# 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:**

- 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 360**º**at O. ∠POQ/360**º**= Arc PQ/Circumference of directing circle=

2πr/2πR = r/R

∴∠POQ = θ = r/R*360**º**= (20/75)*360**º**= 96**º.** - Taking any point O as center and radius (R) 75mm, draw an arc PQ which subtends an angle θ = 96
**º**at O. - 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.
- 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.
- O as center, draw the concentric arcs passing through 1,2,3,…. etc.
- O as center and OC as radius draw an arc to represent the locus of center.
- Divide arc PQ into same number of equal parts (12) and name them as 1′,2′,…,etc.
- Join O1′,O2′,…,etc and extend them to cut the locus of center at C1,C2,C3,..,etc.
- Taking C1 as center and radius equal to r, draw an arc cutting the arc through 1 at P1.
- Similarly, obtain the other points and draw a smooth Epicycloid through them.

**To draw a Tangent and Normal at a given point M** - M as center, and radius r= CP cut the locus of center at the point N.
- Join NO which intersects the base circle arc PQ at S.
- Join MS, the normal and draw the tangent perpendicular to it.

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