Technical Drawing - Cycloids - Basic Cycloid

Construction of a Cycloid

Below is a discription of how to construct a Cycloid for a point Pon a circle as it rotates along a straight line without slipping.

Firstly draw the circle and a line from its base to the left or right. The line should be to the right if the circle is rotating clockwise, and to the left if the circle is rotating anti-clockwise.

In this situation we are going to find the locus of a point P which lies at the bottom of the circle where it touches the straight line.

Now divide the circle into an equal number of parts. Here we have used the 60°-30° set square to give us 12 equal divisions. Lable these sections 0 to 12.

Taking the distance between any two of these points with your compass mark 12 off these distances along the base line along which the circle will rotate. Lable these points 0 to 12 as shown. When taking the distance between any two of the divisions of the circle you should add a little bit to the distance on your compass to account for the curve of the circle.

Project the new points onto a line from the centre of the circle and label then C1 to C12.

As the circle rotates along the line, point 1 on the circle will move to point 1 on the line, and therefore the center of the circle will now be at point C1.

Next project lines from the equal divisions of the circle parallel to the straight line. These are the height lines.

With your compass set to the radius of the circle, place the point of the compass on the point C1 and cut the height line coming from point 1 on the circle. Then repeat this for point C2 and height line 2, etc. until you reach point C12 and line 12.

By joining these new points you will have created the locas of the point P for the circle as it rotates along the straight line without slipping. In other words, you have created a Cycloid.

Essentially what you have done is break up the continuous movement of the circle along the straight line into 12 sections and found the position of point P for each movement.

It would get very cluttered if you drew in every circle along the way.

And your final result is a Cycloid.

Construction of a Tangent and a Normal to a point on a Cycloid

You can construct a Tangent and a Normal to any point on the Cycloid by using this method.

Pick a point.

With the radius of the circle on your compass mark on the centre line of the rotating circle.

Now draw a circle in this position.

Draw a vertical line through the centre of the circle.

Draw a line from the top of the circle to the point and you will have the Tangent.

Draw a line from the bottom of the circle to the point and you will have the Normal.

How to find the Centre of Curvature to a point on a Cycloid

You can find the Centre of Curvature to any point on the Cycloid by using this method.

Pick a point.

With the radius of the circle on your compass mark on the centre line of the rotating circle.

Now draw a circle in this position.

Draw a vertical line through the centre of the circle.

Draw a line from the bottom of the circle to the point which will give you the Normal. Continue this line below the drawing.

Draw a line from the point, through the centre of the circle until it intersects the other side of the circle.

Draw a line straight down until it intersects the Normal. Where this intersection occurs is the Centre of Curvature for the point.