Trochoid

Trochoid is a curve generated by a point fixed to a circle as the circle rolls along a straight line.  If the point is outside the rolling circle,  the curve obtained is called an inferior trochoid and when outside the circle is called superior trochoid.  Figure 4 illustrates  an inferior trochoid and a superior trochoid.

 

Figure 4 illustrating the superior and inferior trochoids.

Construction of an inferior trochoid.
The construction procedure for obtaining an inferior trochoid is shown in figure 5. With centre C’ and radius R, draw a circle.  From A, draw a horizontal line  AB = 2πR. Draw C’- C” parallel and equal to AB and divide it in to 12 equal parts C1, C2, C3 ,…. C12. Draw the generating point Q’ along C’-A. With centre C’ and radius  C’-Q’ draw a circle and divide the circumference in to 12 equal parts and label tham as  1,2,3, ….,12.  With C1, C2, C3,etc., as the centres and radius equal to C’-Q’, cut arcs on the horizontal lines through 1,2,3,etc., to locate  the points Q1, Q2, Q3, etc. Join Q1, Q2, Q3, etc., to obtain the inferior trochoid.

Figure 5. Construction of an inferior trochoid.