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Joel Daniels II Elaine Cohen Overview Research Goal § We provide physical visualizations to modern structural biologists, thus reviving tactile feedback, in addition to immersive environments. § Embed multiple representations within a single form: - Create a solid model by combining a backbone spline model with converted molecular spline surface, originally a triangular mesh - Fabricate the solid model using Stratasys Fused Deposition Model (FDM) Rapid Prototyping Machine - Create a flexible mold from the FDM model - Use Z-Corp 3 D Color Printer to create an opaque color coded backbone - Insert the backbone into the flexible mold and inject clear plastics to fill the remaining volume Research Challenges § Modeling software packages require homogenous representations in order to combine models. § Develop methods to achieve smooth transitions where spline surfaces meet along edges. § Some corners may not be able to be represented smoothly. Research Solutions § Develop a system to convert molecular meshes into spline models § Introduce a novel algorithm that: § Makes all edges smooth except near corners § Minimizes G 1 discontinuities that may occur near corners at which surfaces meet. Conversion System Pipeline 1. The input triangular mesh is a byproduct of molecular sampling techniques. Resulting in large meshes with many data points. 2. The triangles of the mesh are mapped to the faces of the icosahedron, and these faces are paired to form the rectangular data grids 3. Cross-Boundary smoothness is guaranteed in the estimation of differential properties across adjacent data grids. 4. Corner tangents and twists are computed to minimize G 1 discontinuities. 5. Complete spline interpolation converts the data grids and tangents into a smooth spline model. Case Studies/Results I. Model A i. Worst case angle = 1. 74 o i. corresponding boundary on the original mesh = 7. 6 o ii. 92% of the boundary is G 1 continuous iii. 99% of the boundary is within 1 o of G 1 continuity II. Model B i. Worst case angle = 4. 48 o i. corresponding boundary on the original mesh = 12. 4 o ii. 91% of the boundary is G 1 continuous iii. 98% of the boundary is within 1 o of G 1 continuity Quality Guarantee § Ridges are sections of a boundary curve where the surface normals of the two abutting surfaces differ. § Our algorithm picks a solution that confines these regions to corners and minimizes the ridges without modifying the data or adding degrees of freedom. Conclusion § Two models are analyzed, measuring all ridges that occur in the conversion process. The peak to trough variation is less steep in the model A than in model B, especially in areas surrounding corners. § The graphs plot the difference in degrees between the normals of two surfaces along their shared edge. § The ridges are shown as G 1 discontinuities, where the difference of the normals is not 0 degrees. § The algorithm confines the trouble regions to shared boundary curves between the corner and first shared data point. Corner Stitching Algorithm 1. Fit a cubic surface to the neighborhood data points of the corner. 2. Compute the tangent vectors on this surface along each boundary emanating from the corner. 3. Compute cross-boundary information at the corner to optimize smoothness. Constraints include: a. Co-planarity of differential properties. b. Regularity of parameterized corner region. 6. Modeling software packages combine models to form a solid model within a homogenous environment. § The system successfully converts a molecular mesh into a spline model, creating homogenous environments in which modeling software packages can operate. § Leads to being able to produce the desired physical visualizations. § Converted model an accurate representation of the original data, via interpolation, with an added level of smoothness. § A novel corner stitching algorithm is introduced, minimizing creases along boundaries. § Potential ridges are confined to the first and last knot interval for each boundary. § These ridges constitute a small portion of the overall boundary area, and even smaller portion of area of the overall model. § 99% of the boundaries of the flatter model are within 1 o of G 1 continuity. Even in a worse case, as with the high curvature model, similar results are realized as 98% of the boundary curve is within 1 o of G 1 continuity. § Comparisons of the original and converted models demonstrate that even the worst case ridge is significantly smoother than the original dihedral angle. Acknowledgements This work was supported in part by NSF(EIA 0121533) and NIH(573996). References  Sanner, M. , 1999. “Python: A programming language for software integration and development”. J. Mol. Graphics Mod. , 17, February, pp. 57– 61.  M. Sanner, B. Duncan, C. C. , and Olson, A. , 1999. “Integrating computation and visualization for biomolecular analysis: An example using python and avs. ”. Proceedings Pacific Symposium in Biocomputing, pp. 401– 412.  M. Sanner, D. S. , and Olson, A. , 2002. “Viper a visual programming environment for python”. 10 th International Python Conference, February.  Cohen, E. , Riesenfeld, R. , and Elber, G. , 2001. Geometric Modeling with Splines: An Introduction. AK Peters, Ltd.  Loop, C. , 1994. “Smooth spline surfaces over irregular meshes”. In SIGGRAPH ’ 94: Proceedings of the 21 st annual conference on Computer graphics and interactive techniques, pp. 303– 310.  Krishnamurthy, and Levoy, M. , 1996. “Fitting smooth surfaces to dense polygon meshes”. In ACM Transaction on Computer Graphics (SIGGRAPH 1996 Proceedings).  Livingston, J. B. , 1990. “Intersurface continuity of solid models”. Master’s Thesis, University of Utah, Department of Computer Science.