In this video I go over an example on Calculus with Parametric Curves and this time describe the “involute” of a circle as a pair of Parametric Equations. The involute of a circle is the curve formed by tracing the path of the end of string as it is being unwound from the circle, while being held “taut”, or pulled tight so there is no slack. To determine the coordinates of the point on the involute curve, I show how we can use basic trigonometry to create two right-angle triangles, which we then can determine the coordinates by adding or subtracting the vertical or horizontal components. I also use the definition of arc length in establishing the length of the taut string in relation to the radius of the circle. This involute curve is a very interesting curve, and I will be using these derived parametric equations in my next video on the area of grazing a cow has so make sure to watch this video, and stay tuned for the next!
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Example
A String is wound around a circle and then unwound while being held taut.
The curve traced by the point P at the end of the string is called the involute of the circle.
If the circle has radius r and center O and the initial position of P is (r , 0), and if the parameter θ is chosen as in the figure below, show that the parametric equations of the involute are: