## Cycloidal Curves

Cycloidal curves are generated by a fixed point in the circumference of a circle when it rolls without slipping along a fixed straight line or circular path. The rolling circle is called the generating circle, the fixed straight line, the directing line and the fixed circle, the directing circle.

### Cycloid

A cycloid is a curve generated by a fixed point on the circumference of a circle, when it rolls without slipping along a straight line.

**To draw a cycloid, given the radius R of the generating circle.**

**Construction**

1. With centre O and radius R, draw the given generating circle.

2. Assuming point P to be the initial position of the generating point, draw a line PA, tangential and equal to the circumferance of the circle.

3. Divide the line PA and the circle into the same number of equal parts and nuber the points.

4. Draw the line OB, parallel and equal to PA. OB is the locus of the centre of the generating circle.

5. Errect perpendiculars at 1^{1},2^{1},3^{1}, etc., meeting OB at O_{1}, O_{2}, O_{3}, etc.

6. Through the points 1,2,3 etc., draw lines parallel to PA.

7. With centre O, and radius R, draw an arc intersecting the line through 1 at P_{1}‘ P_{1} is the position of the generating point, when the centre of the generating circle moves to O_{1}, S. Similarly locate the points P_{2}, P_{3} etc.

9. A sIIlooth curve passing through the points P,P_{1}‘ P_{2},P_{3} etc., is the required cycloid.

**Note**: T-T is the tangent and NM is the normal to the curve at point M.

### Epi-Cycloid and Hypo-Cycloid

An epi-cycloid is a curve traced by a point on the circumference of a generating circle, when it rolls without slipping on another circle (directing circle) outside it. If the generating circle rolls inside the directing circle, the curve traced by the point in called hypo-cycloid

To draw an epi-cyloid, given the radius ‘r’ of the generating circle and the radious ‘R’ of the directing circle.

**Construction**

1. With centre O’ and radius R, draw a part of the directing circle.

2. Draw the generating circle, by locating the centre 0 of it, on any radial line O^{1} P extended such that OP = r.

3. Assuming P to be the generating point, locate the point, A on the directing circle such that the arc length PA is equal to the circumference of the generating circle. The angle subtended by the arc PA at O’ is given by θ = <PO’ A = 3600 x rlR.

4. With centre 0′ and radius 0′ 0, draw an arc intersecting the line 0′ A produced at B. The arc OB is the locus of the centre of the generating circle.

s. Divide the arc PA and the generating circle into the same number of equal parts and number the points.

6. Join O’-1′, O’-2′, etc., and extend to meet the arc OB at O_{1}‘O_{2} etc.

7. Through the points 1,2,3 etc., draw circular arcs with 0′ as centre.

8. With centre O_{1} and radius r, draw an arc intersecting the arc through 1 at P_{1}.

9. Similarly, locate the points P_{2}, P_{3} etc.

10. A smooth curve through the points P_{1},P_{2}‘P_{3} etc., is the required epi-cycloid.

Notel: The above procedure is to be followed to construct a hypo-cycloid with the generating circle rolling inside the directing circle (Fig. 3).

Note 2 : T-T is the tangent and NM is the normal to the curve at the point M.