Problem 19 - Thomas Calculus

Question

The involute of a circle If a string wound around a fixed circle is unwound while held taut in the plane of the circle, its end P traces an involute of the circle. In the accompanying figure, the circle in question is the circle \(x^{2}+y^{2}=1\) and the tracing point starts at \((1,0) .\) The unwound portion of the string is tangent to the circle at \(Q,\) and \(t\) is the radian measure of the angle from the positive \(x\) -axis to segment \(O Q .\) Derive the parametric equations $$ x=\cos t+t \sin t, \quad y=\sin t-t \cos t, \quad t>0 $$ of the point \(P(x, y)\) for the involute.

Step-by-Step Solution

Answer

The parametric equations are \(x = \cos t + t \sin t\) and \(y = \sin t - t \cos t\).

Step 1: Understand the Problem

We need to derive the parametric equations for the involute of a circle, which is the path traced by the end P of a taut string unwound from a circle. The given circle is described by the equation \(x^2 + y^2 = 1\), and the point starts at \((1, 0)\).

Step 2: Establish Relationship using Tangent Line

Consider that as the string is unwound, the point \(Q\) (where the string is tangent to the circle) moves along the circumference. The angle \(t\) (in radians) corresponds to the rotation from the positive x-axis to the line segment \(OQ\). As \(t\) increases, the arc length from \((1,0)\) to \(Q\) is \(t\). Thus, the arc length creates a correspondence between the angle \(t\) and the distance.

Step 3: Derive Parametric Representation for Tangent Point Q

The coordinates of the tangent point \(Q\) are found using the angle \(t\). For a unit circle, these are given as \(Q = (\cos t, \sin t)\). This is because \(Q\) is a point on the circle at angle \(t\) from the positive x-axis.

Step 4: Determine Vector along Tangent Line

The vector that represents the direction of the line (tangent at \(Q\)) is perpendicular to the radius \(OQ\). The direction of the tangent line is given by vector \((\sin t, -\cos t)\), since this vector is perpendicular to \((\cos t, \sin t)\) (the radius).

Step 5: Compute Involute Coordinates

The end of the string, point \(P(x, y)\), will move along the tangent direction \((\sin t, -\cos t)\), at a distance equivalent to the arc length \(t\). This gives us:- For the \(x\) coordinate: \(x = \cos t + t \cdot \sin t\).- For the \(y\) coordinate: \(y = \sin t - t \cdot \cos t\).

Step 6: Conclude the Parametric Equations

Thus, the parametric equations are derived as follows:\[x = \cos t + t \sin t\]\[y = \sin t - t \cos t\]where \(t > 0\) because the string unwinds from \((1,0)\) as \(t\) increases.

Key Concepts

Parametric Equations

Parametric equations are a way to express a curve by defining both the x and y coordinates as functions of a third variable, often called a parameter (commonly denoted as \(t\)). The beauty of parametric equations lies in their ability to depict complex curves that might be difficult to represent with a single equation. For the involute of a circle, we express both x and y in terms of \(t\) as follows:

\[x = \cos t + t \sin t\]
\[y = \sin t - t \cos t\]

Here, \(t\) represents the angle (in radians) from the positive x-axis to the tangent line at point Q on the unit circle. As the string unwinds, \(t\) increases and the point P traces a complex path (the involute). Each value of \(t\) yields a unique (x, y) coordinate on this path. The use of parametric equations makes it easier to visualize and calculate the properties of such paths that would be cumbersome using a single function approach.

Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing it. It represents the instantaneous direction of the curve at that point. In the context of a circle, the tangent line to a circle is perpendicular to the radius at the point of tangency. This relationship is crucial when considering the involute of a circle since the unwound string is always tangent to the circle.

For our problem, as the string unwinds, the length of the arc from the start point \((1,0)\) to the tangent point \((Q)\) is precisely measured by the parameter \(t\). The tangent direction at any point \(Q = (\cos t, \sin t)\) is given by the vector \((\sin t, -\cos t)\). This vector is perpendicular to the radius direction \((\cos t, \sin t)\) because their dot product is zero:

\[\cos t \cdot \sin t + \sin t \cdot (-\cos t) = 0\]

Thus, the vector \((\sin t, -\cos t)\) is the direction in which the point \(P\) moves as the string continues to unwind, making it a key aspect in deriving the parametric equations for the involute.

Arc Length

Arc length is the distance along a section of a curve. For a circle of radius 1, it relates directly to the central angle in radians. In our scenario, the involute of the circle is affected by the arc length since the length of the string becomes the arc length as it unwinds from the circle.

The arc length from the starting point \((1,0)\) to the point \((Q)\) is \(t\), correspond to the angle \(t\) formed with the x-axis. Since we are dealing with a unit circle (radius 1), the formula to calculate the arc length simplifies to the angle in radians. As \(t\) increases, the length of the unwound string is equal to this arc length \(t\), and it defines how far away point \(P\) is from point \(Q\) along the tangent vector.

Understanding arc length in relation to the involute helps to connect the geometric visualization with the algebraic parametric equations. Each increase in \(t\) not only changes the angle but also extends the path traced by point \(P\), thereby creating the unique shape of the involute.