Question

Sketch the graph of the four-cusped hypocycloid \(x=a \sin ^{3} t, y=a \cos ^{3} t, 0 \leq t \leq 2 \pi\), and find its length. Hint: By symmetry, you can quadruple the length of the first quadrant portion.

Short answer

The length of the four-cusped hypocycloid is \(3a\).

Step-by-step solution

Understand the Parameters

The given equations, \(x = a \sin^3 t\) and \(y = a \cos^3 t\), define a four-cusped hypocycloid. Here, \(a\) is a constant scaling factor, and \(t\) ranges from 0 to \(2\pi\), describing a complete revolution.

Sketch the Graph

To sketch the graph, consider the parametric equations. For \(t = 0\), \((x,y) = (0, a)\). As \(t\) varies from 0 to \(\pi/2\), \(x\) and \(y\) trace one loop in the first quadrant. By symmetry (the function is symmetric with respect to both axes), these loops repeat in the other quadrants.

Calculate the Arc Length for First Quadrant

Focus on calculating the arc length in the first quadrant: \(0 \leq t \leq \pi/2\). Use the arc length formula for parametric equations \(L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\).

Differentiate

Find the derivatives: \(\frac{dx}{dt} = 3a \sin^2 t \cos t\) and \(\frac{dy}{dt} = -3a \cos^2 t \sin t\).

Apply Arc Length Formula

Calculate:\[\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = (3a \sin^2 t \cos t)^2 + (-3a \cos^2 t \sin t)^2 = 9a^2 \sin^4 t \cos^2 t + 9a^2 \cos^4 t \sin^2 t.\]Combine terms:\[9a^2 \sin^2 t \cos^2 t (\sin^2 t + \cos^2 t) = 9a^2 \sin^2 t \cos^2 t.\]

Simplify and Solve Integral

Simplify and solve the integral:\[L = \int_{0}^{\pi/2} \sqrt{9a^2 \sin^2 t \cos^2 t} \, dt = 3a \int_{0}^{\pi/2} \sin t \cos t \, dt.\]Use substitution: Let \(u = \sin t\), \(du = \cos t \, dt\), then:\[L = \frac{3a}{2} \int_{0}^{1} u \, du = \frac{3a}{2} \left[\frac{u^2}{2}\right]_{0}^{1} = \frac{3a}{4}.\]

Calculate Full Length

The full length of the hypocycloid is four times the length of the arc in the first quadrant:\[L_{total} = 4 \times \frac{3a}{4} = 3a.\]

Key concepts

Parametric Equations

Parametric equations allow us to express coordinates as functions of a parameter, often denoted as \( t \). In the case of the four-cusped hypocycloid, the parametric equations are given by \( x = a \sin^3 t \) and \( y = a \cos^3 t \). Here, \( a \) is a constant that scales the size of the hypocycloid. The parameter \( t \) ranges from 0 to \( 2\pi \), sweeping through the entire graph.

These parametric equations describe each coordinate with respect to the parameter, instead of as a function of each other, making it easier to represent complex curves. The beauty of this method is evident in the tracing of complex loops and shapes like the four-cusped hypocycloid. As \( t \) varies from 0 to \( 2\pi \), the equations guide the position on the plane, constructing the full graph.

These specific equations are crucial as they inherently possess characteristic "cusps" or pointed intersections at certain values of \( t \). This happens when both derivatives of \( x \) with respect to \( t \) and \( y \) with respect to \( t \) have critical points. Knowing how to set and manipulate parametric equations is essential to sketching and understanding complex shapes.

Arc Length Calculation

To determine the arc length of a parametric curve, we employ the general formula for the arc length of parametric equations: \[ L = \int_{a}^{b} \sqrt{ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 } \, dt \] where \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) are the derivatives of \( x \) and \( y \) with respect to \( t \).

For the first quadrant of the four-cusped hypocycloid, \( t \) ranges from 0 to \( \pi/2 \). Calculating the derivatives, we find: \( \frac{dx}{dt} = 3a \sin^2 t \cos t \) and \( \frac{dy}{dt} = -3a \cos^2 t \sin t \).

Substituting these into the arc length formula, we obtain: \[ L = 3a \int_{0}^{\pi/2} \sin t \cos t \, dt \] Using the substitution \( u = \sin t \), with \( du = \cos t \, dt \), simplifies the integral to \( L = \frac{3a}{2} \int_{0}^{1} u \, du \), which evaluates to \( \frac{3a}{4} \).

This value represents the arc length of one quadrant. By leveraging the symmetry of the graph, the complete arc length is four times this value, giving \( 3a \) as the full length.

Symmetry in Graphs

Symmetry plays a crucial role in simplifying complex graph analysis such as our four-cusped hypocycloid. The symmetry of a graph indicates that one part of it mirrors another. For this hypocycloid, symmetry is present with respect to both the x-axis and y-axis. This means the shape and size of each quadrant are identical.

In practice, this allows us to calculate properties like arc length more efficiently. By determining the arc length in just one quadrant (from 0 to \( \pi/2 \)), we can simply multiply by four to find the total length. This is highly beneficial as it reduces the complexity required in calculations.

Recognizing and utilizing graphical symmetry minimizes work and potential errors. When sketching the entire hypocycloid, the existence of symmetry obliges us to only sketch a portion, then stabilize it rotationally or reflectively to get the full figure.

Graphical symmetry isn't just about aesthetics – it significantly eases the stress of computational demands, particularly when the shape gets more intricate.

Derivative Calculation

Derivative calculation is an essential aspect of understanding and working with parametric equations. It helps ascertain the rate of change of the coordinates concerning the parameter \( t \). For our hypocycloid, the derivatives are critical in deriving the arc length.

To find the derivative of the function \( x = a \sin^3 t \), the chain rule of differentiation leads us to \( \frac{dx}{dt} = 3a \sin^2 t \cos t \). Similarly, for \( y = a \cos^3 t \), the derivative is \( \frac{dy}{dt} = -3a \cos^2 t \sin t \).

These derivatives not only contribute to our understanding of the curve in terms of offensive and defensive points (cusps) but also play a pivotal part in computing the arc length in parametric forms. Accurate differentiation is indispensable in plot navigations and further characterizes the nature of all points on the hypocycloid.

Learning to differentiate parametric equations effectively provides deep insight into not just tracing curves, but also their velocities and directed paths over specified intervals.