Parametric Equations and Polar Coordinates Study Notes

PARAMETRIZATIONS OF PLANE CURVES

Definition of Parametric Equations

• When the coordinates $x$ and $y$ of a point $P(x, y)$ on a curve are given as functions of a third variable $t$ (called the parameter), the equations are known as parametric equations.

• The set of points $(x, y)$ generated as $t$ varies over an interval is called the graph of the equations.

Algebraic Method: Converting to Cartesian Equations

• To identify a curve defined by parametric equations, one can often eliminate the parameter $t$ to obtain a Cartesian equation relating $x$ and $y$.

• Example 1: Determine the Cartesian equation and sketch the curve defined by $x = t^2$, $y = t + 1$ for -\infty < t < \infty.

  • Solution: From $y = t + 1$, we find $t = y - 1$.

  • Substituting into the expression for $x$ yields: $x = (y - 1)^2$.

  • This is a parabola opening to the right with its vertex at $(0, 1)$.

Examples of Parametric Path Identification

• Example (a): $x = \sqrt{t}$, $y = t$, for $t \geq 0$.

  • Squaring $x$ gives $x^2 = t$. Since $y = t$, the Cartesian equation is $y = x^2$.

  • Because $x = \sqrt{t}$, the domain is restricted to $x \geq 0$.

• Example (b): $x = \cos(t)$, $y = \sin(t)$, for $0 \leq t \leq \pi$.

  • Using the identity $\cos^2(t) + \sin^2(t) = 1$, we get $x^2 + y^2 = 1$.

  • For the interval $0 \leq t \leq \pi$, $y$ remains non-negative ($y \geq 0$), representing the upper semi-circle.

• Example (c): $x = 1 + \sin(2t)$, $y = \cos(2t) - 2$, for $0 \leq t \leq \frac{3\pi}{4}$.

  • Isolate the trigonometric terms: $\sin(2t) = x - 1$ and $\cos(2t) = y + 2$.

  • Apply the identity $\sin^2(2t) + \cos^2(2t) = 1$.

  • The Cartesian equation is $(x - 1)^2 + (y + 2)^2 = 1$.

  • The path is an arc of a circle centered at $(1, -2)$ with radius $1$.

Additional Examples from Practice

• Linear Path: $x = 2t - 5$, $y = 4t - 7$ for - \infty < t < \infty .

  • $t = \frac{x + 5}{2}$.

  • $y = 4\left(\frac{x + 5}{2}\right) - 7 = 2(x + 5) - 7 = 2x + 10 - 7 = 2x + 3$.

• Parabolic Path: $x = 4t + 3$, $y = 16t^2 - 9$ for $-1 \leq t \leq 1$.

  • $t = \frac{x - 3}{4}$.

  • $y = 16\left(\frac{x - 3}{4}\right)^2 - 9 = 16\left(\frac{(x - 3)^2}{16}\right) - 9 = (x - 3)^2 - 9$.

• Exponential Path: $x = e^{2t} + 1$, $y = e^t$.

  • Recognize that $e^{2t} = (e^t)^2$.

  • Therefore, $x = y^2 + 1$.

CALCULUS WITH PARAMETRIC CURVES

Tangents of Parametrized Curves

• If $x = f(t)$ and $y = g(t)$ are differentiable, then the slope $\frac{dy}{dx}$ is calculated using the chain rule:     $\frac{dy}{dx} = \frac{dy / dt}{dx / dt}$, provided $dx / dt \neq 0$.

• Example 1: If $x = t^2 + 3$ and $y = t^2 - 1$, find $\frac{dy}{dx}$ at $t = 6$.

  • $\frac{dx}{dt} = 2t$, $\frac{dy}{dt} = 2t$.

  • $\frac{dy}{dx} = \frac{2t}{2t} = 1$.

  • At $t = 6$, $\frac{dy}{dx} = 1$.

