Cartesian lemniscate Find the equation in Cartesian coordinates of the lemniscate r² = a² cos 2θ, where a is a real number.

Verified step by step guidance

1. Recall that the given lemniscate is expressed in polar coordinates as \(r^{2} = a^{2} \cos 2\theta\), where \(r\) is the radius and \(\theta\) is the angle.
2. Use the double-angle identity for cosine: \(\cos 2\theta = \cos^{2} \theta - \sin^{2} \theta\).
3. Express \(r\), \(\cos \theta\), and \(\sin \theta\) in terms of Cartesian coordinates: \(r = \sqrt{x^{2} + y^{2}}\), \(\cos \theta = \frac{x}{r}\), and \(\sin \theta = \frac{y}{r}\).
4. Substitute these into the equation: \(r^{2} = a^{2} \left( \frac{x^{2}}{r^{2}} - \frac{y^{2}}{r^{2}} \right)\), which simplifies to \(r^{2} = a^{2} \frac{x^{2} - y^{2}}{r^{2}}\).
5. Multiply both sides by \(r^{2}\) to eliminate the denominator, resulting in \(r^{4} = a^{2} (x^{2} - y^{2})\), and then replace \(r^{2}\) by \(x^{2} + y^{2}\) to get the Cartesian form: \((x^{2} + y^{2})^{2} = a^{2} (x^{2} - y^{2})\).

Key Concepts

Polar to Cartesian Coordinate Conversion

Polar coordinates (r, θ) relate to Cartesian coordinates (x, y) through the formulas x = r cos θ and y = r sin θ. Understanding this conversion is essential to rewrite equations given in polar form into Cartesian form by expressing r and θ in terms of x and y.

Double-Angle Trigonometric Identities

The double-angle identity for cosine, cos 2θ = cos² θ - sin² θ, helps express trigonometric functions of 2θ in terms of sin θ and cos θ. This identity is crucial for manipulating the given polar equation to relate it to x and y coordinates.

Equation Manipulation and Substitution

After converting r and θ to x and y, algebraic manipulation and substitution are needed to eliminate θ and express the equation purely in terms of x and y. This process involves squaring, factoring, and rearranging terms to achieve the Cartesian form of the lemniscate.