Involute

What is Involute?

Involute is a special branch of geometry dealing with the study of differential geometry of curves.

Attach an imaginary string to a point on a curve. Extending the string wide and unwinding it on the given curves keeps the string always taut. The locus of all the points formed by the string is called the involute of the original curve and the traced curve is known as the involute of its evolute.

Involute was discovered by a Dutch mathematician and a physicist named Christine Huygens in 1673.

Let us study what is involute, how to draw an involute, involute curve, involute equation, involute of a circle, and involute applications.

What is Involute?

An involute is a particular type of curve that is dependent on another curve. An involute curve is the locus of taut string as the string is either unwind from or wind around the curve.

Involutes of the Curves

The involutes of the different involute curves as given below:

• Involute of a Circle

• Involute of a Catenary

• Involute of a Deltoid

• Involute of a Parabola

• Involute of an Ellipse

1) Involute of a Circle: 

The involute of the circle was first studied by Huygens, he got this idea when he was considering clocks without pendulums to be used on ships at sea. In his first clock without a pendulum, he used the circle involute and tried to force the pendulum to swing in the path of a cycloid.this curve is similar to Archimedes spiral

2) Involute of a Catenary 

Involute of a catenary appears to be a tractrix through the vertex. It looks like a hanging cable supported by its ends.

3) Involute of a Deltoid

4) Involute of a Parabola 

5) Involute of an Ellipse

Involute Equations

Let us study different involute equations.

• Circle Involute

• Catenary Involute

• Deltoid Involute

Circle Involute:

x = r (cos t + t sin t) , 

y = r (sin t – t cos t) ,  where, r = radius of the circle, t = parameter of angle in radian.

Catenary Involute:

x = t – tanh t, 

y = sech t, where t be the parameter.

Deltoid Involute:

 x = 2 r cos t + r cos 2t, 

y = 2 r sin t – r sin 2t

where, r = radius of rolling circle of deltoid.

Involute of a Circle

• In Cartesian Coordinates:

If r is the radius of the circle and the angle parameter is t, then
x = r (cos t + t sin t)
y = r (sin t – t cos t)

• In Polar Coordinates:

If r and θ are the parameters, then r = a sec α
θ = tan α – α, where, a be the radius of the circle.

• Arc length of circle involute:

The length of the arc of the involute of the circle is
L = (r/2) t2

How to Draw Involute

Now let us study how to draw involute by following given steps:

1. Draw a few number of tangents to the points given on the curve

2. Pick two neighboring tangent lines.

1. Extend these in opposite directions

2. Find their intersection point.

3. Now, Take that endpoint as center

4. Take the distance between the given center and the point of 1st tangent.

5. An arc will be drawn.

As shown in the following figure, let L1 and L2 be two successive tangents

Let X be their intersection point and XA be the radius.

So,The arc AA1 is obtained.

3. Let us take another 2 neighboring tangents L2 and L3

1. Take their intersection point Y as center

2. Take distance YA1 as radius

3. Draw an arc A1A2

4. Repeat the same process for the rest of the tangents. This way we will get a curve out of these arcs.. And we get the involute of the curve.

Involute Application

 Some of the involute applications are

• The involutes of the curve is widely used in industries and businesses.

• One of the major applications of Involute of circle is in designing of gears for revolving parts where gear teeth follow the shape of involute.

• This is more meaningful in engineering drawings.

• The basic application of involute usage is in winding clocks & toys wherein a winding key is used to motion the spiral spring in a circular involute.