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 (4)

• 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

7
Local Coordinate Systems (1)

• A local coordinate system is a way of examining
  motion (in our case) at a particular instant.
  The tangent, normal and binormal vectors help to
  examine the forces riders feel at different
  points along a roller coaster track. These
  vectors are mutually perpendicular to each other
  and they change with time. We will be able to
  use them to explain why riders feel weightless at
  certain times and pushed into the seat at other
  times.
• Tangent Vector The 1st derivative of a
  parametric equation shows how the path is
  changing from one instant to the next. Another
  way of saying this is that it gives the
  instantaneous velocity. The tangent vector is
  found at any point by plugging in a value for the
  parameter to the 1st derivative. If you were
  sitting on a roller coaster, the tangent vector
  would describe your instantaneous velocity. It
  points directly forward (or on some roller
  coasters, like the boomerang, directly backwards).

8
Local Coordinate Systems (2)

• Normal Vector When discussing the normal vector
  we must be careful to consider both the normal
  vector (sometimes called normal force) and the
  mathematical definition of the normal direction.
  They are sometimes not the same quantity. The
  normal direction is defined by the curvature of
  the track and will be discussed in a few slides.
  The normal vector, however, is defined as the
  vector perpendicular to the tangent plane of the
  track. It always points straight up or down as
  you are sitting on the coaster car.
• Binormal Vector In the context of a roller
  coaster, the binormal refers to the forces acting
  on a person in the lateral direction. Students
  who have taken multivariable calculus may note
  that you find the direction of the binormal by
  crossing the tangent and normal vectors.

9
Local Coordinate Systems (3)

• Note that how the coordinate system is defined is
  always important. For example, the normal vector
  changes its position with time for an observer
  watching the roller coaster and the normal is
  always either out of or into the track for the
  rider on a roller coaster. For an observer
  watching the roller coaster, the normal can be
  pointing in any direction. This is why the words
  local coordinate system are used. Local refers to
  the fact that we are examining the forces as if
  we were the rider sitting in the cart, not as an
  observer, watching the coaster from a distance.

10
Curvature (1)

• The intuitive meaning of curvature is an adequate
  conceptual definition of the word for our
  purposes. A straight line has no curvature and a
  circle of a small radius is more curved than one
  of a large radius. It therefore makes sense that
  the concept of rate of change of direction can
  be applied to curvature in its definition.
• Another way to look at it is by discussion of
  tangent circles (see picture).

11
Curvature (2)

• However, here is the formal definition
• Curvature, denoted by ?, is the absolute value
  of the rate of change of the angle of inclination
  of the tangent vector. Stated another way, it
  is the magnitude of the rate of change of the
  unit tangent vector with respect to arc length.
• In equation form, if r(t)x(t)i y(t)j
• Curvature is used as a substitute for radius, R,
  when applying physics formulas since curvature ?
  (1/R), where R is the radius of the tangent
  circle
• It was mentioned earlier that the normal
  direction is defined by curvature. While the
  normal force (or vector) is always perpendicular
  to the tangent plane to the track, the normal
  direction always points toward the center of
  curvature. It does not depend on how banked the
  track is. Think of a case when the normal force
  and the normal direction are not the same.

12
Splines

• One way of parameterizing the path of a roller
  coaster is connecting different types of paths
  together. For example, a hill might be modeled
  by a parabola and a loop might be modeled by an
  ellipse. The connection between these two curves
  is very important.
• A cubic spline is a spline constructed of
  piece-wise third-order polynomials which pass
  through a set of points. Where the polynomials
  meet, we set their 1st and 2nd derivatives equal.
  A continuous and smooth transition results.
• Why is continuity important?