Orthogonal Curves

Section 1.13 Orthogonal Curves

Subsection 1.13.1 Recommended Tutorials

Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:

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Subsection 1.13.2 Introduction

Orthogonal curves are curves that are perpendicular whenever they intersect. Perpendicular lines are an elementary example of this.

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Recall that if \(m_1\) and \(m_2\) are the slopes of two perpendicular lines, then \(m_1 m_2 = -1\text{.}\) Similarly, two curves are orthogonal if their derivatives multiply to \(-1\) at every point where they intersect.

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The majority of curves in this activity are implicit functions, so you will be using the implicitplot() command to visualize them. Make sure to include the plots package using the with() command near the top of your Maple worksheet.

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Exercises 1.13.3 Exercises

1.

In this exercise, you will show that

\begin{equation*} y^2 - x^2 = 3 \text{ and } xy = 2 \end{equation*}

are orthogonal curves. Start by assigning each equation a name, such as curve1 and curve2.

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(a)

Using a single implicitplot() command, plot both curves. The two curves should intersect at two points. If you enforce \(1:1\) scaling, do they appear to be perpendicular at the intersection points?

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Hint 1.

Do not forget to include multiplication between \(x\) and \(y\) when defining the equation of the second curve.

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Hint 2.

\(1:1\) scaling can be enfored by adding the optional scaling=constrained parameter to the implicitplot() command.

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Hint 3.

When plotting an equation using the implicitplot() command, the graph may not always appear smooth. This was especially common in earlier editions of Maple. If this is ever the case, you may include either numpoints=30000 or grid=[250,250] as a parameter in the implicitplot() command to increase the number of plot points.

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(b)

Solve the system of two equations to find the \(x\)- and \(y\)-coordinates of the two points of intersection.

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Hint.

In order to find points of intersection, Maple can solve a system of equations in one solve() or fsolve() command for both \(x\) and \(y\text{.}\) If you choose to use the solve() command, you may need to include the optional parameter explicit=true to avoid the RootOf() output.

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A similar example is detailed in Example 13.3.

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(c)

Use the implicitdiff() command to determine the derivatives of each curve at the first intersection point. Do these slopes multiply to \(-1\text{?}\)

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Hint.

You will need to substitute both an \(x\)-coordinate and a \(y\)-coordinate into the derivative for each point when using the subs() command.

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(d)

Use the implicitdiff() command to determine the derivatives of each curve at the second intersection point. Do these slopes multiply to \(-1\text{?}\)

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2.

In this exercise, you will investigate two families of curves given by

\begin{align} \amp y = cx^2 \tag{1.2}\\ \amp x^2 + 2y^2 = k \amp k \amp> 0 \tag{1.3} \end{align}

where \(c\) and \(k\) are constants. Each choice of \(c\) and \(k\) gives a slightly different curve of the same “family”.

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(a)

Using a single plot() command, visualize these families of curves by plotting all of the following examples in one graph:

\begin{align*} y \amp = -2x^2 \amp x^2 + 2y^2 \amp = 1\\ y \amp = -x^2 \amp x^2 + 2y^2 \amp = 4\\ y \amp = 0 \amp x^2 + 2y^2 \amp = 9\\ y \amp = x^2\\ y \amp = 2x^2 \end{align*}

Ensure that all five curves of the \(y=cx^2\) family are the same colour and all three curves of the \(x^2 + 2y^2 = k\) family are a single, different colour.

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Hint.

Be sure to include the y= in each of the curves from the first family when using implicitplot(). This command only accepts equations as curves to be plotted.

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(b)

Assign equation (1.2) to family1 and equation (1.3) to family2. Solve the system of equations to find the \(x\)- and \(y\)-coordinates of the four intersection points of the two families of curves. The coordinates of the points should be given as expressions involving \(c\) and \(k\text{.}\)

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Hint.

When using the solve() command, you may need to include the optional parameter explicit=true to avoid the RootOf() output.

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(c)

Compute the derivative of family1 and assign it to dydx1. Compute the derivative of family2 and assign it to dydx2.

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(d)

Substitute each of the four points from part (b) into the product dydx1*dydx2 to show that the curves are orthogonal at each of the four intersection points.

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