Feature Based Modeling and Design

1. Feature Based Modeling and Design Alyn Rockwood Kun Gao KAUST

2. Greetings from Saudi Arabia

3. Greetings from Journal of Graphics ToolsAnnouncing a special issue of Geometric Algebra and Graphics Applications!

4. Modeling with rectangular patches:Problems “… current digital tools are unable to decouple the creative process from the underlying mathematical attributes of the surface.” -K. Singh Laying out patches Non-intuitive design curves: NOT what an artist would choose to represent face. Awkward patch layout

5. What is a “Feature?” • Die Mathematiker sind eine Art Franzosen; redet man mit ihnen, so übersetzen sie es in ihre Sprache, und dann ist es also bald ganz etwas anderes. • Mathematicians are like Frenchmen; if you talk to them, they translate it into their own language, and then it is immediately completely different. • Johann Wolfgang von Goethe

6. What is a “Feature?” Boundaries, G1 discontinuities, creases, high curvature regions, ridges, peaks…

7. Modeling with features A model that more closely reflects the artist’s conception

8. Desiderata • Multisided patches. Feature curves are not always rectangular. • Freeform topology. Does not constrain design by forcing on how to layout patch networks, rather than features. • Floating curves and points. Interior attributes allow fine-tuning and richness with minimal input. • General curve Input. trig functions and fractals, isolated points, derivative information such as slopes and curvature enhance modeling effects. • G2 continuity. Connecting patches smoothly • Functionality. Compact database, rapidly computed, analytic surfaces and supports rendering.

9. Desiderata

10. A New Approach - foundations • Discrete least squares minimizes • (x - xi)2 + (y - yi)2 + (z - zi)2 Weighted least squares minimizes • wi(x,y,z) [(x - xi)2 + (y - yi)2 + (z - zi)2] Where, for example wi(x,y,z) = 1 / [(x - xi)2 + (y - yi)2 + (z - zi)2+ i ]

11. u1 u u3 u2 A New Approach In parameter space, find ui, the closest point on the ithfootprint to given point u

12. u A New Approach In parameter space, find di, the “distance” to closest point on the ithfootprint d1 d3 d2

13. x1 f1 u1 x3 x2 f3 f2 u3 u2 A New Approach Footprints are pre-images of features in object space via feature maps fi . on theith footprint. Define xi so that fi (ui) = xi

14. A New Approach In object space, find weighted least squares solution of the xi where the weights wi(x,y,z) = 1 / di x1 x x3 x2

15. u A New Approach • F(u) = x F(u) = x

16. A New Approach To summarize: From u find point on ith footprint and compute point on attribute to use it in least squares where weight is determined as reciprocal distance. Do it for all footprints

17. Example x1 x x3 Move u See x move u1 u u3 u2

18. Example x1 When u is close to the footprint then the corresponding feature dominates because its distance is small. The interpolation property x x3 u1 u u3 u2

19. Solving the least squares Let fi: (u,v)  (x,y,z), be the attribute functions Let wi: (u,v)  wiR be the weight functions. For F = (xi(u,v) , yi(u,v) , zi(u,v) ) minimize: E = i ||fi(u,v) - F|| 2 wi(u,v) Hence without loss E/x = i [-2(xi(u,v) - x)wi(u,v) + (xi(u,v) - x )2wi(u,v) /x] = i [-2(xi(u,v) - x )wi(u,v)] . Minimizing by setting it to 0 ixi(u,v) wi(u,v) = xi wi(u,v).   implies   x = ixi(u,v) wi(u,v) / i wi(u,v). Putting it together:

20. Mildly surprising discovery F(u) =  wi(u)fi(u) /  wi(u) generalizes Shepard’s formula Weights and interpolants are defined in a separate parameter space with general distances

21. Convex combination The weights wi(u) /  wj(u) sum to 1 (partition of unity). • F(u) is affinely invariant. • The surface lies within the convex hull of the fi(u) • The surface reproduces the plane/line.

22. Minimal energy “soap film” effect Three lines and a sine curve; no connection needed

23. Higher order continuity Let F(u) = S Ŵi(u)2Li(u) where Ŵi(u) = wi(u)/Sjwj(u) and the loft Li(u) = (1-si) fi(ti) + sigi(ti). (si and ti are distance and footprint parameter functions of u) F(u) is cotangent to the linear loft Li(u) along the attribute curve fi(ti). Tangency is determined by gi(ti).

