1.1 Parametric Equations

Parametric equations offer a powerful way to describe curves using a parameter. They allow us to represent complex shapes and motions that might be difficult to express with traditional functions. From simple lines to intricate cycloids, parametric equations provide a versatile tool for mathematical modeling.

Converting between parametric and rectangular forms, analyzing curve properties, and applying these concepts to real-world scenarios are key skills. Understanding parametric equations opens up new possibilities for describing and studying various mathematical and physical phenomena.

Parametric Equations

Plotting parametric curves

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• Define a curve using a parameter, typically denoted as $t$, where the x-coordinate is a function of $t$, $x = f(t)$, and the y-coordinate is a function of $t$, $y = g(t)$
• Evaluate the functions for various values of $t$ within a suitable range based on the given context or problem to plot the curve
• Substitute the $t$ values into both functions to find the corresponding $x$ and $y$ coordinates and plot the points, connecting them smoothly to form the curve (e.g., for $t$ ranging from 0 to 2π)
• These functions can be expressed as vector-valued functions, combining both x and y components into a single vector function

Conversion to rectangular form

• Eliminate the parameter $t$ to convert parametric equations to rectangular form $y = f(x)$ (or $x = f(y)$)
• Solve one of the equations for $t$ in terms of $x$ or $y$ and substitute the expression for $t$ into the other equation
• Simplify the resulting equation to obtain $y$ as a function of $x$ (or $x$ as a function of $y$), keeping in mind that not all parametric equations can be converted to rectangular form (e.g., cycloids or trochoids)

Parametric equations of common shapes

• Lines passing through a point $(x_0, y_0)$ with direction vector $(a, b)$ have parametric equations $x = x_0 + at$ and $y = y_0 + bt$
• Circles with center $(h, k)$ and radius $r$ have parametric equations $x = h + r\cos(t)$ and $y = k + r\sin(t)$, where $t$ typically ranges from 0 to $2\pi$ to trace the entire circle
• These equations can also be expressed in polar coordinates, which relate the distance from the origin to a point on the curve with the angle from the positive x-axis

Cycloids in parametric form

• A cycloid is the curve traced by a point on the circumference of a circle as it rolls along a straight line, with parametric equations $x = r(t - \sin(t))$ and $y = r(1 - \cos(t))$ for a cycloid with radius $r$ starting at the origin
• The parameter $t$ represents the angle through which the circle has rotated, and as $t$ increases, the circle rolls forward, and the point traces the cycloid curve
• Cycloids have periodic curves repeating every $2\pi$ units, cusps (sharp points) where the tracing point touches the straight line, and the area under one arch equal to three times the area of the rolling circle

Analysis of parametric curves

• The arc length of a parametric curve can be calculated using integration techniques
• Tangent lines to parametric curves can be determined by finding the derivative of the curve at a specific point
• For curves representing motion, the velocity and acceleration of an object can be derived from the parametric equations, providing information about the object's movement along the curve