Construction of Involute of a Circle

INVOLUTE OF A CIRCLE

Example 6: Draw an involute of a circle of diameter 50 mm. Also draw normal and tangent at any point M on the curve.

Construction:

1) With O as centre draw a circle of diameter 50 mm. 

2) Draw a tangent AB of length equal to the circumference of circles (π × 50). 

( 3) Divide the circle into 12 equal parts and name the divisions in clockwise direction.

4) Divide the tangent AB also into 12 equal parts and name the parts as 1', 2', 3' etc., 

5) Let Po be the starting point, ie., one end of the curve which coincides with the point A at initial position and coincides with the point B after one complete turn. 

 (Note that Po coincides with A and P12 coincides with B and 12') 

To locate the other points P1, P2 etc.,

6) To locate the intermediate positions of the point Po, draw tangents at points 1, 2, .... etc marked on the circumference of circle.

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

8) Similarly obtain the other intermediate points P2, P3 … taking 2, 3.... as centres and A-2, A-3' … as radius.

9) Draw a smooth curve passing through the points P0, P1, P2, … P11, P12, which is the required involute of the circle.

To draw Normal and Tangent at any point

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

i) Join the points M and centre of the circle O.

ii) Locate the mid point of the line MO at C.

iii) Taking C as centre and CM as radius draw a semicircle to cut the circle at N.

iv) Joint N and M and extend, which is the required normal drawn through M.

v) Draw a line perpendicular to NM. Let this line be TT' which is the required tangent drawn through the point M.