Curves in Space

1. Curves in Space “flying around”

2. Flying Around • Suppose we have a friendly fly buzzing around the room. • How do we describe its motion?

3. The fly at time t = 0.5 sec

4. The fly at time t = 2 sec

5. The fly at time t = 4 sec

6. Describing the motion • We give the coordinates of the fly’s position at each point in time. • The x-coordinate, the y-coordinate and the z-coordinate are functions of t (time).

7. Parametrically defined curves • We can (in principle) define any curve in the plane or in space by thinking of a fly flying along that trajectory and specifying the coordinates of its position at time t. • You will learn to think about parametric curves with the parametric plots project.

8. A familiar example • You already know one of the most useful sets of parametric equations! Suppose our fly is constrained to move in two dimensions and is tied to a point on the floor by a “tether” of length one meter? It will then fly around in a circle. What if it revolves once every 2 seconds? t

9. Why do people care about parametric equations? Describing curves in space. Finding the intersections of parametric curves--- intersections in time vs. intersections in space.

10. Design Pierre Étienne Bézier (1910-1999) • French Engineer and Mathematician • Created Bezier curves and Bezier Surfaces that are now used in most computer aided design and computer graphics • His interest in computer assisted design was automobile design. He worked as a designer for Renault (French Automobile designer.) • Check out Bezier curves on wikipedia. There’s a cool animation!