Joel Daniels II University of Utah GDC Group Converting Molecular Meshes into Smooth Interpolatory...
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Transcript of Joel Daniels II University of Utah GDC Group Converting Molecular Meshes into Smooth Interpolatory...
Joel Daniels II • University of Utah • GDC Group
Converting Molecular Meshes into Smooth Interpolatory Spline
Solid Models
Joel Daniels IIElaine Cohen
David Johnson
Joel Daniels II • University of Utah • GDC Group
Physical Visualizations
• Conventional techniques unable to represent complex molecular data in physical form
• Immersive computer environments– Tactile feedback lost!– Haptics?
• Embed multiple representations within a single physical form
Joel Daniels II • University of Utah • GDC Group
Challenges
• Convert the triangle mesh into spline model
• Data segmentation
• Inter-surface continuity
Joel Daniels II • University of Utah • GDC Group
Conversion System
• Data segmentation
• Cross boundary tangents
• Corner tangents and twists
• Complete spline interpolation
Joel Daniels II • University of Utah • GDC Group
Data Segmentation
• Inherent nature of input mesh
• Map triangles to an icosahedron
• Walk vertices extracting rows and columns• Fill in the complete
spline interpolation matrix
Joel Daniels II • University of Utah • GDC Group
Boundary Tangents
• Align rows and columns of adjacent surfaces
• Fit quadratic function to indicated points
• Evaluate tangent at the boundary point
• Both surfaces influence tangent direction equally
• Fill in the complete spline interpolation matrix
Joel Daniels II • University of Utah • GDC Group
Stitching Corners
• Two corner scenarios– 3-way corners around equator
– 5-way corners at poles
• Impossible to ensure parametric continuity
• Instead strive for geometric continuity
• Discussion done in terms of the 3-way corner, but it is extendable to N-way corners
Joel Daniels II • University of Utah • GDC Group
• Compute tangent control points along each boundary
Joel Daniels II • University of Utah • GDC Group
• Compute twist control points satisfying constraints
Joel Daniels II • University of Utah • GDC Group
• Compute tangent and twist values that realize the given control points
Joel Daniels II • University of Utah • GDC Group
Discontinuity Minimization Analysis
γ´(u) = (c1 – c0) β0(u) + (c2 – c1) β1(u) + (c3 – c2) β2(u)
γ´(u) x L(u) = γ´(u) x R(u)γ´(u) x (L(u) – R(u)) = 0
L(u) – R(u) = [(l0 – c0) – (c0 – r0)] θ0(u) + [(l1 – c1) – (c1 – r1)] θ1(u) + [(l2 – c2) – (c2 – r2)] θ2(u) + [(l3 – c3) – (c3 – r3)] θ3(u)= 2 [(l0 + r0)/2 – c0] θ0(u) = 2 (M – c0) θ0(u)
θ0(u) [(M – c0) x (c1 – c0) β0(u) + (M – c0) x (c2 – c1) β1(u) + (M – c0) x (c3 – c2) β2(u)] = 0
• When (c2 – c1) and (c3 – c2) do not align with (c1 – c0) then the neighborhood is curved and no value of ‘M’ will satisfy the equation
• The algorithm aligns (M – c0) with (c1 – c0) eliminating the largest contributor of the error
Joel Daniels II • University of Utah • GDC Group
Case Studies
• Lower Curvature Model– Worst angle = 1.74º
• original angle = 7.6º
– 99% of the boundaries within 1º of G1 continuity
• Higher Curvature Model– Worst angle = 4.48º
• original angle = 12.4º
– 98% of boundaries within 1º of G1 continuity
Joel Daniels II • University of Utah • GDC Group
Conclusion
• Internally bi-cubic spline surfaces are C2 continuous
• G1 discontinuities confined to first and last knot intervals
• Algorithm guarantees G1 continuous corner when possible, otherwise it attempts to minimize created features
• Successfully convert mesh into a smooth spline model