Question

Conditions for a circular/elliptical trajectory in the plane An object moves along a path given by \(\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t\rangle,\) for \(0 \leq t \leq 2 \pi\) a. What conditions on \(a, b, c,\) and \(d\) guarantee that the path is a circle? b. What conditions on \(a, b, c,\) and \(d\) guarantee that the path is an ellipse?

Short answer

Answer: For a circular trajectory, the conditions are: 1. \(ab + cd = 0\) 2. \(a^2 + c^2 = b^2 + d^2\) For an elliptical trajectory, the conditions are: 1. \(ab + cd = 0\) 2. \((a^2 + c^2)(b^2+d^2) > 0\)

Step-by-step solution

Extract parametric equations for x and y from the given vector equation.

We begin by extracting the parametric equations for x and y from the given vector equation. The given vector is: \(\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t\rangle\) So, the parametric equations for x and y are: \(x(t) = a \cos{t} + b \sin{t}\) \(y(t) = c \cos{t} + d \sin{t}\)

Squaring and adding the parametric equations.

We can obtain an equation for \(x^2 + y^2\) from the parametric equations by squaring both equations and adding them: \(x^2(t) = (a \cos{t} + b \sin{t})^2\) \(y^2(t) = (c \cos{t} + d \sin{t})^2\) Adding these equations, we get: \(r^2(t) = x^2(t) + y^2(t) = (a \cos{t} + b \sin{t})^2 + (c \cos{t} + d \sin{t})^2\)

Expanding the equation and gathering the trigonometric terms.

We continue by expanding the equation and collecting the trigonometric terms: \(r^2(t) = a^2 \cos^2{t} + 2ab \cos{t}\sin{t} + b^2 \sin^2{t} + c^2 \cos^2{t} + 2cd \cos{t}\sin{t} + d^2 \sin^2{t}\) \(r^2(t) = (a^2 + c^2)\cos^2{t} + 2(ab + cd)\cos{t}\sin{t} + (b^2 + d^2)\sin^2{t}\)

Finding the conditions for a circle.

For the path to be a circle, \(r^2(t)\) must be constant (i.e., independent of \(t\)). The path is a circle if and only if the coefficients of all trigonometric terms (those involving \(\cos{t}\) and \(\sin{t}\)) are zero: 1. \(ab + cd = 0\) 2. \(a^2 + c^2 = b^2 + d^2\) These conditions guarantee that the trajectory of the moving object is a circle.

Finding the conditions for an ellipse.

For the path to be an ellipse, we need to make sure that the coefficients of the trigonometric terms (those involving \(\cos{t}\) and \(\sin{t}\)) are zero, and the coefficients of \(\cos^2{t}\) and \(\sin^2{t}\) both have the same sign: 1. \(ab + cd = 0\) 2. \((a^2 + c^2)(b^2+d^2) > 0\) These conditions guarantee that the trajectory of the moving object is an ellipse. Note that since a circle is a special case of an ellipse, a circular trajectory also satisfies these conditions, but the additional condition ensures that it only covers elliptical and not circular trajectories.

Key concepts

Circular Trajectory

Understanding circular trajectories in parametric equations involves ensuring that the object moves in a perfect circle. When the path of a moving object is described by a parametric equation, such as \[\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t\rangle, \] we aim to identify the values of parameters \(a, b, c,\) and \(d\) that make the trajectory a circle. For the path to actually define a circle:

• The coefficients of terms involving both \(\cos{t}\) and \(\sin{t}\) should cancel out. This means \(ab + cd = 0\).
• The sum of squared coefficients must remain equivalent, described as \(a^2 + c^2 = b^2 + d^2\). This ensures the circular trajectory has a constant radius.

These conditions ensure the path represents a perfect circular path without any distortion from trigonometric fluctuations.

Elliptical Trajectory

For an elliptical trajectory, similar parametric expressions describe how an object moves along an ellipse. Continuing from the parametric form:\[\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t \rangle,\]the conditions required for an elliptical path slightly differ from those of a circle. To form an ellipse:

• As in circular constraint, the interference from \(\cos{t}\) and \(\sin{t}\) needs nullification, which maintains \(ab + cd = 0\).
• Both coefficient expressions from the squared terms must be positive, which is ensured by \((a^2 + c^2)(b^2+d^2) > 0\).

In essence, an ellipse may appear elongated in one or multiple directions; this condition guarantees a non-circular shape while allowing symmetrical semi-major and semi-minor axes based on magnitudes of coefficients.

Trigonometric Terms

Trigonometric terms play a pivotal role in defining the shape of trajectories when expressed parametrically. The terms \(\cos{t}\) and \(\sin{t}\) appear in the equation in complex expressions. They are multiplied by constants \(a, b, c,\) and \(d\), which link to geometric properties of circles and ellipses. These terms, when expanded and combined, form expressions like:\[a^2 \cos^2{t} + 2ab \cos{t}\sin{t} + b^2 \sin^2{t},\]showcasing the interactions among cosine and sine components. The combined terms \(2(ab + cd)\cos{t}\sin{t}\) must equate to zero for maintaining zero interaction over a full period. This interaction essentially dictates the geometric structure—without it, trajectories would not remain closed like circles or ellipses.

Conditions for Circle and Ellipse

Ensuring a path is a circle or an ellipse involves satisfying specific mathematical conditions within the parametric form. For a circular path, ensuring that each coefficient condition holds guarantees the path's uniformity and constancy in radius:

• \(ab + cd = 0\)
• \(a^2 + c^2 = b^2 + d^2\)

These result in simplifying the terms, making \(r^2,\) which involves both \(x(t)\) and \(y(t)\), independent of \(t\).On the other hand, an elliptical path's conditions ensure a broader journey:

• \(ab + cd = 0\)
• \((a^2 + c^2)(b^2+d^2) > 0\)

Elliptical trajectories provide flexibility across their axes, allowing varied lengths between major and minor directions. Understanding these criteria helps define the trajectory's overall shape and its ultimate path in the plane.