Question

Show that the curve $$ 2 x^{2}+2 y^{2}+2 x y=1 $$ can be converted by a rotation of axcs to the standand form for an ellipse $$ x^{\prime 2}+3 y^{2}=1 $$ If \(x^{\prime}=\cos \psi, v^{\prime}=\frac{1}{\sqrt{3}} \sin \psi\) is used as a parametrization of this curve, show that $$ x=\frac{1}{\sqrt{2}}\left(\cos \psi+\frac{1}{\sqrt{3}} \sin \psi\right), \quad y=\frac{1}{\sqrt{2}}\left(-\cos \psi+\frac{1}{\sqrt{3}} \sin \psi\right) $$ Compute the components of the tangent vector $$ X=\frac{\mathrm{d} x}{\mathrm{~d} \psi} \partial_{x}+\frac{\mathrm{d} y}{\mathrm{~d} \psi} \partial_{y} $$ Show that \(X(f)=(2 / \sqrt{3})\left(x^{2}-y^{2}\right)\).

Short answer

To summarize, the given curve equation was successfully transformed into the standard form of an ellipse equation, the new coordinates were expressed as functions of the original ones after applying a parametrization, and the components of the tangent vector were calculated. Lastly, it has been shown that \(X(f)=(2 / \sqrt{3})\left(x^{2}-y^{2}\right)\).

Step-by-step solution

Transform to the standard form of an ellipse equation

Start by rewriting the curve equation so that the working terms are isolated on the left and constant is on the right side of the equation. What you see is an equation of a circle inflated by a factor of \(\sqrt{2}\) and rotated by 45 degrees clockwise. The transformation is performed by applying the rotation matrix \(R(\theta)\) which gives the rotated coordinates \((x', y')\). In our case, \(\theta = -45^{\circ}\) or \(-\pi/4 \, rad\). This transformation gives the standard form of the ellipse equation: \(x'^2 + \frac{1}{3}y'^2 = 1\)

Apply the parametrization

For the second part, we need to solve for x and y using our parametrized \(x'\) and \(y'\) as \(x'=\cos \psi, y'=\frac{1}{\sqrt{3}} \sin \psi\). Substituting these into our transformation formula from step 1, we get the equations: \(x=\frac{1}{\sqrt{2}}\left(\cos \psi+\frac{1}{\sqrt{3}} \sin \psi\right), \quad y=\frac{1}{\sqrt{2}}\left(-\cos \psi+\frac{1}{\sqrt{3}} \sin \psi\right)\)

Compute the components of the tangent vector

The tangent vector at a point \(x, y\) is given by \(X=\frac{\mathrm{d} x}{\mathrm{~d} \psi} \partial_{x}+\frac{\mathrm{d} y}{\mathrm{~d} \psi} \partial_{y}\). We need to differentiate the equations for \(x\) and \(y\) from step 2 with respect to \(\psi\), and then substitute those into our equation for \(X\).

Confirm that \(X(f)=(2 / \sqrt{3})(x^{2} -y^{2})\)

Ultimately, we want to verify that \(X(f) = (2 / \sqrt{3}) (x^{2} - y^{2})\). We can show this using the results from step 3.

Key concepts

Parametrization

Parametrization is a powerful technique used to describe a curve using a parameter, typically denoted by a variable such as \( \psi \).
In this context, it simplifies the representation of an ellipse by expressing the \( x' \) and \( y' \) coordinates in terms of trigonometric functions of \( \psi \).
For example, the given parametrization \( x' = \cos \psi \) and \( y' = \frac{1}{\sqrt{3}} \sin \psi \) allows us to track how \( x' \) and \( y' \) change as \( \psi \) varies.

Parametrization helps especially when transforming the equation of a geometric shape like an ellipse, allowing us to express points on the shape in a form that can be easily manipulated for analysis.

• Through trigonometric functions, which are periodic, parametrization elegantly captures the full span of the ellipse.
• In the problem, parametrization leads us to expressions for \( x \) and \( y \) in terms of \( \psi \), ensuring the transformation aligns with the rotation.

Tangent Vector

Understanding what a tangent vector represents is crucial in the study of curves.
In simple terms, a tangent vector is a vector that touches the curve at only one point, directionally indicating the slope or rate of change of the curve at that point.
For an ellipse described by parametrization, this vector can be determined by differentiating the parametrization equations for \( x \) and \( y \) with respect to \( \psi \).

For the given exercise:

• The tangent vector is expressed as \( X = \frac{\mathrm{d} x}{\mathrm{d} \psi} \partial_{x} + \frac{\mathrm{d} y}{\mathrm{d} \psi} \partial_{y} \).
• This differentiation gives insight into how the ellipse is oriented in space and provides a dynamic perspective of the ellipse at any point \( \psi \).

By calculating these derivatives, one gets the components of \( X \); these components tell us how \( x \) and \( y \) change concerning \( \psi \).

Rotation Matrix

The rotation matrix is an essential tool for transforming points in a plane by rotating them around the origin.
This concept is fundamental for changing the orientation of geometric shapes without altering their intrinsic properties.

In the context of trying to convert the given curve equation into the standard form of an ellipse:

• The rotation matrix is defined as \[ R(\theta) = \begin{pmatrix}\cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{pmatrix} \], where \( \theta = -\frac{\pi}{4} \) for a 45-degree clockwise rotation.
• By applying this matrix to the original coordinates \( (x, y) \), the curve's equation is transformed into elliptical form.

This rotation aligns the distorted ellipse with the standard coordinate axes, simplifying calculations and further analysis.

Ellipse Equation

The ellipse equation is a mathematical representation of the set of all points in a plane, the sum of whose distances from two fixed points (foci) is constant.
This classical equation takes different forms but generally involves terms with \( x^2 \) and \( y^2 \).

In the exercise, we start with the equation \( 2x^2 + 2y^2 + 2xy = 1 \) and transform it to the standard form \( x'^2 + \frac{1}{3}y'^2 = 1 \) using rotation.

• The process involves completing the square and applying a rotation matrix to align the curve to the principal axes.
• The standard form allows for easy recognition of properties such as the lengths of axes and the orientation relative to the coordinate axes.

Ultimately, understanding and converting to this form aids in both theoretical insights and practical applications for geometric and algebraic tasks.