Involute: Introduction, Equations & Sample Questions

Involute is a curve, derived by an imaginary attached string which winds and unwinds it tautly on the curve. It is counted as the locus of the free end of this particular string. So, involute basically deals with the study of differential geometrical curves. It can be explained with the concept of Permutations and Combinations. Let’s take an example; there is a type of number lock that would only open if you insert the perfect number combination. We can see this kind of lock on briefcases mostly. Now, suppose you do not know the exact combination of the lock and you have to find it, then you must go through all the possible combinations right? Now, that’s exactly what happens in the case of Involute. In this article, we will discuss this concept in detail.

Key Takeaways: involute, circle, curves, lines, imaginary, string

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What is Involute?

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In differential geometry, an Involute is a particular kind of a curve that depends on another curve. It is the Locus of the free end of that particular imaginary string which is attached to the curve, winding and unwinding tautly on the mentioned curve.

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Involute of Curves

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This includes- 

• Involute of a Circle
• Involute of a Catenary
• Involute of a Deltoid
• Involute of a Parabola
• Involute of a Ellipse

Now we will discuss these in detail.

Involute Of a Circle

This structure exactly looks like the Archimedes spiral. According to Encyclopedia Britannica- “Spiral of Archimedes- Archimedes only used geometry to study the curve that bears his name. In modern notation it is given by the equation r = aθ, in which a is a constant, r is the length of the radius from the center, or beginning, of the spiral, and θ is the angular position (amount of rotation) of the radius.)

It looks just like the picture below-

Involute Of a Circle

A spiral staircase can be a similar structure with this.

Involute of a Catenary

While walking through a narrow lane, we can see a few ‘U’-shaped hanging cables that are only supported by their both ends. This Involute looks just like those hanging cables.

Involute of a Catenary

In this case, we need to learn about the term- ‘Tractrix’ as well. The Tractrix can be defined as the catenary insolute, through the vertex.

Involute of a Deltoid

The Involute of a Deltoid is actually a tricuspid curve which has three cusps. It looks like the Greek word delta. Let’s have a look at the picture below.

Involute of a Deltoid

Involute of a Parabola 

Involute of a Parabola

Involute of a Ellipse

Involute of a Ellipse

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Equations

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Here we will mainly discuss three involute equations,

• Circle Involute: 

x = r(cos t + t sin t)

y= r(sin t – t cos t)

Here, r = radius of the circle

t = perimeter of angle in radian

• Catenary Involute:

x = t – tanh t

y = sech t

Here, t = the perimeter

• Deltoid Involute:

x = 2 r cos t + r cos 2t

y = 2 r sin t – r sin 2t

Here, r = radius of rolling circle that is involved in formation of the deltoid.

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Process of drawing an Involute

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1. In the beginning, we need to draw a curve and decide some points on it. In addition to this, we need to draw a few tangents on that curve from those previously decided points.
2. After that, we need to choose two nearest tangent lines and extend them to opposite directions until they intersect each other. Mark the point where they intersect each other.

Now, taking that intersecting point as the center and the distance between that center and the first tangent point, we have to draw an arc.

Here, L₁ and L₂ are the two chosen tangents, X is the radius and AA₁ is the arc we get.

3. Similarly, by taking L₂ and L₃ tangents, we get their intersecting point – Y. And considering Y as the center along with YA1 as the radiu, we can draw the arc A₁A₂.
4. Finally, by repeating the same process mentioned before, we need to create arcs for every tangent. As a result we will get the Involute of the main curve as required.

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Some real-life application of Involute

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1. This concept of Involute is hugely used while building various gears for machines.
2. The concept of Involute is used in various architectural drawings, specially while designing a modern spiral staircase.

Also Read:

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Things to Remember

• Involute is a curve that is derived by an imaginary attached string which winds and unwinds it tautly on the mentioned curve.
• Involute deals with the differential geometrical curves. While evolute is the original curve of Involute.
• Tractrix is the catenary insolute, through the vertex.
• Circle Involute: x = r(cos t + t sin t) and y= r(sin t – t cos t) {Where,r = radius of the circle and t = perimeter of angle in radian}
• Catenary Involute: x = t – tanh t and y = sech t {Here, t = the perimeter}
• Deltoid Involute: x = 2 r cos t + r cos 2t and y = 2 r sin t – r sin 2t {Here, r = radius of rolling circle that is involved in formation of the deltoid.}
• Business application of Involute can be seen while making various gears.

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