This file contains items of small print: technical points which describe more precisely issues which are glossed over in the main notes, where sometimes half-truths have to be perpetrated in the interests of coherence and even clarity.
Where there is such an issue in the main notes you will find an SP link, which leads you to the paragraph filling in more accurate details.
None of the small print is examinable, but the alert student may find comfort here when the lecturer appears to have pulled a fast one or ignored an important distinction.
There is an index of small print terms at the end of this file.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.
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.
Orientation 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.
The distinction between points and displacements is only half of the story. In fact there are three sorts of displacements.
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 RT for mapping in, instead of R and R -1 respectively.
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.
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 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 Real to Point to Real to (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.
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.
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.
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.
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 = u2-v2; y = 2uv; z = u2+v2 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.