Parametric equations and polar coordinates offer new ways to represent curves beyond traditional Cartesian methods. These tools allow us to describe more complex shapes and movements, opening up exciting possibilities in math and physics.

By expressing x and y in terms of a parameter t, or using distance r and angle θ, we can model circles, spirals, and other intricate curves. This approach proves invaluable for analyzing motion and solving real-world problems.

Parametric Equations for Curves

Representing Curves with Parametric Equations

• Parametric equations express the coordinates of points on a plane curve in terms of an independent parameter, often denoted as $t$
• In parametric form, a curve is defined by two equations: $x = f(t)$ and $y = g(t)$, where $f$ and $g$ are functions of the parameter $t$
• Parametric equations are useful for representing curves that are not functions, such as circles or curves that have multiple $y$-values for a single $x$-value
• The direction of a curve represented by parametric equations can be determined by the values of the parameter $t$ and the signs of the derivatives of the $x$ and $y$ components with respect to $t$

Common Curves Represented by Parametric Equations

• Circles: $x = r\cos(t)$, $y = r\sin(t)$, where $r$ is the radius and $t$ is the angle in radians
• Ellipses: $x = a\cos(t)$, $y = b\sin(t)$, where $a$ and $b$ are the semi-major and semi-minor axes, respectively
• Cycloids: $x = a(t - \sin(t))$, $y = a(1 - \cos(t))$, where $a$ is the radius of the generating circle and $t$ is the angle of rotation
• Lissajous curves: $x = A\sin(at + \delta)$, $y = B\sin(bt)$, where $A$, $B$, $a$, $b$, and $\delta$ are constants that determine the shape of the curve

Parametric vs Cartesian Forms

Converting from Parametric to Cartesian Form

• To convert from parametric form to Cartesian form, eliminate the parameter $t$ by solving one equation for $t$ and substituting the result into the other equation
• When converting from parametric to Cartesian form, the resulting equation may not always be a function, as it could represent a curve that fails the vertical line test
• Consider the domain of the parameter $t$ to ensure that the entire curve is represented when converting between parametric and Cartesian forms

Converting from Cartesian to Parametric Form

• To convert from Cartesian form to parametric form, introduce a parameter $t$ and express $x$ and $y$ in terms of $t$, ensuring that the resulting parametric equations satisfy the original Cartesian equation
• There are infinitely many possible parametrizations for a given Cartesian equation, as the choice of the parameter $t$ is arbitrary
• Example: The circle $x^2 + y^2 = r^2$ can be parameterized as $x = r\cos(t)$, $y = r\sin(t)$, where $t$ is the angle in radians

Curves in Polar Coordinates

Graphing Curves in Polar Coordinates

• Polar coordinates represent a point on a plane using a distance $r$ from the origin (called the pole) and an angle $\theta$ from the polar axis (usually the positive $x$-axis)
• The polar equation $r = f(\theta)$ defines a curve in polar coordinates, where $r$ is the distance from the origin and $\theta$ is the angle from the polar axis
• To graph a curve in polar coordinates, create a table of values for $\theta$ (usually in radians) and calculate the corresponding $r$ values using the polar equation
• Plot the points $(r, \theta)$ in the polar coordinate system by measuring the angle $\theta$ from the polar axis and the distance $r$ from the origin

Symmetry in Polar Curves

• If $f(\theta) = f(-\theta)$, the curve is symmetric about the polar axis
• If $f(\theta) = f(\pi - \theta)$, the curve is symmetric about the vertical line $\theta = \pi/2$
• If $f(\theta) = f(\theta + \pi)$, the curve is symmetric about the origin (pole)
• Example: The cardioid $r = a(1 + \cos(\theta))$ is symmetric about the polar axis because $f(\theta) = f(-\theta)$

Common Curves in Polar Coordinates

• Circles: $r = a$, where $a$ is the radius
• Cardioids: $r = a(1 + \cos(\theta))$, where $a$ is a constant that determines the size of the cardioid
• Limaçons: $r = a + b\cos(\theta)$ or $r = a + b\sin(\theta)$, where $a$ and $b$ are constants that determine the shape of the limaçon
• Rose curves: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$, where $a$ is a constant that determines the size of the rose curve and $n$ is a positive integer that determines the number of petals

Applications of Parametric and Polar Equations

Modeling Real-World Phenomena

• Parametric equations can model the path of projectiles, describe the motion of particles in physics, and represent curves in computer graphics and animation
• Polar equations can model real-world phenomena such as the patterns of petals on a flower, the shape of a microphone's pickup pattern, or the path of a pendulum
• Example: The path of a projectile launched with an initial velocity $v$ at an angle $\theta$ can be modeled using the parametric equations $x = (v\cos(\theta))t$ and $y = (v\sin(\theta))t - \frac{1}{2}gt^2$, where $g$ is the acceleration due to gravity and $t$ is time

Solving Problems Involving Parametric and Polar Equations

• To find the intersection points of two curves given in parametric form, set their $x$ and $y$ components equal to each other and solve the resulting system of equations for the parameter $t$
• To find the area enclosed by a polar curve, use the formula $A = \frac{1}{2}\int_{a}^{b} r^2 d\theta$, where $[a, b]$ is the interval of $\theta$ over which the area is being calculated
• When solving problems involving parametric and polar equations, it may be helpful to convert between parametric, polar, and Cartesian forms to simplify calculations or gain insights into the problem
• Example: To find the area enclosed by the cardioid $r = a(1 + \cos(\theta))$, use the formula $A = \frac{1}{2}\int_{0}^{2\pi} a^2(1 + \cos(\theta))^2 d\theta$, which evaluates to $3\pi a^2$