Outline for CAD conference submission
Introduction
* Counting parameters
* interpolating spline
* variational definitions - notion of optimality
* Extensionality (follows from variational def)
Key result: two parameters + extensional -> cut from rigid curve
So problem reduces to finding best curve.
Counting parameters
* Curve segments in splines are potentially from infinite dimensional
space
* In practice, most splines use curves drawn from finite-dimensional
manifold
* Subtract out translation, rotation, uniform scaling (i.e. align to
endpoints of chord)
* Observation: dimensionality never > 2 * (num control pts - 2), and
most splines are defined as a composition
Concrete representation of parameter space is tangent at endpoints.
Defn of 2 parameter: segment is uniquely determined by tangents at
endpoints. (maybe need to
Catalog of existing splines that fit into 2 parameter framework:
* Euler spiral spline (Mehlum Kurgla 2)
* MEC (Birkhoff, etc etc)
* IKARUS
* Hobby
* Circle spline
Observation: 2-parameter is sufficient for G2 continuity, without
cheating.
Variational defs and extensionality
* Cite Knuth for extensionality def'n.
* MEC is defined by minimizing L2 norm of curvature.
* Any spline which is defined variationally is by construction
extensional. Argument by adding point - if curve is different it
would have had lower fnl, and that could have served in first place.
2 parameters + extensional -> cut from rigid curve
Start by giving examples: MEC, Euler spiral. Also parameter-counting
argument. s0 and s1 <-> th0 and th1.
Give argument in terms of extending the curve (figure
two_continue.pdf). This needs carefully crafted assumptions as far as
uniqueness and existence of curve segment given th0, th1.
Properties of curve <-> properties of spline
Just about any curve can be used. Some (Euler spiral) have unique
s0,s1 soln for given th0,th1; others (MEC) need disambiguating
rule. Properties of the resulting spline follow from properties of the
generating curve.
* Inflection point (odd symmetry) -> spline can contain inflection pt.
* Monotonic curvature <-> monotonic curvature ("salience")
+ might mention monotonic curvature -> no self-intersection within
segment. Or is this a digression?
* lim{n->\infty} k'/k = 0 -> roundness
* k' near 0 -> locality. Less means better locality. This is
empirical.
Aesthetic curves and user study
* Aesthetic curve family is good candidate for our generating curve -
it has monotonic curvature, shallow k' near 0 (for higher
exponents), and a good rationale that the eye is more sensitive to
k' in low k regions.
* Present user study results.
* Result that \alpha = 1.5 is preferred proves that MEC is not
accurate predictor of user choice. (note, optimizing MEC also
doesn't give roundness)
Conclusion
This is a pretty good approach to the best possible spline. G2
continuity is enough (for human visual perception, anyway), and
2-parameter is sufficient to get that. Any reasonable definition of
"best" will imply extensionality (certainly any1 definition in terms of
variational principle implies it formally).
We don't need a variational formulation to arrive at a "best"
spline. Can do it empirically by choosing the most pleasing curve.
(good opportunity to recap user study).
Two-parameter is a useful taxonomic organizing principle. It covers a
range of existing important splines.