Intermediate Math

1
Intermediate Math

• Parametric Equations
• Local Coordinate Systems
• Curvature
• Splines

2
Parametric Equations (1)

• We are used to seeing an equation of a curve
  defined by expressing one variable as a function
  of the other.
• Ex. y f(x)
• Ex. y
• A parameter is a third, independent variable (for
  example, time).
• By introducing a parameter, x and y can be
  expressed as a function of the parameter, as
  opposed to functions of each other.
• Ex. F(t) ltf(t), g(t)gt, where x f(t) and y
  g(t)
• F(t) ltcos(t), sin(t)gt - what is this curve and
  why is this parameterization useful?

3
Parametric Equations (2)

• Each value of the parameter t determines a point,
  (f(t), g(t)), and the set of all points is the
  graph of the curve.
• Complicated curves are easily dealt with since
  the components f(t) and g(t) are each functions.
• Ex. F(t)ltsin(3t), sin(4t)gt
• Sometimes the parameter can be eliminated by
  solving one equation (say, xf(t)) for the
  parameter t and substituting this expression into
  the other equation yg(t). The result will be
  the parametric curve.

4
Parametric Equations (3)

• Using parametric equations, we can easily add a
  3rd dimension
• A conceptual example
• Picture the xy-plane to be on the table and the
  z-axis coming straight up out of the table
• Picture the parameterized 2-D path (cos(t),
  sin(t)) which is a circle on the table
• Add a simple z-component such that the circle
  climbs off the table to form a helix (or
  corkscrew), zt
• Mathematically
• Add a simple linear term in the z-direction
• F(t)ltcos(t), sin(t), tgt

5
Parametric Equations (4)
6
Parametric Equations (5)

• The calculus we use for parametric equations is
  very similar to that in single-variable calculus.
• As with regular curves, parametric curves are
  smooth if the derivatives of the components are
  continuous and are never simultaneously zero.
• To take the derivative of a parametric equation,
  take the derivative of each of the components.
• If F(t)ltcos(t), sin(t), tgt, then F(t)lt-sin(t),
  cos(t), 1gt
• As with single variable calculus, the 1st
  derivative indicates how the path changes with
  time.
• Note that another way to represent parametric
  equations is to use unit vectors. From the above
  example
• F(t)ltcos(t), sin(t), tgt turns into F(t)
  cos(t)i sin(t)j tk