Question

Besides the usual Cartesian coordinates \((u, v)\) with \(u, v \in \mathbb{R}\), we represent the points of the plane by polar coordinates \((r, \varphi)\) with \(r \in \mathbb{R}\) and \(0 \leq \varphi<2 \pi\). This representation is not unique; for example, when \(\varphi<\pi\) then \((r, \varphi)\) and \((-r, \varphi+\pi)\) represent the same point. We obtain the polar coordinates from the Cartesian ones by the formulas \(u=r \cos \varphi\), and \(v=r \sin \varphi\). Now consider the curve \(C=\\{(r, \varphi): 0 \leq \varphi<2 \pi\) and \(r=\sin 2 \varphi\\} \subseteq \mathbb{R}^{2}\), and let \(I=\left\langle\left(x^{2}+y^{2}\right)^{3}-4 x^{2} y^{2}\right\rangle \subseteq \mathbb{R}[x, y]\). (i) Create a plot of \(C\). (ii) Using the addition formulas for sine and cosine, show that \(C \subseteq V(I)\). (iii) Prove that also the reverse inclusion \(V(I) \subseteq C\) holds (be careful with the signs).

Short answer

Plot a lemniscate, verify \(C \subseteq V(I)\) using trigonometric identities, and confirm \(V(I) \subseteq C\) by reverse calculation.

Step-by-step solution

Understanding Polar Coordinates

Polar coordinates \((r, \varphi)\) are a different way to describe a point in the plane. Here, \(r\) is the radial distance from the origin, and \(\varphi\) is the angle from the positive \(u\)-axis. When converting from Cartesian to polar coordinates, use the transformations: \(u = r \cos \varphi\) and \(v = r \sin \varphi\).

Plotting the Curve C

The curve \(C\) is given by \((r, \varphi): 0 \leq \varphi < 2\pi, r = \sin 2\varphi\). To plot \(C\), note that \(r = \sin 2\varphi\) oscillates between -1 and 1. Therefore, plot the points in polar coordinates where the angle \(\varphi\) varies, and mark the radius equal to \(\sin 2\varphi\). This results in a figure-eight pattern, known as a lemniscate.

Using Addition Formulas

The curve transforms to Cartesian coordinates using \(u = \sin 2\varphi \cos \varphi\) and \(v= \sin 2\varphi \sin \varphi\). Using the angle addition formula for sine, we apply \(\sin 2\varphi = 2 \sin \varphi \cos \varphi\). Substitute \(r = \sin 2\varphi\) into these expressions.

Simplifying the Curve Equation

Substitute \(u = (2\sin^2 \varphi)(\cos \varphi)\) and \(v = (2\cos^2 \varphi)\sin \varphi\). From this, \(x = r \cos \varphi\) and \(y = r \sin \varphi\). Thus, \(x^2 + y^2 = r^2 = \sin^2 2\varphi\). Verify that \((x^2 + y^2)^3 = 4x^2 y^2\). This confirms \(C \subseteq V(I)\).

Prove Reverse Inclusion V(I) ⊆ C

The reverse inclusion \(V(I) \subseteq C\) needs verifying points \((x,y)\) satisfying \(\left(x^2 + y^2\right)^3 = 4 x^2 y^2\) lead back to the polar condition. Express \(\tan \varphi = y/x\), check \(r = \sin 2\varphi = \sqrt{x^2 + y^2}\), following a similar method for angles and using the identities. Verify \( r=(\sin 2\varphi) = y/x\) leads points back to criterion pattern for \(C\). This checks reverse inclusion holds too.

Key concepts

Cartesian coordinates

In the world of geometry, Cartesian coordinates give us a straightforward way to pinpoint a spot on a plane, using two numbers. These numbers are typically referred to as \((u, v)\), where both \(u\) and \(v\) are real numbers. This coordinate system is attributed to the French mathematician Ren\'e Descartes, and it works by assigning each point a unique pair of numbers representing horizontal (\(u\)) and vertical (\(v\)) distances from a fixed point known as the origin.

In simpler terms, imagine a grid with horizontal and vertical lines where each intersection point is determined by how far you are from the axes.

• The decimal values of \(u\) describe how far left or right you go.
• The decimal values of \(v\) describe how far up or down you go.

Understanding Cartesian coordinates is crucial for converting between different systems—like turning Cartesian coordinates into polar coordinates, which involve a radius and angle instead of two distances.

lemniscate

A lemniscate is a beautiful, symmetrical figure that resembles the shape of a figure-eight or an infinity symbol. This mesmerizing looped shape can be represented mathematically, and one way is via the polar coordinate equation \(r = \sin 2\varphi\). In polar coordinates, \(r\) represents the radius—how far a point is from the center—and \(\varphi\) is the angle. As the angle \(\varphi\) changes from 0 to \(2\pi\), \(r = \sin 2\varphi\) creates the up-and-down swing that traces the loops of the lemniscate.

The curve oscillates between 1 and -1, giving it its distinct looped structure. This pattern forms beautifully, similar to how harmonic waves would look if plotted in a circle. It's interesting to see this form naturally appear in various contexts, from certain route calculations to even in art and architecture, due to its pleasant symmetry and balance.

angle addition formulas

Angle addition formulas are essential tools in trigonometry, used for calculating the sine or cosine of two angles combined. They provide a simpler way to handle complex trigonometric expressions and are super helpful when switching between coordinate systems, especially when working with curves in polar and Cartesian coordinates.

For instance, the sine angle addition formula is

• \(\sin(a + b) = \sin a \cos b + \cos a \sin b\)

This helps to expand or transform expressions involving sums of angles. Similarly, the cosine angle addition formula is expressed as:

• \(\cos(a + b) = \cos a \cos b - \sin a \sin b\)

These formulas become especially handy when proving properties of complex geometrical shapes, like the lemniscate, in different coordinate systems. By using these formulas, you can reduce the intricacies of the polar curve's behavior into manageable pieces, which can then be translated into Cartesian terms or vice versa.