1. Involutes
An involute is the locus of a point on a line as the line rolls along a shape
It can also be thought of as the locus of the end of a piece of string as the string is wound/unwound around the circumference of a plane figure

2. Applications of Involutes
Involutes are used to determine the length of belts used in pulleys and other machines
Involutes are also used to calculate the amount of material required to create tyres and wheels

3. Involute of a Square

4. C B A B A C B D A A

5. Involute of a Triangle

6. B A B C A A

7. Involute of a Hexagon

8. A A B B C C D D A B C D E C E B F B A A A

9. Involute of a Circle

10. 9 8 8 10 7 7 9 6 6 8 7 5 6 5 7 5 4 4 4 6 11 3 3 2 3 5 2 10 4 1 9 6 5 3 2 1 4 1 2 8 1 3 7 1 2 6 2 3 5 1 4 4 1 5 2 3 1 2 3 1 2 1 1 2 3 4 5 6 7 8 9 10 11 12 4 1 2 3

11. Making Involute models

12. Tangents to Involutes

13. Tangents to Involutes
Normal
Tangent
Involutes are curves, and as with all curves a tangent can be drawn to the involute

14. Tangent to an involute at point P

15. Normal
Tangent

16. Tangent to an involute at point P

17. Normal
Tangent