Objectives
- Understand how curves in space are described with parametric equations.
- Explore the concept of orientation of parametric equations.
- Look at how parametric curves can be naturally formed through the intersection of surfaces.
Section 4.1 Parameterizations of Curves in Space
This lesson focuses on the correspondences between parametric equations and the curves that make up their graphs. The first example recalls the parametric form of a line and its description for how to move through the points on the line. As part of the discussion of this example, the non-uniqueness of parameterizations is shown by giving several parameterizations for the same curve in space. Other examples include a circle in the yz-plane (with multiple parameterizations), a helix that moves around the z-axis, a curve involving exponentials (as demonstrated by showing the shadow of the curve, and a curve that is defined by the intersection of two surfaces. The intersection of two surfaces example is also used to show how some of the naïve guesses involved in coming up with a parameterization can lead to algebraic roadblocks.
This lesson should be done after completing the lesson on lines in space, the quiz on lines in space, and the lesson on plotting a vector valued function of one variable.
Subsection 4.1.1 Parametric Curves in Space
