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.