Cycloidal curves, involutes and spirals: a solved example
A circle of 50 mm diameter rolls along a straight line without slipping. Draw the curve traced by a point on its circumference for one complete revolution. Draw a tangent and normal to the curve at any point.
Every line below was drawn and checked by Graspline's own geometry engine, by the textbook method, one step at a time with the reason for each.
The construction, step by step
What the question gives
Draw the generating circle of diameter 50 mm, and the base line, one circumference (157.1 mm) long.
Divide and construct
Divide the circle and the base line into 12 equal parts. At each division of the rolled circle, the point that started in contact with the line is a point of the curve.
The curve
Join the constructed points with a smooth curve. It is the cycloid.
Tangent and normal
The normal at the point passes through where the generating circle touches the base line at that same instant; the tangent is at right angles to it.
Result
The cycloid, drawn from 13 constructed points, with the tangent and normal drawn.
✓ Verified: every constructed point lies on the curve, and the tangent is at right angles to the normal.
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