This worksheet deals with special types of curves called involutes and evolutes. These are curves which can be associated to any given curve alpha. An involute to alpha may be thought of as any curve which is always perpendicular to the tangent lines of alpha . If you think of unwinding string tautly from around a curve, you will get a picture of an involute. An evolute of is a curve whose involute is alpha, so evolutes and involutes are, in a sense, inverses to each other. An evolute of may also be thought of as the curve determined by the centers of curvature of alpha. Involutes are used in gear tooth design to eliminate vibration as much as possible, while the involute and evolute of a very special curve can be used to design a special type of clock.