\footnote{$^1$}
{{\bf Polyhedron}: \qquad Traditionally geometric polyhedra have plane
faces, while topological polyhedra are merely collections of vertices,
edges and faces.

In subdivision theory we find it convenient to deal with objects which
are some kind of mongrel.  The vertices are geometric, having
coordinates, but the edges and faces are topological, merely
indicating the manifold connectivity of the configuration.

It is therefore possible to have a `polyhedron' whose faces are not plane.
}

\footnote{$^2$}
{{\bf Dimension of a Pointset}:\qquad

Cantor produced a lovely example of a 1:1 map between sets of
different dimensions.

Take a number $x$ between 0 and 1, and express it as an infinite
sequence of digits after the decimal point.

Take all the digits in even positions in this number, and concatenate
them to form another fraction $y$, and similarly concatenate all
the digits in odd positions to form $z$

Now treat $y$ and $z$ as coordinates of a point in the unit
square.  This gives a reversible map between all points on the unit
line and all points of the unit square.

We disregard this kind of map because it is not continuous.
Continuous maps do preserve dimension.
}


footnote{$^3$}
{{\bf Orientation}\quad
is not a property of point sets, but all our Euclidean
representations seem to support it for free.  The point set for a
plane does not distinguish between the two sides it divides space
into; the point set for a line does not know which direction you are
going to run along it.

This means that the geometric entities which we are representing are
not quite the point sets, but something more, which is not quite so
simple to describe.
 
However, it is really nice to be able to use orientation in
algorithms, so don't complain.  The half-space (everything on one side
of a plane) is a lovely primitive heavily used in the sister subject
of solid modelling.


footnote{$^4$}
{{\bf Displacements}:\qquad
The distinction between points and displacements is only half of the
story.  In fact there are three sorts of displacements.

 \noindent * One sort are true displacements, like the vectors U and V
 in the surface representation.

 \noindent * The second sort are dual vectors, like the surface normal
 vector N.

The first sort have components which are essentially distances, the
second sort have components which are reciprocals of distances.  This
makes a difference when they are to be transformed by a scaling.  When
we apply a similarity transform to the second kind of vector we need
to use $R^{-T}$ for mapping out and $R^T$
for mapping in, instead of $R$ and $R^{-1}$
respectively.

 \noindent * The third sort are the unit vectors, (which model pure
directions) which lie somewhere in between.

The distinctions between the types of displacement are even more
pronounced when transformations other than solid body (congruence)
transforms are used.  All is made clear if you apply tensor notation,
with superscripts for the first type of vector and subscripts for the
second.  Contraction applies only between scripts of opposite kind and
the alternating tensor (which gives the vector product has three
scripts of the same kind).

However, as long as only congruence transforms are used, 
$R^{-1} = R^{T}$ and so the distinctions do not cause any difference.
}


\footnote{$^5$}
{{\bf Non-Manifoldness of Parametric Curves}:\qquad
A parametric curve satisfies the definition of a univariate manifold
only if it nowhere intersects itself. At a self-intersection it is NOT
manifold, and really careful mathematics has to take care of such
points by a process of {\sl stratification}, which divides the curve
into manifold pieces by such self-intersections.  Stratification is
really important in the theory of cellular solid models.

In CAGD it is less critical, and in practice we just ignore this
problem because we can always restore manifoldness by expanding the
defining map from ({\sl Real} to {\sl Point}) to ({\sl Real} to
\sl{(Real,Point)}).  This is sufficient to give us as much regularity
as we need for analysis.

Another technical fix is to regard the coefficients and the curve as
lying in a much higher dimensional space (3 + number of coefficients),
where each coefficient has a unit coordinate in its own dimension, all
the others having a zero value.  This is slightly less powerful, and
it gets rid of non-manifoldness whenever the existence and location of
a non-manifold point depends only the coordinates of the coefficients.
If there is a non-manifold point inherent in the basis, which always
occurs, whatever the coefficient values, it leaves it there.

For nice univariate manifoldness, we also need the basis functions to
be more or less continuous.  A discontinuity in the basis functions
leads to a jump in the curve.  A really wild basis -- for example,
containing a function which takes one value for rational abscissae and
a different value for irrational ones -- could totally prevent the
curve from being even piecewise manifold.  

Again we ignore this in the theory of how to represent parametric
curves, because we stay with basis functions which are at least C0.
The issue will crop up again in a milder form as a result of
refinement definitions later in the course.
}

\footnote{$^6$}
{{\bf Hermite Interpolation}:\qquad
What Hermite actually invented was a bit more complicated, as he
supported interpolation through any collection of data points and
derivatives at them.  The 'Hermite' we use is a tiny subset of this.


\footnote{$^7$}
{{\bf Unequal Interval Splines}:\qquad 
In this course we focus on splines where the polynomial pieces all
have the same length (in parameter).  Spline theory is much much
richer, covering the cases where the pieces have different lengths,
and even when some pieces have a length which tends to (and reaches)
zero.

When an interval is of zero length, what happens is that the
continuity at the join drops by one degree, so that a cubic spline
would acquire a point where the curve was only C1 instead of C2, for
example.

When multiple consecutive intervals are of zero length, the continuity
drops by a corresponding number of levels.

We actually use high multiplicity at the ends, which changes the
shapes of the basis functions near the ends, to give nice
end-conditions, but this issue will be skimmed over during this
course, because we shall focus on closed curves and the middles of big
surfaces.

The unequal interval theory has a beautiful exposition in the book 'A
practical guide to Splines' by Carl de Boor.  
}

\footnote{$^8$}
{{\bf Unequal Interval Lagrange Polynomials}: \qquad
Lagrange polynomials are well defined if the interpolation points are
not equally spaced.  The formula in the notes applies to the general
case, only failing if points actually become coincident.

If two points do tend to coincidence, in the limit, a specified
derivative can also be interpolated, and so we can consider Hermite
interpolation to be a limiting case of Lagrange.

If the interpolation points are not equally spaced, but bunch together
towards the ends (to find the distribution, project points equally
spaced round a semicircle on to a diameter) the ripples do not blow up
at the ends, but remain constant size.  This behaviour is still bad
enough.
}


\footnote{$^9$}
{{\bf Arc Length Parametrizations}:\qquad There are some very lovely recent
results (look up Rida Farouki at UC Davis) which show that it is
impossible for any parametric polynomial apart from the straight line
to have arc length parametrization.

However, it is possible for a parametric polynomial to have an arc
length which is a rational polynomial in terms of parameter.  These
'rational speed curves' have a number of degrees of freedom the same
as the ordinary rational parametric polynomials of just over half the
degree.  The theory ties together Pythagorean triangles whose sides
are polynomials, not numbers, (the formula $x = u^2-v^2; y = 2uv; z =
u^2+v^2$ still works when $u,v,x,y,z$ are all functions) and the
squaring of complex numbers.

The interesting, but unexplained, fact is that aesthetically they are
rather more beautiful than their rational equivalents.
}