24. Interpolation to derivatives Five sided, horizontal slope, varying loftfi(ti) gi(ti)

25. Interpolation to derivatives Five sided and slopes, linear lofts

26. Interpolation to derivatives Five sided and slopes, linear lofts

27. G1 continuity Theorem 1. If F(u) = i [ Ri(si, ti)] (Wi(u)/ i Wi(u))2are defined with separate footprints, where Ri(si, ti) = (1- si) fi(ti) + sigi(ti), then, F(u0, v0)/u = Ri(0, t0)/uandF(u0, v0)/v = Ri(0, t0)/v for point (u0, v0) on the footprint at parameter t0. Parameter si=si(u,v)is the distance to the ith footprint Parameter ti=ti(u,v)is then parametric value of the nearest point on the ith footprint Theorem guarantees that if two patches share a common curve fi(t) and have two lofts that share tangent planes at the common curve, then the surface patches also share common tangent planes; they are G1 at fi(t).

28. G1 continuity Contouring of three and four-sided patch configuration. Matched lofts across curves and at vertices.

29. Higher order continuity Let F(u) = S Ŵi(u)3Qi(u) where Ŵi(u) = wi(u)/Sjwj(u) and Qi(u) = (1-si)2fi(ti) + 2 (1-si) sigi(ti) + si 2hi(ti) . F(u) is cotangent to the parabolic loft Qi(u) along the attribute curve fi(ti). Curvature is determined by gi(ti) and hi(ti) .

30. G2 continuity Let Qi(s,t) = (1-s)2fi(t) + 2(1-s)s gi (t) + s2hi(t) be a parabolic loft. Consider two such lofts for each ith footprint, namely QLi(s,t) and QRi(s,t), Theorem 2. Given surfaces L(u) = i QLi(s,t) [Wi(u)/ i Wi(u) ]3,and R(u) = i QRi(s,t) [Wi(u)/ i Wi(u) ]3 inwhich QLi(s,t) and QRi(s,t) meet withG2 continuity on the boundary curves of L(u) and R(u), then L(u) and R(u) areG2 continuous. Theorem 3. Given surfaces as in Theorem 2 where QLi(s,t) and QRi(s,t) meet with twist continuity on the boundaries of L(u) and R(u), then L(u) and R(u) are twist continuous.

31. G2 Continuity • Set of 2, 3, 4 and 5-sided patches • Isophote showing curvature continuity

32. G2 Continuity Highly reflective, aesthetic surfaces

33. Editability Multi-sided patches: Car with 2-, 3- 4- and 5-sided patches A-pillar a single 7-sided patch

34. Editability Minimal curve input for high expressive content

35. Editability Editting cuves across interior, arbitrary parameter position.

36. Editability Editting cuves across interior, arbitrary parameter position.

37. Editability– auto footprint Footprint space inferred from the shape of the patch. Automobile A-pillar: 7-sided and lengths …and flattening!

38. Editability– demo

39. Floating edges Unattached footprint maps to unattached attribute curve; surface interpolates floating curve

40. Floating edges – topology? • A topologist is one who doesn't know the difference between a doughnut and a coffee cup. • John Kelley

41. Floating edges Footprints are orthogonal projections of (red) attribute curves. Appalachian mountain trimmed to square (arbitrary, no polygon!)

42. Template parameter spaces Cylindrical footprint space. Circular footprints. Distance is is vertical height from point to circle. distance u

43. Template parameter spaces Similar shapes. Cylindrical footprint space reapplied to daffodil- twice. distance u

44. Interpolation to fractals Properly defined lofts yields slope of ridge brown = fi(ti), red = gi(ti).

45. Two-sided attributes, single patch Floating edge with two lofts. Switch loft on footprint – C0 continuity glefti(ti) grighti(ti)

46. Two-sided attributes, single patch Several floating edges (7) with paired lofts. Switch lofts on footprint – C0 continuity

47. Two-sided attributes, curves 7 Bezier curves with fractal noise is total data base

48. Two-sided attributes, single patch Floating edge with two attribute functions. Switch functions again – C-1 continuity! flefti(ti) frighti(ti)

49. Two-sided attributes, single patch Several floating edges (3) with paired lofts. Antelope Island. Cliff

50. Closure of Parametric Curves Attributes include any parametric curve, which allows adding trigonometric noise, for example: finew(t) = fiold(t) + [0, 0.3*sin(5*pi*t), 0.2*sin(4*pi*t)],