How to Solve Involute Problems with Examples
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:
Draw a few number of tangents to the points given on the curve
Pick two neighboring tangent lines.
Extend these in opposite directions
Find their intersection point.
Now, Take that endpoint as center
Take the distance between the given center and the point of 1st tangent.
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.
Let us take another 2 neighboring tangents L2 and L3
Take their intersection point Y as center
Take distance YA1 as radius
Draw an arc A1A2
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.
FAQs on What Is an Involute in Maths?
1. What is an involute in mathematics?
In mathematics, an involute is a specific type of curve traced by a point on a taut string as it is unwrapped from a fixed curve. Imagine a string tightly wound around a circle; the path that the end of the string follows as it is unwound is the involute of the circle. The original curve from which the string is unwrapped is called the evolute of the involute.
2. What are the most important applications of an involute curve?
The most significant real-world application of the involute curve is in mechanical engineering, specifically in the design of gear teeth. Other important applications include:
Gear Systems: The profile of most modern gear teeth is an involute, which ensures smooth and efficient power transmission.
Pumps: In gerotor pumps and scroll compressors, involute shapes are used to create chambers that change volume, enabling fluid to be moved effectively.
Computer-Aided Design (CAD): The involute curve is a fundamental shape used in software for designing mechanical parts and systems.
3. How is an involute different from an evolute?
An involute and an evolute are two curves with a reciprocal relationship. The primary difference lies in their definition and geometric role:
Involute: It is the curve traced by the end of a string being unwrapped from another curve. It is essentially 'generated from' the base curve.
Evolute: It is the locus of all the centres of curvature of another curve. The evolute is the curve 'that generates' the involute.
In simple terms, if curve A is the evolute of curve B, then curve B is the involute of curve A.
4. Why is the involute shape considered ideal for gear teeth design?
The involute shape is ideal for gear teeth due to several key advantages. The primary reason is that it maintains a constant pressure angle between mating gears. This results in a smooth, uniform transfer of rotational motion and power. Furthermore, involute gears are less sensitive to small variations in the distance between their centres, making manufacturing and assembly more tolerant to minor errors.
5. What are the parametric equations for the involute of a circle?
For a circle with radius r centred at the origin (0,0), the parametric equations that describe its involute are:
x(t) = r (cos(t) + t sin(t))
y(t) = r (sin(t) - t cos(t))
Here, t is the parameter, which represents the angle (in radians) through which the taut string has been unwrapped from the circle.
6. Can an involute be generated from shapes other than a circle, for example, a square?
Yes, an involute can be generated from any convex shape, not just a circle. If you were to unwind a taut string from the perimeter of a square, the path traced by the end of the string would be the involute of the square. This path would consist of a series of connected circular arcs. Each arc is generated as the string unwraps from one side of the square, pivoting at each corner.
7. What is a key geometric property of the relationship between an involute and its original curve?
A fundamental geometric property is that the normal at any point on the involute curve is always tangent to the original base curve (the evolute). This means if you draw a line perpendicular to the involute at any point, that line will touch the original curve at a single point (the point of tangency). The length of this normal segment is equal to the length of the string that has been unwrapped.
