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

Physical Visualizations

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

Why Convert the Mesh?

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

• Extract the corner neighborhood

Joel Daniels II • University of Utah • GDC Group

• Assign a parameterization and weights

Joel Daniels II • University of Utah • GDC Group

• Fit a cubic surface to the neighborhood

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

Joel Daniels II • University of Utah • GDC Group

Questions?