• Example 2: Find the tangent to the curve $x = \sec(t)$, $y = \tan(t)$ for \frac{-\pi}{2} < t < \frac{\pi}{2} at the point $(\sqrt{2}, 1)$ where $t = \frac{\pi}{4}$.

  • $\frac{dx}{dt} = \sec(t)\tan(t)$, $\frac{dy}{dt} = \sec^2(t)$.

  • $\frac{dy}{dx} = \frac{\sec^2(t)}{\sec(t)\tan(t)} = \frac{\sec(t)}{\tan(t)} = \csc(t)$.

  • At $t = \frac{\pi}{4}$, $\frac{dy}{dx} = \csc\left(\frac{\pi}{4}\right) = \sqrt{2}$.

  • Equation of tangent: $y - 1 = \sqrt{2}(x - \sqrt{2}) \Rightarrow y = \sqrt{2}x - 2 + 1 \Rightarrow y = \sqrt{2}x - 1$.

Second Order Derivatives

• The second derivative $\frac{d^2y}{dx^2}$ for a parametric curve is defined as:     $\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right) = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$.

• Example 3: Find $\frac{d^2y}{dx^2}$ as a function of $t$ if $x = t^2 - t$ and $y = t^3 - t$.

  • $\frac{dx}{dt} = 2t - 1$, $\frac{dy}{dt} = 3t^2 - 1$.

  • $\frac{dy}{dx} = \frac{3t^2 - 1}{2t - 1}$.

  • $\frac{d}{dt}\left(\frac{3t^2 - 1}{2t - 1}\right) = \frac{(2t - 1)(6t) - (3t^2 - 1)(2)}{(2t - 1)^2} = \frac{12t^2 - 6t - 6t^2 + 2}{(2t - 1)^2} = \frac{6t^2 - 6t + 2}{(2t - 1)^2}$.

  • $\frac{d^2y}{dx^2} = \frac{6t^2 - 6t + 2}{(2t - 1)^3}$.

Length of a Parametrically Defined Curve (Arc Length)

• If $x = f(t)$ and $y = g(t)$ for $a \leq t \leq b$, and the derivatives are continuous, the arc length $L$ is given by:     $L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$.

• Example 4: Find the length of a circle of radius $r$ defined by $x = r\cos(t)$, $y = r\sin(t)$ for 0 \leq t \n\leq 2\pi .

  • $\frac{dx}{dt} = -r\sin(t)$, $\frac{dy}{dt} = r\cos(t)$.

  • $\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 = r^2\sin^2(t) + r^2\cos^2(t) = r^2$.

  • $L = \int_0^{2\pi} \sqrt{r^2}\,dt = \int_0^{2\pi} r\,dt = [rt]_0^{2\pi} = 2\pi r$.

POLAR COORDINATES

Fundamentals and Representation

• Definition: Points are represented as $(r, \theta)$, where $r$ is the directed distance from the origin (pole) and $\theta$ is the directed angle from the positive x-axis (initial ray).

• Non-uniqueness: A single point in the plane can have multiple polar representations:

  • $(r, \theta)$ is the same as $(r, \theta + 2n\pi)$ for any integer $n$.

  • $(r, \theta)$ is the same as $(-r, \theta + \pi)$.

• Sign Conventions:

  • $\theta$ is positive in the anticlockwise direction.

  • $\theta$ is negative in the clockwise direction.

  • A point $(-r, \theta)$ lies in the opposite direction of $(r, \theta)$.

Exercises: Polar Representations

• Point (a): (3, 225°)

  • Equivalent representations: $(3, -135\text{\u0026deg;})$, $(-3, 45\text{\u0026deg;})$, $(-3, -315\text{\u0026deg;})$.

• Point (d): (2, -225°)

  • Visualized as $135\text{\u0026deg;}$ anticlockwise.

  • Equivalent: $(2, 135\text{\u0026deg;})$, $(-2, -45\text{\u0026deg;})$, $(-2, 315\text{\u0026deg;})$.

