the motion starts when `P` is in contact with the fixed circle (figure on the left), we choose a coordinate system in which the origin is at `O` and the x-axis points to `P`. The coordinates of `P` relative to `C` are:

`(r cos b; -r sin b)`

The minus sign in the second coordinate is there because `b` is measured clockwise. Coordinates of `C` relative to `O` are:

`((R - r) cos t, (R - r) sin t)`

Note, angle `b` may be expressed as:

`b = t + β; β = b - t`

Thus, the coordinates of `P` relative to `O` are:

`((R - r) cos t + r cos β, (R - r) sin t - r sin β) (1)`

But the angles `t` and `b` are not independent: as the motion progresses, the arcs of the fixed and moving circles that come in contact must be of equal length `L`.

`L = R t L = r b`

Using this relation to express `b` in terms of `t`, we get

`b = R t / r`

Equations (1) become:

```x = (R - r) cos t + r cos ((R / r - 1) t) (2)
y = (R - r) sin t - r sin ((R / r - 1) t)```

Equations (2) are the parametric equations of the hypocycloid, the angle `t` being the parameter (if the rolling circle rotates with constant angular velocity, `t` will be proportional to the elapsed time since the motion began). The general shape of the curve depends on the ratio `R/r`. If this ratio is a fraction `m/n` in lowest terms, the curve will have `m` cusps (corners), and it will be completely traced after moving the wheel `n` times around the inner rim. If `R/r` is irrational, the curve will never close, although going around the rim many times will nearly close it.

Using the code

The demo application provided with this article uses a `Hypocycloid` control derived from `UserControl` to model a behaviour of a hypocycloid described above.

The functionality of the hypocycloid is implemented in the `Hypocycloid` class. It has a `GraphicsPath path` data field that helps to render the hypocycloid path over time. A floating point variable, `angle`, corresponds to the angle `t` described earlier.

• Variable `ratio = R / r`
• `delta = R - r`

All the math is done within the timer `Tick` event handler.

```void timer_Tick(object sender, EventArgs e)
{
angle += step;
double
cosa = Math.Cos(angle),
sina = Math.Sin(angle),
ct = ratio * angle;

movingCenter.X = (float)(centerX + delta * cosa);
movingCenter.Y = (float)(centerY + delta * sina);
PointF old = point;
point = new PointF(
movingCenter.X + r * (float)Math.Cos(ct),
movingCenter.Y - r * (float)Math.Sin(ct));
int n = (int)(angle / pi2);
if (n > round)
{
round = n;
ParentNotify(msg + ";" + round);
}
if (round < nRounds)
else if (!stopPath)
{
ParentNotify(msg + ";" + round + ";" + path.PointCount);
stopPath = true;
}
parent.Invalidate();
}```

`ParentNotify` is the event of the generic delegate type `Action<string>`.

`public event Action<string> ParentNotify;`

We use it to notify a parent control of a current angle (round).

Besides a constructor, the class has the following public methods: `Reset`, `Draw`, `Start`, `Stop`, and `SaveToFile`. Remember also that the Y axis in a Windows window goes down.

Russian Federation
St-Petersburg State Technical University professor:

Lecture on OOP, C# and C++.

Microsoft Authorized Educational Center:

Lecture on Windows programming with C# and C++.

Microsoft Certified Professional (C# Desktop Apps and MFC).

Have long practice and experience in finding the right way to formulate and numerically solve differential equations.

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