Question

The Lamé curve described by $$\left|\frac{x}{a}\right|^{n}+\left|\frac{y}{b}\right|^{n}=1$$ where \(a, b,\) and \(n\) are positive real numbers, is a generalization of an ellipse. a. Express this equation in parametric form (four sets of equations are needed). b. Graph the curve for \(a=4\) and \(b=2,\) for various values of \(n\) c. Describe how the curves change as \(n\) increases.

Short answer

Question: Express the Lamé curve equation in parametric form, graph the curve for specific values of \(a\) and \(b\), and for various values of \(n\). Describe how the curves change as \(n\) increases. Answer: The parametric equations for the Lamé curve in different quadrants are: Quadrant I: $$x_1(t) = a t^n$$ $$y_1(t) = b (1-t)^n$$ Quadrant II: $$x_2(t) = a t^n$$ $$y_2(t) = -b (1-t)^n$$ Quadrant III: $$x_3(t) = -a t^n$$ $$y_3(t) = b (1-t)^n$$ Quadrant IV: $$x_4(t) = -a t^n$$ $$y_4(t) = -b (1-t)^n$$ When graphing the curve for \(a = 4\) and \(b = 2\), and various values of \(n\), we observe the following: - When \(n = 1\), the curve is a diamond shape with the vertices touching the coordinate axes. - When \(n = 2\), the curve is an ellipse. - As \(n\) increases, the curves become more rectangular with rounded corners. - As \(n\) approaches infinity, the curve tends to a rectangle with vertices at \((\pm a, \pm b)\).

Step-by-step solution

Convert the Lamé curve equation into parametric form

Observe that the given equation can be split into four cases based on the signs of x and y. Let's consider the parameter \(t\) such that \(t \in [0, 1]\). Case 1: \(x \ge 0\) and \(y \ge 0\). In this case, we have: $$(\frac{x}{a})^{n}+(\frac{y}{b})^{n}=1$$ Let \(x = a t^n\) and \(y = b (1-t)^n\). Substituting these into the equation gives: $$(\frac{a t^n}{a})^{n}+(\frac{b (1-t)^n}{b})^{n} = t^n+(1-t)^n=1$$ So, the parametric equations for this case are: $$x_1(t) = a t^n$$ $$y_1(t) = b (1-t)^n$$ Case 2: \(x \ge 0\) and \(y \le 0\). Following a similar process as before, the parametric equations for this case are: $$x_2(t) = a t^n$$ $$y_2(t) = -b (1-t)^n$$ Case 3: \(x \le 0\) and \(y \ge 0\). The parametric equations for this case are: $$x_3(t) = -a t^n$$ $$y_3(t) = b (1-t)^n$$ Case 4: \(x \le 0\) and \(y \le 0\). The parametric equations for this case are: $$x_4(t) = -a t^n$$ $$y_4(t) = -b (1-t)^n$$

Graph the curve for the given values of \(a\) and \(b\), and various values of \(n\)

In this step, plot the parametric equations for each case at \(a = 4\) and \(b = 2\), for various values of \(n\), such as \(n = 1, 2, 3, \dots\) Using graphing software or tools will make this step easier.

Describe how the curves change as \(n\) increases

Observe the plotted curves, and describe how the shape of the Lamé curve changes as \(n\) increases. Notice that: - When \(n = 1\), the curve is a diamond shape with the vertices touching the coordinate axes. - When \(n = 2\), the curve is an ellipse. - As \(n\) increases, the curves become more rectangular with rounded corners. - As \(n\) approaches infinity, the curve tends to a rectangle with vertices at \((\pm a, \pm b)\).

Key concepts

Parametric Equations

Parametric equations play a crucial role in representing curves by expressing the coordinates of the points that make up the curve as functions of a variable, typically referred to as the parameter. In the case of the Lamé curve, which is defined by the equation:\[\left|\frac{x}{a}\right|^{n}+\left|\frac{y}{b}\right|^{n}=1\]where variables such as \(a\), \(b\), and \(n\) are involved, rewriting this in terms of a parameter allows for easier graphing and analysis. The Lamé curve involves four cases, based on the signs of \(x\) and \(y\):

• Case 1: Both \(x\) and \(y\) are non-negative, leading to the parametric forms \(x_1(t) = a t^n\) and \(y_1(t) = b (1-t)^n\).
• Case 2: \(x\) is non-negative while \(y\) is non-positive, resulting in \(x_2(t) = a t^n\) and \(y_2(t) = -b (1-t)^n\).
• Case 3: \(x\) is non-positive and \(y\) is non-negative, giving \(x_3(t) = -a t^n\) and \(y_3(t) = b (1-t)^n\).
• Case 4: Both \(x\) and \(y\) are non-positive, producing the equations \(x_4(t) = -a t^n\) and \(y_4(t) = -b (1-t)^n\).

These parametric equations make it possible to plot each segment of the curve effectively using a single parameter \(t\), typically ranging within 0 to 1.

Graphing Lamé Curves

Graphing Lamé curves involves plotting each set of parametric equations to see the full curve in the coordinate plane. With the given values \(a = 4\) and \(b = 2\), each quadrant of the Lamé curve can be graphed individually by plotting each pair of parametric equations for varying \(n\) values. A graphing software or tool is a helpful resource to visualize these changes accurately.
Start with a specific value for \(n\), such as 1, which gives a distinct shape. Each increase in \(n\) modifies the curve, transforming it from a diamond-like form to shapes with more rounded edges.

• For \(n = 1\), the graph appears as a diamond, with points touching the axes.
• At \(n = 2\), the figure becomes an ellipse, traditionally characterized by smoother shapes.
• As \(n\) grows (e.g., 3, 4, etc.), the resulting shapes increasingly resemble a rectangle with curved edges.

Using the parametric equations helps in systematically drawing fractions of the curve and thus completing the graph representation step by step.

Ellipse Generalization

The Lamé curve is recognized as an ellipse generalization, which means it extends the concept of an ellipse by modifying certain characteristics. When analyzing the equation \(\left|\frac{x}{a}\right|^{n}+\left|\frac{y}{b}\right|^{n}=1\), it can be seen that it reduces to the standard ellipse equation when \(n = 2\). This relationship forms the basis for calling the Lamé curve a generalization.

• With \(n = 2\), the curve satisfies the ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
• For \( n < 2 \), parts of the curve appear pointed or more diamond-like as seen with \(n = 1\).
• For \(n > 2\), the structure transitions into more rectangular forms, underscoring the influence of \(n\) as it molds the curve from an ellipse to potentially a rectangle.

Understanding this progression helps in comprehending how the Lamé curve modifies classical curves by tweaking the power \(n\). Essentially, these modifications reflect a continuum from traditional ellipses to what might resemble a rectangle, showing how adjusting parameters can lead to a wide array of curve shapes.