Construction of Cycloid

Sample Problem 1:
A coin of 40mm diameter rolls over a horizontal table without slipping. A point on the circumference of the coin is in contact with the table surface in the beginning and after one complete revolution.Draw the path traced by the point.Draw a tangent and normal at any point on the curve.

Steps for Construction of Cycloid:

1. Draw the rolling circle of diameter (2r) 40mm. Draw the base line PQ equal to the circumference of the rolling circle at P. Divide the rolling circle into 12 equal parts as 1,2,etc.Draw horizontal lines through the points 1,2,etc.Note: As the rolling circle is assumed to roll Clock-Wise(CW), numbering of the division points on it should be in Counter Clock Wise (CCW) direction.
2. Divide the base line PQ into the same number of equal parts (12) at 1′,2′,3′,… etc.Draw lines perpendicular to PQ at 1′,2′,….,etc, to cut the horizontal line drawn through C (Called Locus of Center) at C1,C2,…,Etc.., respectively.
3. C1,C2,C3,… etc. as centers and radius equal to the rolling circle (20mm), draw arcs to cut the horizontal lines through the points 1,2,… etc. at P1,P2,..,etc. Draw a smooth Cycloid curve through the points P,P1,P2,..,etc.Note:When the circle rolls forward CW by 1/12 of a revolution , the division point 1 on the rolling circle will come in contact with 1′ and the center of the rolling circle moves to the new position C1 which is vertically above 1′.To draw the Normal and Tangent at a point D
4. D as center and radius = radius of the rolling circle, cut the line of locus of center at C’, draw a perpendicular line to PQ to get E on the base line.
5. Connect DE, the normal. At D, draw a line perpendicular to DE to get Tangent TT.

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