Intermediate Math
Parametric Equations
Local Coordinate Systems
Curvature
Splines
Parametric Equations (1)
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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= 4  x 2
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) = <f(t), g(t)>, where x= f(t) and y= g(t)
F(t) = <cos(t), sin(t)> - what is this curve and why is
this parameterization useful?
Parametric Equations (2)
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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.
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Ex. F(t)=<sin(3t), sin(4t)>
Sometimes the parameter can be
eliminated by solving one equation
(say, x=f(t)) for the parameter t
and substituting this expression
into the other equation y=g(t). The
result will be the parametric curve.
Parametric Equations (3)
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Using parametric equations, we can easily add a
3rd dimension:
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A conceptual example:
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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), z=t
Mathematically:
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Add a simple linear term in the z-direction:
F(t)=<cos(t), sin(t), t>
Parametric Equations (4)
Parametric Equations (4)
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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.
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If F(t)=<cos(t), sin(t), t>, then F’(t)=<-sin(t), cos(t), 1>
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:
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F(t)=<cos(t), sin(t), t> turns into: F(t) = cos(t)i +sin(t)j +tk
Local Coordinate Systems (1)
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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).
Local Coordinate Systems (2)
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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.
Local Coordinate Systems (3)
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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.
Curvature (1)
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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).
Curvature (2)
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However, here is the formal definition:
Curvature, denoted by κ, 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:
 
x ' y ' ' x ' ' y '
( x' )
2
 ( y' )
2

3

2
x ' y ' ' x ' ' y '
v3
Curvature is used as a substitute for radius, R, when applying physics formulas since
curvature κ =(1/R), where R is the radius of the tangent circle
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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.
Splines
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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?