h1. S-Power-Basis-Forms

2Geom provides a very powerful algebra for modifying paths.  Although
paths are kept in an extended SVG native form where possible, many
operations require a more mathematical form.  Our prefferred form is
a sequence of s-power basis polynomials, henceforth referred to as
s-basis.  We may convert to this form, perform the required operations
and convert back, approximating to a requested tolerance as required.

The precise details of the s-basis form are beyond the scope of this
manual - the interested reader should consult \cite{SanchezReyes1997,SanchezReyes2000,SanchezReyes2001,SanchezReyes2003,SanchezReyes2004}.
An elementary, functional description is given in Appendix C.

(TODO: work out textile citations, math inclusion)

Geometrically important properties:
* exact representation of bezier segments
* low condition number on bezier conversion.
* strong convergence guarantees
* $C^0$ continuity guarantee

The following operations are directly implementable and are very efficient:
* fast conversion from all svg elements
* basic arithmetic - @+@, @-@, $\times$, $\div$
* algebraic derivative and integral
* elementary trigonometric functions: $\sqrt{\cdot}$, $\sin(\cdot)$, $\cos(\cdot)$, $\exp(\cdot)$
* efficient degree elevation and reduction
* function inversion
* exact solutions for many non trivial operations
* root finding
* composition

All of these operations are fast.  For example, multiplication of two
beziers by converting to s-basis form, multiplying and converting back
takes roughly the same time as performing the bezier multiplication
directly, and furthermore, subdivision and degree reduction are
straightforward in this form.

h2. Implementation

h3. *Linear*

The *Linear* class represents a linear function, mostly for use as a
building block for *SBasis*.  *Linear* fully implements *AddableConcept*,
*OffsetableConcept*, and *ScalableConcept* yielding the following operators:

<pre><code>
    AddableConcept:    x + y, x - y, x += y, x -= y
    OffsetableConcept: x + d, x - d, x += d, x -= d 
    ScalableConcept:   x * d, x / d, x *= d, x /= d, -x
</code></pre>

(where @x@ and @y@ are *Linear*, d is *Coord*, and all return *Linear*)

As *Linear* is a basic function type, it also implements the *FragmentConcept*.

The main *Linear* constructor accepts two *Coord* values, one for the Linear's
value at 0, and one for its value at 1.  These may then later be accessed and
modified with the indexing operator, @[]@, with a value of 0 or 1.

h3. *SBasis*

The *SBasis* class provides the most basic function form,
$f(t) \rightarrow y$.  *SBasis* are made up of multiple *Linear* elements,
which store to/from values for each polynomial coefficient.

*SBasis*, like *Linear*, above, fully implements *AddableConcept*,
*OffsetableConcept*, and *ScalableConcept*.

As *SBasis* is a basic function type, it implements the *FragmentConcept*.

Usually you do not have to directly construct SBasis, as they are obtained
one way or another, and many of the operations are defined, however, *SBasis*
may be constructed as an implicit *Linear* cast, as a copy, or as a blank.
The class is actually an extension of @std::vector<Linear>@.  This provides
the primary method of raw *SBasis* construction -- @push_back(Linear)@, which
adds another coefficient to the *SBasis*.

*SBasis* also provides the indexing accessor/mutator, and due to its vector
nature, iteration.

(TODO: wouldn't the indexing be provided by vector any way?)

h3. *SBasis2D*

SBasis2D provides a multivariate form - functions of the form
$f(u,v) \rightarrow z$.  These can be used for arbitrary distortion
functions (take a path $p(t) \rightarrow (u,v)$ and a pair of surfaces
$f(u,v),g(u,v)$ and compose: $q(t) = (f(p(t)), g(p(t)))$.

(TODO: flesh out this section)