Basic Polar Graphs

1. Circles: If $r = a$ (constant), the graph is a circle of radius $|a|$ centered at the origin.

2. Lines: If $\theta = \theta_0$ (constant), the graph is a line passing through the origin making an angle $\theta_0$ with the initial ray.

Relating Polar and Cartesian Coordinates

• Equations for Conversion:

  • $x = r\cos(\theta)$

  • $y = r\sin(\theta)$

  • $r^2 = x^2 + y^2$

  • $\tan(\theta) = \frac{y}{x}$

• Example 1b: Convert $(4, \frac{-2\pi}{3})$ to Cartesian.

  • $x = 4\cos\left(\frac{-2\pi}{3}\right) = 4\left(\frac{-1}{2}\right) = -2$.

  • $y = 4\sin\left(\frac{-2\pi}{3}\right) = 4\left(\frac{-\sqrt{3}}{2}\right) = -2\sqrt{3}$.

• Equation Conversion Example 3a: $r = 4\csc(\theta)$.

  • $r = \frac{4}{\sin(\theta)} \Rightarrow r\sin(\theta) = 4 \Rightarrow y = 4$. (A horizontal line).

• Equation Conversion Example 4d: $(x^2 + y^2 - 2y)^2 = 4(x^2 + y^2)$.

  • $(r^2 - 2r\sin(\theta))^2 = 4r^2$.

  • $r^2 - 2r\sin(\theta) = \pm 2r \Rightarrow r - 2\sin(\theta) = \pm 2$.

  • Resulting in equations like $r = 2 + 2\sin(\theta)$ (a cardioid).

GRAPHING IN POLAR COORDINATES

Symmetries in Polar Graphs

• Symmetry about the x-axis: Equation remains unchanged when $\theta$ is replaced by $-\theta$ (or $(r, -\theta)$).

• Symmetry about the y-axis: Equation remains unchanged when $\theta$ is replaced by $\pi - \theta$ (or $(-r, -\theta)$).

• Symmetry about the Origin: Equation remains unchanged when $r$ is replaced by $-r$ (or $\theta$ is replaced by $\pi + \theta$).

Common Curves

• Cardioids: e.g., $r = a(1 \pm \cos(\theta))$ or $r = a(1 \pm \sin(\theta))$.

• Limaçons: $r = a \pm b\cos(\theta)$ or $r = a \pm b\sin(\theta)$.

• Rose Petals: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$.

  • If $n$ is odd, there are $n$ petals.

  • If $n$ is even, there are $2n$ petals.

• Lemniscates: $r^2 = a^2\cos(2\theta)$ or $r^2 = a^2\sin(2\theta)$.

Slope of a Polar Curve

• The slope of the curve $r = f(\theta)$ at the point $(r, \theta)$ is:     $\frac{dy}{dx} = \frac{\frac{df}{d\theta}\sin(\theta) + f(\theta)\cos(\theta)}{\frac{df}{d\theta}\cos(\theta) - f(\theta)\sin(\theta)}$.

AREAS AND LENGTHS IN POLAR COORDINATES

Area in the Plane

• Area of a Fan-Shaped Region: The area of the region between the origin and the curve $r = f(\theta)$ from $\theta = \alpha$ to $\theta = \beta$ is:     $A = \int_{\alpha}^{\beta} \frac{1}{2}r^2\,d\theta = \int_{\alpha}^{\beta} \frac{1}{2}(f(\theta))^2\,d\theta$.

• Area Between Two Curves: If $0 \leq r_1(\theta) \leq r \leq r_2(\theta)$, the area is:     $A = \int_{\alpha}^{\beta} \frac{1}{2}(r_2^2 - r_1^2)\,d\theta$.

Length of a Polar Curve

• If $r = f(\theta)$ has a continuous derivative for $\alpha \leq \theta \leq \beta$, the arc length $L$ is:     $L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2}\,d\theta$.

