Technical Drawing - Cycloids - Superior Troichoid

What is a Superior Trochoid ?

A Superior Trochoid is the path or Locus of a point which lies on the outside of a circle when it rotates along a straight line without slipping.

Construction of a Superior Trochoid

Below is a discription of how to construct a Superior Trochoid for a point P outside a circle as it rotates along a straight line without slipping.

Procede for the first two steps as you would for a Cycloid.

We now need to create the height lines for the Superior Trochoid, and this is where things are a little different from the construction of a Cycloid.

Draw a circle that runs through the point P. We get our height lines from where the extended division lines of the circle cut this new circle.

You can continue by setting your compass to the radius of the new circle, placing the point of the compass on C1 and cutting height line 1. Continue on as with the Cycloid.

Now join up the points and you have a Superior Trochoid.

Construction of a Tangent and a Normal to a point on a Superior Trochoid

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

Pick a point.

With the radius of the circle which passes through point P on your compass mark on the centre line of the rotating circle.

Now draw a circle, with the same radius, 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 Superior Trochoid

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

Pick a point.

With the radius of the circle which passes through the point P on your compass mark on the centre line of the rotating circle.

Now draw a circle, with the same radius, 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.

Basic Cycloid
Inferior Troichoid
Superior Troichoid