Parametric Curves

Introduction to Parametric Curves

Parametric curves are a fundamental concept in calculus, allowing us to describe curves in the plane that cannot be represented as the graph of a function y = f(x) or x = g(y). Instead, both x and y are expressed as functions of a third variable, typically t, which often represents time or another parameter. This approach is especially useful for describing trajectories and motion in two dimensions.

• Parametric equations define a curve by specifying both x and y as functions of t: , where and is an interval.

• The parameter t provides a direction along the curve, indicating how the curve is traced as t increases.

• Parametric curves can represent complex shapes and motions that are not possible with standard function graphs.

Example 1: Parametric Representation of a Circle Segment

Consider the parametric equations for . This describes a segment of a circle of radius 1 centered at the origin.

• By eliminating t, we find , confirming the curve lies on the unit circle.

• The direction of increasing t traces the curve from point A () to point B ().

Example 2: Parametric Representation of a Parabola

Given for , we can eliminate t to find the Cartesian equation.

• Solving for t:

• Substitute into x:

• This is a parabola, and the direction of increasing t is indicated by arrows on the curve.

Example 3: Parameterization of a Straight Line

To parameterize a straight line passing through points and , with at and at :

• The parameterization is for .

• Both x and y are linear functions of t.

Example 4: Parametric Representation of an Ellipse

Consider for , . Eliminating t gives the equation of an ellipse.

• ,

• The curve is an ellipse centered at the origin, with axes of length 2a and 2b.

Definition: Plane Curves and Parameterization

A plane curve is a set of points such that and for in some interval , where and are continuous functions. The choice of , , and is called a parameterization of the curve.

• A single curve can have many different parameterizations.

• Parameterizations can affect the speed and direction with which the curve is traced.

Example 5: Multiple Parameterizations of a Circle

The unit circle can be parameterized in different ways:

• Standard: ,

• 

• Alternative: ,

• 

Example 6: Parametric Circle with Shifted Center

Consider for . Eliminating t gives:

• Rewriting:

• This is a circle of radius centered at .

• The curve is traced counterclockwise, starting from as .

• The point does not belong to the curve.

Summary Table: Common Parametric Curves

Additional info: The notes above expand on the original lecture content by providing full definitions, step-by-step elimination of parameters, and a summary table for common parametric curves. All included images directly illustrate the parametric equations and their geometric interpretations.