• Example: Find the length of the cardioid $r = 1 - \cos(\theta)$.

  • $\frac{dr}{d\theta} = \sin(\theta)$.

  • $r^2 + (\frac{dr}{d\theta})^2 = (1 - \cos(\theta))^2 + \sin^2(\theta) = 1 - 2\cos(\theta) + \cos^2(\theta) + \sin^2(\theta) = 2 - 2\cos(\theta)$.

  • Use identity: $2 - 2\cos(\theta) = 4\sin^2\left(\frac{\theta}{2}\right)$.

  • $L = \int_0^{2\pi} \sqrt{4\sin^2(\theta/2)}\,d\theta = \int_0^{2\pi} 2\sin(\theta/2)\,d\theta = [-4\cos(\theta/2)]_0^{2\pi} = 4 - (-4) = 8$.

CONIC SECTIONS

Parabolas

• Standard Forms (Vertex at $(0, 0)$):

  • $x^2 = 4py$: Opens Up (Focus at $(0, p)$, Directrix $y = -p$).

  • $x^2 = -4py$: Opens Down (Focus at $(0, -p)$, Directrix $y = p$).

  • $y^2 = 4px$: Opens Right (Focus at $(p, 0)$, Directrix $x = -p$).

  • $y^2 = -4px$: Opens Left (Focus at $(-p, 0)$, Directrix $x = p$).

• Parameter $p$: Distance from the vertex to the focus or directrix.

Ellipses

• Definition: The set of points where the sum of distances from two fixed foci is constant ($2a$).

• Standard Equations (Center at Origin):

  • Horizontal Major Axis: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (a > b).

  • Vertical Major Axis: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ (a > b).

• Constants:

  • $c = \sqrt{a^2 - b^2}$ (Center-to-focus distance).

  • Vertices: $(\pm a, 0)$ or $(0, \pm a)$.

  • Foci: $(\pm c, 0)$ or $(0, \pm c)$.

Hyperbolas

• Standard Equations (Center at Origin):

  • Horizontal Transverse Axis: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.

  • Vertical Transverse Axis: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$.

• Constants:

  • $c = \sqrt{a^2 + b^2}$ (Center-to-focus distance).

  • Asymptotes: $y = \pm \frac{b}{a}x$ (horizontal) or $y = \pm \frac{a}{b}x$ (vertical).

CONIC SECTIONS IN POLAR COORDINATES

Eccentricity ($e$)

• Defines the type of conic based on the ratio of the distance from a point $P$ to the focus $F$ ($PF$) versus the distance to the directrix $D$ ($PD$):     $PF = e \cdot PD$.

  • Parabola: $e = 1$.

  • Ellipse: e < 1 .

  • Hyperbola: e > 1 .

• For ellipses and hyperbolas, $e = \frac{c}{a} = \frac{\text{distance between foci}}{\text{distance between vertices}}$.

Polar Equations of Conics

• A conic with a focus at the origin and a directrix at $x = k$ (to the right) has the equation:     $r = \frac{ke}{1 + e\cos(\theta)}$.

• General variations depending on directrix position (left, right, above, below):     $r = \frac{ke}{1 \pm e\cos(\theta)}$ or $r = \frac{ke}{1 \pm e\sin(\theta)}$.

Lines and Circles in Polar Form

• Lines: A line through $P_0(r_0, \theta_0)$ perpendicular to the ray from the origin to $P_0$ has equation:     $r\cos(\theta - \theta_0) = r_0$.

• Circles: A circle of radius $a$ centered at $P_0(r_0, \theta_0)$ follows the law of cosines:     $a^2 = r^2 + r_0^2 - 2rr_0\cos(\theta - \theta_0)$.

  • If the center is on the positive x-axis $(a, 0)$: $r = 2a\cos(\theta)$.

  • If the center is on the positive y-axis $(0, a)$: $r = 2a\sin(\theta)$.