Hypocycloid

• Download source code - 20.4 KB

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)
    path.AddLine(old, point);
   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.