Procedure

1. To draw a cycloid, given the radius R of the generating circle. Also, draw a tangent and a normal at a given height from the baseline

1. Construct the circle with the given radius R. Mark its centre as C and assume a point P on it as the tangential point.
2. From P draw tangential line PA whose length is equal to the circumference of the circle.
3. Divide the circle and the tangential line into an equal number of parts and mark these points.
4. From C draw a line parallel to PA and produce perpendicular lines from the division of the line. Mark the points where they intersect as C1, C2, C3, C4 and so on...
5. Similarly, draw parallel lines to PA from the divisions on the circle.
6. With C1 as the centre and R as the radius, draw an arc on line the parallel line that is passing through the 1st division of the circle. Label the point as P1.
7. Join P1, P2, P3, etc. to obtain a smooth curve which is the required cycloid.
8. Locate a point M on the cycloid at a given height from the baseline.
9. From M draw an arc of radius R on the centre line produced from C. Drop a line perpendicular to PA from the arc.
10. Join this line with M and produce it. This is the required normal.
11. Draw a line perpendicular to normal that is passing through M. This is the required tangent.

2. To draw an epicycloid, given the radius r of the generating circle and the radius R of the rolling circle. Also, draw a tangent and a normal to the epicycloid. .

1. Draw a part of the directing circle with radius R. Mark its centre as O’.

2. Extend the radial line O’P and draw the generating circle on it. Locate it’s centre as C such that CP= r.

3. On the directing circle, locate point A such that the circumference of the generating circle is equal to the arc length PA. The angle subjected by the arc PA at the centre O’ gives the position of point A. This is calculated by the following formula: θ = (r/R)×360°

4. Divide the generating circle into any number of equal parts and label them.

5. With centre O’, draw a circular arc having radius O’P.

6. Similarly , draw circular arcs through the points 1,2,3,etc. of the circle with O’ as the centre.

7. Draw a circular arc through the centre of generating circle C with O’ as the centre. This is locus of the centre of generating circle.

8. Divide the arc PA into same number of parts as the generating circle. Extend the divisions to meet the locus at C1, C2, C3, etc.

9. With C1 as the centre having radius equal to that of the generating circle’s (r), draw an arc on the circular arc that is passing through the 1st division of the generating circle.

10. Mark arc for the rest of the divisions and obtain the points P1, P2, P3, etc.

11. Join the points P1, P2,P3,P4 and so on to obtain a smooth curve which is known as the epicycloid.

12. Locate a point M on the epicycloid from the centre of the directing circle

13. With M as centre, draw an arc of radius r on the locus of the generating circle

14. Join the arc with O’ and mark their point of intersection on the arc PA.

15. Join this point with M and extend it. This is the required normal.

16. Draw a line perpendicular to normal that is also passing through the required point M . This is the required tangent.

3. To draw a hypocycloid, given the radius r of the generating circle and the radius R of the directing circle. Also, draw a tangent and a normal to the given hypocycloid.

1. With centre O’ draw a part of the directing circle with radius R. Label it as O’P.

2. With O’ as the centre, locate point A by the following formula: θ = (r/R)×360°

3. On the radial line O’P, Mark PC such that it is equal to the radius of the generating circle (r). Construct the generating circle with C as the centre.

4. Draw circular arcs through each of the point on the circle with O’ as the centre.

5. Draw circular arc through C which is the locus of the centre C of generating circle

6. Divide the arc PA into same number of equal parts as the generating circle.

7. Locate the centre points C1, C2 ,C3 on the locus of the generating circle.

8. With C1 as the centre and radius equal to that of the generating circle, mark off an arc on the circular arc passing through the 1st division point of the circle.

9. Similarly, mark off arcs for the rest of the points and obtain the points P1,P2,P3 and so on.

10. Join P1,P2,P3,P4,etc. The smooth curve so obtained is known the hypocycloid.

11. Locate a point M on the hypocycloid which is at the required height from the centre of the directing circle.

12. From M , draw an arc on the locus of C , having radius r.

13. Join the arc with O’ and extend it to meet the arc PA at S.

14. Join S with M. This is the required normal.

15. Draw a line perpendicular to normal that is also passing through the Required point M. This line is known as the tangent.