Day 25 in MAT229

Today:

• Announcements:
  • You have a homework due (7.8).

  • The Sehnert banquet (6 pm) and lecture is coming up, Monday, October 6th. I'll pass around a sheet -- sign up if you'd like to come (and you may bring a guest).

  • Want to study in Belize next summer? I would, but I'm not a student! Please go for this....

  • Please read section 10.3 for next time, on polar coordinates.

  • Reprise: a note about last Friday's help session: Excel spreadsheets and approximate integration

• Section 10.1: Parametric Curves
  • And now for something completely different....

    A good introduction to parametric curves is given by ballistics. If we shoot a bullet into the air with speed v horizontally, and we neglect all forces but gravity, then the bullet will trace out a parabola (some bullets are larger than others):

    

    Now how might we characterize the path of the bullet? The answer is a parametric curve, of the form C(t)=(x(t), y(t)).

    or
    If we wish, we can solve for t in the equation for x and use that to eliminate the parameter t from the equation for y, hence getting an equation for the parabola traced out:

    According to this parameterization, where is the "bullet" at time t=0? In which direction is the motion occuring -- left to right, or right to left? (You have a parametric graphing mode on your calculator -- let's try it out.)

    Orbits of planets in the heavens, movements of ants on a hill, a robotic arm in an assembly plant: all these can be described by parametric curves.

  • A couple of famous examples involving the cycloid:
    
    • The Brachistochrone Problem (1696, solved by Newton in a single day after he heard the challenge)
    • The Tautochrone Problem (1673)

  • Parametrizing a line:
    

  • Parametrizing a circle:
    

  • There are two distinctively different kinds of derivatives that we're going to have to consider here:
    • There are the time-derivatives, which give us the actual speeds of the "particles" as they zip along in time. For the bullet, the speed will be
      

    • Then there's the standard "derivative to the curve", dy/dx, which, as we will see in the next section, works out to
      

  • One of the most important applications of parametric curves is in the Bezier curve lab at the end of 10.2. There are lots of important instances where we need to be able to design custom curves, and this is one of the most common ways of creating them. For example, you can use them to create your own "sigmac", using control points. (I had all my students create their own in numerical analysis.)

  • Examples:
    • #21, p. 665
    • #14
    • #27

• Section 10.2: Calculus with Parametric Equations
  • We start with the the chain rule, and the derivative dy/dx. We must express this derivative in terms of time, which is the actual independent variable in parametric curves:
    

  • Tangent lines and Arc Length
    Suppose a particle has followed the parametric curve : then we can compute how far the particle has travelled during the interval easily using the dirt formula, (in its modified form ).

    In this case, the rate is just the speed. So we compute the integral

    
    or, in the limit,
    

    This is actually just a re-expression of the arc length formula:

    and then we "factor out the x'(t)" to get where and .

    But arc length may be different from the distance the particle travelled: a particle can revisit many sections of the curve y(x) -- so once again we need to be careful to distinguish between the independent variable of interest (whether x or t).

    Let's calculate how far a particle travelled when parameterized by and , with

    What happens to our integral if we double the time interval?

    • Examples:
      • #1, p. 675
      • #3
      • #20
      • #41