Construction of Involute of a Regular Polygon

INVOLUTE OF A REGULAR POLYGON

Example 8: Construct an involute of a regular pentagon of 25mm side. Also draw a tangent and a normal at any point on the curve.

Procedure:

1) Draw a regular pentagon of side 25 mm and name the vertices as A, B, C, D and E in clockwise direction.

2) Extend the sides AE, BA, CB, DC and ED.

3) With A as centre and radius AE, draw an arc to cut the side BA extended at P1,

4) With B as centres and B – P1 as radius draw an arc to cut the side CB extended at P2. 

5) Similarly obtain the points P3, P4 and P5 as shown in Fig. 7.16. The curve passing through E, P1, P2.... P5 is the required involute.

To draw normal and tangent at any point M

Let M be a point on the curve through which normal and tangent are to be drawn.

i) since the point M is on the curve which is drawn with D as centre, Join D and M and extend, which is the required normal NN'. 

ii) Draw a line perpendicular to NN' at M. Let this line be TT', which is the required tangent.

Example 9: Construct an involute of a hexagon of side 20mm. Also draw a tangent and normal at any point on the curve.

Procedure:

1) Draw a regular hexagon ABCDEF of side 20mm.

2) Extend the sides of hexagon BA, CB, DC..... etc.

3) With A as centre and radius AF draw an arc to cut the side BA extended at P1.

4) With B as centre and radius B-P1 draw an arc to cut the side CB extended at P2. 

5) Similarly obtain the points P3, P4, P5 and P6.

6) Join the points F, P1, P2....P6 by a smooth curve which is the required involute. 

To draw a normal and tangent at any point M.

Let M be the point on the curve through which normal and tangent are to be drawn. 

i) since the point M is on the curve which is drawn with E as centre, Join E and M and extend, which is the required normal NN'.

ii) Draw a line perpendicular to NN' through M. Let this line be TT', which is the required tangent.