Parametric and Polar Curves

Introduction to Parametric Curves

Parametric equations provide a powerful way to describe curves in the plane by expressing the coordinates as functions of a parameter, typically denoted as t. This approach is especially useful for representing curves that cannot be described as functions in the form y = f(x).

• Parametric equations define a curve by specifying both x and y as functions of a third variable (the parameter): for .

• Curves such as circles, ellipses, and more complex paths are naturally described using parametric equations.

Basic Ideas and Example: Circular Motion

Consider a motor boat traveling counterclockwise around a circular course with a radius of 4 miles, completing one lap every hours at a constant speed. The position of the boat at any time can be described parametrically:

• x-coordinate:

• y-coordinate:

These equations trace out a circle of radius 4 centered at the origin as varies from $0.

Parametric vs. Standard Descriptions

There are two main ways to describe curves:

• Standard (explicit) form:

• Parametric form:

Parametric equations allow for more flexibility, especially for curves that loop or have multiple y-values for a single x-value.

Eliminating the Parameter

To convert parametric equations to a standard form, solve one equation for and substitute into the other. For example, given:

Solving for from gives . Substitute into :

This is the equation of a parabola in standard form.

Orientation of Parametric Curves

The positive orientation of a parametric curve is the direction in which the curve is traced as the parameter increases. This is typically indicated by arrows on the curve.

Example: Parametric Equations for a Circle

Consider the parametric equations:

• for

These describe a circle of radius 4 centered at the origin. The table below shows key points as varies:

Application: Turtle on a Circular Path

If a turtle walks with constant speed in the counterclockwise direction on the same circle, starting at (4, 0) and completing one lap in 30 minutes, the parametric equations become:

• for

General Parametric Equations for Circles

The general form for a circle of radius centered at is:

If , the curve is traced in the counterclockwise direction as increases.

Parametric Equations of a Line

The parametric equations for a line passing through with direction vector are:

• for

The slope of the line is (if ). If and , the line is vertical.

Example: Parametric Equations of Lines

Consider the parametric equations for :

• To find the slope-intercept form, solve for in terms of and substitute into :

For a line with slope passing through , choose :

For a line segment from to , the slope is . To traverse from to , let , :

Summary Table: Key Parametric Forms