Introduction Transfinite Surface Interpolation New Representations Results Conclusion Ribbon-based Transfinite Surfaces eter Salvi † , Tam´ arady † , Alyn Rockwood ‡ P´ as V´ † Budapest University of Technology and Economics ‡ King Abdullah University of Science and Technology CAGD 31(9), pp. 613–630, 2014. GMP 2015 P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Outline Introduction 1 Curvenet-based Design Coons Patches Transfinite Surface Interpolation 2 Ribbons Domain Polygons Parameterizations Simple Parameterizations Constrained Parameterizations Blending Functions New Representations 3 Generalized Coons Patch Composite Ribbon Patch Results 4 Conclusion 5 P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Curvenet-based Design Motivation Free-form surface design based on feature curves Hand-drawn sketches or images as input Tools for 3D curve / cross-derivative generation Semi-automatically generated surfaces Key issue: n -sided surface representation P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Curvenet-based Design Conventional Surfacing Methods Trimming Defining the quadrilateral? Boundary modification? Stitching? Quadrilaterals Creating smooth divisions? Modification – effect on the dividing curves? Recursive subdivision Initial polyhedra? Cross-derivative constraints? P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Curvenet-based Design Transfinite Surface Interpolation Avoid dealing with control points or polyhedra No need for interior data Exact boundary interpolation Real-time editing of complex free-form models Smooth connections Previous work: Coons ’67 Charrot–Gregory ’84 Kato ’91 Sabin ’96 etc. P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Coons Patches C 1 Coons Patch Boundary curves: S ( u , 0), S ( u , 1), S (0 , v ), S (1 , v ) Cross-derivatives: S v ( u , 0), S v ( u , 1), S u (0 , v ), S u (1 , v ) Hermite blends: α 0 , α 1 , β 0 , β 1 � α 0 ( u ) β 1 ( u ) � = β 0 ( u ) α 1 ( u ) U � α 0 ( v ) β 1 ( v ) � V = β 0 ( v ) α 1 ( v ) � S ( u , 0) S u = S v ( u , 0) S ( u , 1) S v ( u , 1) � � S (0 , v ) S v S u (1 , v ) � = S u (0 , v ) S (1 , v ) S (0 , 0) S u (0 , 0) S (1 , 0) S u (1 , 0) S v (0 , 0) S uv (0 , 0) S v (1 , 0) S uv (1 , 0) S uv = S (0 , 1) S u (0 , 1) S (1 , 1) S u (1 , 1) S v (0 , 1) S uv (0 , 1) S v (1 , 1) S uv (1 , 1) V ( S u ) T + S v U T − VS uv U T S ( u , v ) = P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Coons Patches Reformulation s 4 Positional and tangential P (s ) 3 3 P (s ) 4 4 T (s ) s 3 constraints: P i ( s i ) and T i ( s i ) 3 3 T (s ) 4 4 T (s ) Assume compatible twists: 2 2 T (s ) 1 1 s 1 W i , i − 1 = T ′ i (0) = − T ′ P (s ) i − 1 (1) 2 2 P (s ) 1 1 s 2 � α 0 ( s i +1 ) � T � P i ( s i ) 4 � � S ( u , v ) = − β 0 ( s i +1 ) T i ( s i ) i =1 � α 0 ( s i +1 ) � � α 0 ( s i ) � T � P i (0) 4 � P ′ i (0) � β 0 ( s i +1 ) T i (0) T ′ i (0) β 0 ( s i ) i =1 P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Coons Patches Ribbon-based Coons Patch Linear interpolants (ribbons): P i+1 R i ( s i , d i ) = P i ( s i ) + γ ( d i ) T i ( s i ) d i γ ( d i ) = β 0 ( d i ) /α 0 ( d i ) = 2 d i +1 Distance parameter d i = s i +1 d i Corner correction patch s i Q i , i − 1 ( s i , s i − 1 ) = R i (s i ,d i ) P i-1 P i P i (0) + γ (1 − s i − 1 ) T i (0) + γ ( s i ) T i − 1 (1) + γ ( s i ) γ (1 − s i − 1 ) W i , i − 1 4 4 � � S ( u , v ) = R i ( s i , d i ) α 0 ( d i ) − Q i , i − 1 ( s i , s i − 1 ) α 0 ( s i ) α 0 ( s i − 1 ) i =1 i =1 P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Overview Transfinite Surface Interpolation Input: Hermite data ( P i , T i ) Surface S ( u , v ) = n � Interpolant i ( s i , d i ) · i =1 Blend i ( d 1 , . . . , d n ) Constituents Ribbons Parameterization functions Domain polygon Blending functions P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Ribbons Ribbon Construction Continuous normal vector Given: boundary curves P i ( s i ) and normal vectors at some points function N i ( s i ) by RMF T i ( s i ) ⊥ N i ( s i ) Resulting surface P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Domain Polygons Domain Construction Regular n -sided polygon (good most of the time) Domain “similar” to the boundary curves Similarity of... Arc lengths Angles Measure of similarity: Deviation of arc length / angle ratios Use heuristics if measure > threshold (see V´ arady et al. ’11) P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Parameterizations Ribbon Mapping – Parameterization Constraints s i ∈ [0 , 1] For a point on side i ... Simple parameterization: d i = 0 s i − 1 = 1 s i +1 = 0 d i − 1 = s i d i +1 = 1 − s i Constrained parameterization: ∂ d i − 1 = ∂ s i ∂ d i − 1 = ∂ s i ∂ u ∂ u ∂ v ∂ v ∂ d i +1 = − ∂ s i ∂ d i +1 = − ∂ s i ∂ u ∂ u ∂ v ∂ v P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Parameterizations Bilinear Line Sweep (simple) P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Parameterizations Wachspress Distance Parameters (simple) For convex polygons λ i ( u , v ) = w i ( u , v ) / � k w k ( u , v ) w i ( u , v ) = C i / ( A i − 1 ( u , v ) · A i ( u , v )) ⇒ d i ( u , v ) = 1 − λ i − 1 ( u , v ) − λ i ( u , v ) P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Parameterizations Interconnected Parameterization (constrained) Let s i ( u , v ) be a line sweep (e.g. bilinear) d i ( u , v ) = (1 − s i − 1 ( u , v )) · α 0 ( s i ) + s i +1 ( u , v ) · α 1 ( s i ) P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Parameterizations Cubic Parameterization (constrained) Based on bilinear Constant parameter lines defined by cubic B´ ezier curves λ : fullness parameter Leads to a sixth-degree equation Only fourth-degree when λ = 1 3 Precomputable P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Parameterizations Example Using λ = 1 3 : P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Parameterizations Example Using λ = 1 2 : P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
Introduction Transfinite Surface Interpolation New Representations Results Conclusion Blending Functions Blending Side Interpolants ⇒ SB Patch “Side-based” (SB) patch [Kato ’91] n S SB ( u , v ) = � R i ( s i , d i ) · B ∗ i ( d 1 , . . . , d n ) i =1 k � = i d 2 � 1 / d 2 k B ∗ i i ( d 1 , . . . , d n ) = = j 1 / d 2 k � = j d 2 � � � j j k Blend function singular in the corners P. Salvi, T. V´ arady, A. Rockwood BME & KAUST Ribbon-based Transfinite Surfaces
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