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There is a special branch of geometry known as Differential Geometry of The Curves. In this subject, the students study about the smooth curves lying in Euclidean space and the application of various methods of integral and differential calculus on them. In differential geometry, we come across with some different types of curves that are known as involutes. Actually, the involutes are the shapes associated with some other curves. These are also known as evolvent. An involute of a given curve is some other curve that always remains perpendicular to the tangent lines to that given curve. This can also be thought as the process of winding or unwinding a string tautly around a curve.

The involute was discovered in 1673 by Christiaan Huygens, a Dutch mathematician and physicist. In one of his book, he focused upon the theories about the motion of pendulum. On the basis of which, he introduced the concept of involute or evolvent of the curve. In this page, we are going to study about involute of different curves and their properties.



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In differential geometry, the involute is defined as a curve that is obtained by attaching an imaginary string and winding or unwinding it tautly on the given curve. The locus of the free end of this attached taut string is known as an involute or an evolvent.
The involute can also be defined in the following way :
The involute of a given curve is the locus of the free end of a line segment whose another end is tangent to the given curve. This line segment is wound or reversely unwound along with the continuous variation in the length of line segment by a certain amount. The curve so traced by free end is called an involute.
The evolute of an involute of a curve is referred to that original curve. In other words, the locus of the center of curvature of a curve is called evolute and the traced curve itself is known as the involute of its evolute.

Involutes of the Curves

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Involutes of the different curves are shown below.

1) Involute of a Circle:
The involute of a circle is similar to Archimedes spiral as given below.
Involute of a Circle

2) Involute of a Catenary:
The catenary is said to be curve which resembles with a hanging cable or chain, supported only by its ends. A catenary is a U-shaped curve, looks as parabola hanging on its own weight. The involute of the catenary through the vertex is called tractrix. It is shown in the following figure :
Involute of Catenary

3) Involute of a Deltoid:
A deltoid is a tricuspoid curve i.e. a curve with three cusps. This curve looks like Greek letter delta. The involute of deltoid is shown in the following figure :
Involute of Deltoid

4) Involute of a Parabola:
It is demonstrated by the diagram below :
Parabola Semicubical Parabola

5) Involute of an Ellipse:
The involute of an ellipse is shown in the following figure :
Ellipse Evolute


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The equations of involutes of different shapes are explained as below:

Circle Involute:

The parametric equations of the involute of a circle in the Cartesian coordinates are :
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:

In Cartesian coordinates, the parametric equations of the involute of a catenary are as follows:
x = t - tanh t
y = sech t
Where, t be the parameter.

Deltoid Involute:

The parametric equations of a deltoid are represented as the following :
x = 2 r cos t + r cos 2t
y = 2 r sin t - r sin 2t
Where, r = radius of rolling circle involved in formation of deltoid.

Involute of a Circle

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The involute of a circle is not only a theoretical concept, but also it has various applications in real life. One of the most common application is the use of this concept is design of the teeth of cogwheel or tooth-wheel used in the rotating machines.

This involute looks same as the Archimedes spiral. It has successive turns which are traced by keeping a constant distance among the parallel curves. Its parametric equations are given below :

In Cartesian Coordinates:

Let r be the radius of the circle and angle parameter be t, then

x = r (cos t + t sin t)

y = r (sin t - t cos t)

In Polar Coordinates:

If r and $\theta$ be the parameters, then

r = a sec $\alpha$

$\theta$ = tan $\alpha$ - $\alpha$

Where, a be the radius of circle.

Arc length
of circle involute:

The length of the arc of the involute of the circle is

L = $\frac{r}{2}$ $ t^{2}$

How to Draw Involute

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The involute of a curve may be drawn by the instructions given below in the following steps:
Step 1: Draw few number of tangents to the points on the given curve.

Step 2: First, pick two neighboring tangent lines. Extend them in opposite direction and find their intersection point. Now, with that endpoint as center and taking the distance between this center and the point of first tangent, an arc is drawn. In the following figure, let L$_{1}$ and L$_{2}$ be two successive tangents whose intersection point is X and XA be the radius. The arc AA$_{1}$ is obtained.
Involute Example

Step 3: Again, choose another two neighboring tangents L$_{2}$ and L$_{3}$. Taking their intersection point Y as center and distance YA$_{1}$ as radius, draw an arc A$_{1}$A$_{2}$. Refer the following image.
Involute Curve

Step 4: For the rest of the tangents, repeat the same process. In this way, we get a curve out of the arcs so constructed. This curve is the required involute of the given curve.


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The involutes of the curve have many applications. They are extremely useful for various industries and businesses. Some special properties of involutes make them very important for the gear industries. If two revolving machines and gears do have teeth following the shape of involute (not just a common triangular shape), the involute gear system is formed. In this case, the relative rotation rates must be constant at the time the teeth were engaged. Also, the gears should always make a contact with a single steady line of force.

When the teeth are of different shapes, the relative force and speed are not constant ; they rise and fall as the teeth engage. This results in noise, vibration and excessive wear of the teeth. Hence, almost all the modern gear teeth have the involute shape.

Especially, the involute of circle has important applications in scroll compressing and gas compressing. Scroll and gas compressors are build in this shape. Such equipments make less noise and so are very efficient.
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