Table of Contents

## Explain in Detail :What is Involute?

The involute is a curve that is found by tracing the path of a point on the edge of a gear as it rotates. The curve generated by the point moving along the gear’s circumference while the gear itself rotates around a fixed point.

## Involutes of the Curves

- The involutes of the curves are the curves that obtained by tracing the points of the curves with a fixed radius.
- involutes of the circle are the straight lines that obtained by tracing the points of the circle with a fixed radius.
- The involutes of the ellipse are the hyperbolas that obtained by tracing the points of the ellipse with a fixed radius.

## 1) Involute of a Circle:

The involute of a circle is a curve that is tangent to the circle at the point of contact and is generated by a point on the circle that moves so that its distance from the center of the circle always equal to the radius of the circle.

## 2) Involute of a Catenary

The catenary the curve formed a freely hanging chain or cable. The curve is not a straight line, but has the shape of a parabola.

To generate the curve, we first need to find the equation of a parabola. We can do this by using the method of completing the square.

The equation of a parabola can written in the form:

y = ax2 + bx + c

We can use this equation to find the equation of the catenary.

To do this, we first need to find the value of a. This can done by using the formula:

a = –b / (2x)

We can then use this value to find the equation of the catenary.

The equation of the catenary can written in the form:

y = ax2 + bx

We can use this equation to find the coordinates of the points on the catenary.

## 3) Involute of a Deltoid

The deltoid is a three-dimensional muscle that is triangular in shape and has origins along the clavicle, the acromion, and the spine of the scapula. It inserts along the deltoid tuberosity of the humerus. The deltoid is responsible for shoulder abduction (lifting the arm away from the body), and it also assists with shoulder flexion (lifting the arm forward) and shoulder extension (lifting the arm back).