Last time:
- curves
- dense representations (signal processing, filtering)
- subdivision representations
- variational representations
- spline representations
some things to notice:
- cubic polynomials = linear functions of control points
- basis functions
- controllability
- arc length vs. natural parameters
Fitting
set of points, want a “smoother” description
- filter (averaging)
- smoothing
- fitting a lower-order curve
- doesn’t matter what the representation is
- fitting polynomials is a linear least-squares problem
- even if its cubic – assuming fixed parameterizations
fit, not interpolate
- implicit constraints on interpolation (build curve through points)
- implicit constraints on curve properties (b-spline for smoothness)
- fit curve to constraints
Multi-Dimensional Interpolation
not just independent dimensions – surfaces
for every point in the plane
- triangle – barycentric (generalized baricentric for the future)
- grid
- triangles (linear – break on diagonal – grain matters)
- quads bi-linear
- higher order patches
Scattered Data Interpolation
- nearest neighbor
- tesselation (vornoi)
- radial basis function
- we’ll come back to this…
Why?
- warps / morphs / deformations
- surface patches
- multi-way blends
Special case of 2D Warping/Morphing
- how to specify map
- grid
- scatter point constraints
- as interpolation problem
- line constraints (feature-based morphs) – Bier Neely
