# Construction of Epicycloid

### Sample Problem 1: Draw an Epicycloid of rolling circle 40mm (2r), which rolls outside another circle (base circle) of 150mm diameter (2R) for one revolution. Draw a tangent and normal at any point on the curve.

Steps for Construction of Epicycloid:

1. In one revolution of the generating circle, the generating point P will move to a point Q, so that the arc PQ is equal to the circumference of the generating circle. θ is the angle subtended by the arc PQ at the center O.To calculate θ: Arc 2πR subtends an angle of 360º at O.                             ∠POQ/360º = Arc PQ/Circumference of directing circle=
2πr/2πR =  r/R
∴∠POQ = θ = r/R*360º = (20/75)*360º = 96º.
2. Taking any point O as center and radius (R) 75mm, draw an arc PQ which subtends an angle θ = 96º at O.
3. Let P be the generating point. On OP produced, mark PC = r = 20mm = radius of rolling circle.Taking Center C and radius r (20mm) draw the rolling circle.
4. Divide the rolling circle into 12 equal parts and name them as 1,2,3,etc,. in the CCW direction , since the rolling circle is assumed to roll clockwise.
5. O as center, draw the concentric arcs passing through 1,2,3,…. etc.
6. O as center and OC as radius draw an arc to represent the locus of center.
7. Divide arc PQ into same number of equal parts (12) and name them as 1′,2′,…,etc.
8. Join O1′,O2′,…,etc and extend them to cut the locus of center at C1,C2,C3,..,etc.
9. Taking C1 as center and radius equal to r, draw an arc cutting the arc through 1 at P1.
10. Similarly, obtain the other points and draw a smooth Epicycloid through them.
To draw a Tangent and Normal at a given point M
11. M as center, and radius r= CP cut the locus of center at the point N.
12. Join NO which intersects the base circle arc PQ at S.
13. Join MS, the normal and draw the tangent perpendicular to it.

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