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Involute curve for gears
Involute curve also known as evolvent curve is a kind of differential curve. In mechanical engineering this curve is used a lot in gear design. This curve is used to get the gear tooth profile and when two gears of this profile mesh, the instantaneous contact point is only one and it moves along the curve profile. This property is known as line of action or line of Contact or pressure Line and this is why it is used extensively in gear design.

There are many ways to define an involute curve. I take a simple trigonometry method in cartesian coordinates system to explain this. To start let me explain how we define an involute curve.

Just see the picture on the left side. There is pole at the center and a sting is attached to it.
Involute curves
Involute curves
When you wind the string around the pole keeping the unwounded string straight, the path followed by the free end is defined as the involute curve. The same path is followed by the free end when it is wind off (see the picture on the right).

So to define this path followed by the free end using trigonometry equations we follow some simple steps given below. First we need to define the coordinates of the blue points shown on the below pictures at every instantaneous angles with the help of red points and using the red arc lengths.

To define coordinate points for every instantaneous angles we use equations. Here we will see how to find the points for a given angle. To explain the idea, I take 20, 45, 60 and 80 degree angles.
To get the arc lengths, here is the formula.

Arc length (s) = (2 π r/360) x angle (Where, r is the radius)

Lets take the center of the circle as coordinates 0,0,0 in Cartesian system.
Basic trigonometry functions
x ordinate of point ‘x’ (A)= cos20 * 6
y ordinates of point ‘x’ (B)= sin20 * 6

So the final ordinates are,

x ordinate of point ‘point 1’ = A + sin20 * s
y ordinates of point ‘point 1’ = B - cos20 * s

The result (x,y) is (6.355, 0.084)
In our case the hypotenuse is the radius of the circle and lets keep it as 6. So in total,
r = hypotenuse = radius = 6
s = arc length = (2 π r/360) x angle
x ordinate of point ‘x’ (A)= cos45 * 6
y ordinates of point ‘x’ (B)= sin45 * 6

So the final ordinates are,

x ordinate of point ‘point 2’ = A + sin45 * s
y ordinates of point ‘point 2 = B - cos45 * s

The result (x,y) is (7.574, 0.911)
Similarly do it for 60 and 80 degree angles. If you plot the final x,y coordinates of points 1, 2, 3 and 4 and join it as shown in the picture below, you get the involute curve…
That's all for this topic. I hope this helped you.