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EXAMPLES

Involute of a Circle

The involute of a circle is a Spiral . In Cartesian Coordinates the curve follows:


x=a(\cos(t)+t\sin(t))\,

y=a(\sin(t)-t\cos(t))\,

Where: ''t'' is the angle and ''a'' the Radius

Involute of a Catenary

The involute of a catenary through its Vertex is a Tractrix . In Cartesian Coordinates the curve follows:


x=t- anh(t)\,

y= m sech(t)\,

Where: ''t'' is the angle and Sech is the hyperbolic secant (1/cosh(x))
''Derivative''

With r(s)=(\sinh^{-1}(s),\cosh(\sinh^{-1}(s)))\, we have r^\prime(s)=(1,s)/\sqrt{1+s^2}\, and
r(t)-tr^\prime(t)=(\sinh^{-1}(t)-t/\sqrt{1+t^2},1/\sqrt{1+t^2})
substitute t=\sqrt{1-y^2}/y to get
({ m sech}^{-1}(y)-\sqrt{1-y^2},y)

Involute of a Cycloid

One involute of a cycloid is a Congruent cycloid. In Cartesian Coordinates the curve follows:


x=a(t+\sin(t))\,

y=a(3+\cos(t))\,

Where ''t'' is the angle and ''a'' the Radius


APPLICATION

The Involute of a circle has a property that makes it important to the Gear industry: if the teeth of two mating gears have the shape of an involute, their relative rates of rotation are constant while the teeth are engaged. With teeth of other shapes, the relative speeds rise and fall as successive teeth engage, resulting in vibration, noise, and excessive wear. For this reason, nearly all modern gear teeth bear the involute shape.


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