Involute of a Circle Formula Derivation and Properties
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 Involute Curve in Geometry and Calculus
1. What is an involute in mathematics?
An involute is a curve traced by the end of a taut string as it is unwound from another curve. In geometry, the involute depends on the original curve (called the evolute when reversed). For example, the involute of a circle forms the basis of involute gear teeth in engineering. The curve always remains perpendicular to the tangent of the original curve at the point of contact.
2. What is the formula of the involute of a circle?
The parametric equations of the involute of a circle of radius r are x = r(cos t + t sin t) and y = r(sin t − t cos t). Here, t is the parameter (angle in radians). These equations are derived by unwrapping a taut string from a circle and using trigonometric relationships.
3. How do you find the involute of a given curve?
The involute of a curve is found by subtracting the arc length along the curve in the direction of its unit tangent vector. The general formula is:
- If the curve is r(s), then the involute is R(s) = r(s) − (s − s₀)T(s)
- T(s) is the unit tangent vector
- s is the arc length parameter
4. What is the difference between evolute and involute?
An involute is formed by unwrapping a string from a curve, while an evolute is the locus of centers of curvature of a curve. Key differences include:
- Involute depends on arc length unwinding
- Evolute represents centers of curvature
- The involute of a curve has the original curve as its evolute
5. Why is the involute of a circle important in gear design?
The involute of a circle is used in gear teeth design because it ensures constant angular velocity ratio between meshing gears. Important reasons include:
- Maintains smooth power transmission
- Allows slight variation in center distance
- Reduces noise and wear
6. Can you give a simple example of an involute of a circle?
For a circle of radius r = 2, the involute equations are x = 2(cos t + t sin t) and y = 2(sin t − t cos t). For example, when t = 0:
- x = 2(1 + 0) = 2
- y = 2(0 − 0) = 0
7. What are the properties of an involute curve?
An involute curve has geometric properties related to tangency and curvature. Key properties include:
- The tangent to the involute is perpendicular to the string
- The original curve is the evolute of its involute
- The arc length equals the length of the unwound string
8. What is the arc length relationship in an involute?
In an involute, the arc length from the starting point equals the length of the string unwound from the original curve. If s is the arc length parameter, then the displacement along the tangent is proportional to (s − s₀). This direct relationship makes arc length central to involute construction.
9. Is the involute of a circle a spiral?
The involute of a circle resembles a spiral but is not a true spiral like an Archimedean spiral. Unlike standard spirals, its shape depends on the unwinding of a taut string from a circle. It gradually moves away from the circle while maintaining a perpendicular tangent relationship.
10. What are common mistakes when studying involutes?
Common mistakes in learning involute curves involve confusion with related concepts and incorrect formulas. Typical errors include:
- Mixing up involute and evolute definitions
- Forgetting to parameterize by arc length
- Using incorrect parametric equations for a circle
