Construction of Involute of a Square

INVOLUTE OF A SQUARE

Example 7: Construct an Involute of a square of side 25 mm. Also draw a tangent and normal at any point M.

Construction:

1) Draw a square of side 25 mm and name the corners as 1, 2, 3 and 4 in clockwise direction. 

2) Draw a tangent AB coinciding A at 4 of length equal to the perimeter of square.

ie., AB = 4 × 25 = 100 mm. Divide AB into four parts and name the parts as 1', 2', 3′ and 4'.

3) Let P0 be the starting point of involute which coincides with 4 (and A).

4) Draw tangents through the points 1, 2 and 3.

5) With 1 as centre and A-1' as radius draw an arc to cut the tangent drawn through the point 1 to get P1.

6) Similarly obtain the other points P2, P3 and P4 by taking 2, 3 and 4 as centres and A-2', A-3′ and A-4' as radii respectively. (Note that P4 coincides with 4' and B).

7) Draw a smooth curve by joining the points P0, P1, P2, P3 and P4, which is the required involute of the square.

Note :

To join the points P1, P2 etc. arcs may be draw as below.

with 1 as centre, and radius A-1′, draw an arct to join P0 and P1

with 2 as centre, and radius A – 2′, draw an arct to join P1 and P2 and so on.

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) Join A and M and extend it, which is the required normal.

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