Refinable Polycube G-Splines - PDF Free Download

Refinable Polycube G-Splines Martin Sarov Jörg Peters University of Florida NSF CCF-0728797 M. Sarov, J. Peters (UF) SMI 2016 1 / 21

Background Quest Surfaces with rational linear reparameterization = as close as possible to regular splines M. Sarov, J. Peters (UF) SMI 2016 2 / 21

Geometric Continuity Geometric Continuity G 1 = C 1 continuity after change of variables M. Sarov, J. Peters (UF) SMI 2016 3 / 21

Geometric Continuity Geometric Continuity G 1 = C 1 continuity after change of variables Simplest non-affine change of variables: rational linear q = p(ρ) M. Sarov, J. Peters (UF) SMI 2016 3 / 21

Geometric Continuity Geometric Continuity G 1 = C 1 continuity after change of variables Simplest non-affine change of variables: rational linear q = p(ρ) Implies linear scaling by ω: p(t, 0) = q(0, t), ω(t) 1 p (t, 0) = 2 p (t, 0) + 1 q (0, t), ( ω(t) := (1 t)ω 0 + tω 1, ω 0, ω 1 R, : rational linear = closest to regular spline (ω = const) : maximizes the number of control handles M. Sarov, J. Peters (UF) SMI 2016 3 / 21

Polycube connectivity and control handles Outline 1 Polycube connectivity and control handles 2 Surface Construction 3 Multi-resolution M. Sarov, J. Peters (UF) SMI 2016 4 / 21

Polycube connectivity and control handles Restricted Connectivity: Polycube Configurations n {3, 4, 5, 6}: M. Sarov, J. Peters (UF) SMI 2016 5 / 21

Polycube connectivity and control handles 2x2 split: Control Handles c i M. Sarov, J. Peters (UF) SMI 2016 6 / 21

Polycube connectivity and control handles 2x2 split: Control Handles c i Control handles c i Boundary points b i (Bézier) M. Sarov, J. Peters (UF) SMI 2016 6 / 21

Polycube connectivity and control handles 2x2 split: Control Handles c i Edge Recovery: c i b i M. Sarov, J. Peters (UF) SMI 2016 6 / 21

Polycube connectivity and control handles 2x2 split: Control Handles c i Edge Recovery: c i b i smooth? M. Sarov, J. Peters (UF) SMI 2016 6 / 21

Polycube connectivity and control handles 2x2 split: Control Handles c i M. Sarov, J. Peters (UF) SMI 2016 6 / 21

Surface Construction Outline 1 Polycube connectivity and control handles 2 Surface Construction 3 Multi-resolution M. Sarov, J. Peters (UF) SMI 2016 7 / 21

Surface Construction PGS: G 1 construction for vertex valences 3, 4, 5, 6 G 1 constraints on BB-coefficients (technical). M. Sarov, J. Peters (UF) SMI 2016 8 / 21

Surface Construction PGS: G 1 construction for vertex valences 3, 4, 5, 6 G 1 constraints on BB-coefficients Apparent valence (mimic 3 6). 2c p b α,11 10 = b α+1,11 10 + b α 1,11 10, c p := cos 2π p. M. Sarov, J. Peters (UF) SMI 2016 9 / 21

Surface Construction PGS: G 1 construction for vertex valences 3, 4, 5, 6 G 1 constraints on BB-coefficients Apparent valence Assign labels: n, m edge. M. Sarov, J. Peters (UF) SMI 2016 10 / 21

Surface Construction PGS: G 1 construction for vertex valences 3, 4, 5, 6 G 1 constraints on BB-coefficients Apparent valence Assign labels: n, m edge Simple G 1 constraints work for n, m {3, 4, 5, 6} except n, 4, 4, m with n m. M. Sarov, J. Peters (UF) SMI 2016 11 / 21

Surface Construction Smooth PGS surfaces M. Sarov, J. Peters (UF) SMI 2016 12 / 21

Surface Construction Smooth PGS surfaces M. Sarov, J. Peters (UF) SMI 2016 13 / 21

Multi-resolution Outline 1 Polycube connectivity and control handles 2 Surface Construction 3 Multi-resolution M. Sarov, J. Peters (UF) SMI 2016 14 / 21

Multi-resolution Multi-resolution of Polycube G1 -splines M. Sarov, J. Peters (UF) SMI 2016 15 / 21

Multi-resolution Multi-resolution of Polycube G 1 -splines How to refine the control handles? M. Sarov, J. Peters (UF) SMI 2016 16 / 21

Multi-resolution Multi-resolution of Polycube G 1 -splines How to refine the control handles? : [PS15] add (refined) surfaces M. Sarov, J. Peters (UF) SMI 2016 16 / 21

Multi-resolution Multi-resolution of Polycube G 1 -splines How to refine the control handles? : [PS15] add (refined) surfaces : Catmull-Clark on polycube quad net M. Sarov, J. Peters (UF) SMI 2016 16 / 21

Multi-resolution Multi-resolution of Polycube G 1 -splines How to refine the control handles? : [PS15] add (refined) surfaces : Catmull-Clark on polycube quad net : C 1 bicubic spline refinement: depends on b i : if c i is set by PGS (= satisfies G1) then c i is unchanged by Edge Recovery + PGS. M. Sarov, J. Peters (UF) SMI 2016 16 / 21

Multi-resolution Summary Smooth surface covering irregularities of valence n = 3, 4, 5, 6 in a C 1 bi-3 spline surface M. Sarov, J. Peters (UF) SMI 2016 17 / 21

Multi-resolution Summary Smooth surface covering irregularities of valence n = 3, 4, 5, 6 in a C 1 bi-3 spline surface Control Handles = as for C 1 bi-3 spline (larger support) M. Sarov, J. Peters (UF) SMI 2016 17 / 21

Multi-resolution Summary Smooth surface covering irregularities of valence n = 3, 4, 5, 6 in a C 1 bi-3 spline surface Control Handles = as for C 1 bi-3 spline (larger support) Refinable reproduces surfaces at finer level QUESTIONS? M. Sarov, J. Peters (UF) SMI 2016 17 / 21

Multi-resolution Multiresolution: algebraic challenge Challenge: global system, : additional constraint ω( k 2 l ) ( bα;k 1,1 10 + 2b α;k 1,1 20 2b α;k1 10 + b α;k,1 20 ) = 0. }{{} C 2 boundary : large support of c i handles : decay fast M. Sarov, J. Peters (UF) SMI 2016 18 / 21

Multi-resolution PGSER M. Sarov, J. Peters (UF) SMI 2016 19 / 21

Multi-resolution PGS: G 1 construction for vertex valences 3, 4, 5, 6 G 1 constraints on BB-coefficients: d d b(u, v) := b ij Bi d (u)bj d (v), i=0 j=0 p 01 + q 10 = ω 0 p 10 + (2 ω 0 )p 00, (1) p 11 + q 11 = ρ(ω 0, ω 1 ), (2) p 21 + q 12 = σ(ω 0, ω 1 ), (3) p 31 + q 13 = (2 + ω 1 )p 30 ω 1 p 20. (4) ρ(ω 0, ω 1 ) := 1 ( ) 2ω0 p 20 ω 1 p 00 + (6 2ω 0 + ω 1 )p 10 3 σ(ω 0, ω 1 ) := 1 ) ω0 p 30 2ω 1 p 10 + (6 ω 0 + 2ω 1 )p 20. 3( M. Sarov, J. Peters (UF) SMI 2016 20 / 21

Multi-resolution What goes wrong when n, 4, 4, m with n m? M. Sarov, J. Peters (UF) SMI 2016 21 / 21