Feature Based Modeling and Design

Feature Based Modeling and Design

Alyn Rockwood Kun GaoKAUST

Greetings from Saudi Arabia

Greetings from Journal of Graphics Tools

Announcing a special issue of

Geometric Algebra and Graphics Applications!

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 chooseto represent face.Awkward patch layout

What is a Feature?Die Mathematiker sind eine Art Franzosen; redet man mit ihnen, so bersetzen 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

What is a Feature?Boundaries, G1 discontinuities, creases, high curvature regions, ridges, peaks

Modeling with featuresA model that more closely reflects the artists conception

DesiderataMultisided 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 smoothlyFunctionality. Compact database, rapidly computed, analytic surfaces and supports rendering.

Desiderata

A New Approach - foundationsDiscrete 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 ]

A New Approach In parameter space, find ui, the closest point on the ith footprint to given point u

A New ApproachIn parameter space, find di, the distance to closest point on the ith footprintd3d2d1

A New ApproachFootprints are pre-images of features in object space via feature maps fi . on the ith footprint. Define xi so that fi (ui) = xi

A New ApproachIn object space, find weighted least squares solution of the xi

where the weights wi(x,y,z) = 1 / dix1x2x3x

A New ApproachF(u) = xF(u) = x

A New ApproachTo summarize:From u find point on ith footprintand compute point on attributeto use it in least squares

where weight is determined as reciprocal distance.

Do it for all footprints

ExampleMove uSee x moveuxu2u1u3x3x1

ExampleWhen u is close to the footprint then thecorresponding feature dominates because its distance is small.The interpolation propertyuxx1x3u3u2u1

Solving the least squaresLet 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 lossE/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 0i xi(u,v) wi(u,v) = x i wi(u,v).impliesx = i xi(u,v) wi(u,v) / i wi(u,v).Putting it together:

Mildly surprising discovery

F(u) = wi(u)fi(u) / wi(u)

generalizes Shepards formula

Weights and interpolants are defined in a separate parameter space with general distances

Convex combination

The weightswi(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.

Minimal energy soap film effectThree lines and a sine curve; no connection needed

Higher order continuityLetF(u) = S i(u)2Li(u)

where i(u) = wi(u)/Sj wj(u) and the loft

Li(u) = (1-si) fi(ti) + si gi(ti).

(si and ti are distance and footprintparameter functions of u) F(u) is cotangent to the linear loft Li(u) alongthe attribute curve fi(ti).

Tangency is determined by gi(ti).

Interpolation to derivatives

Five sided, horizontal slope, varying loft fi(ti) gi(ti)

Interpolation to derivatives

Five sided and slopes, linear lofts

G1 continuityTheorem 1. If F(u) = i [ Ri(si, ti)] (Wi(u)/ i Wi(u))2 are defined with separate footprints, where Ri(si, ti) = (1- si) fi(ti) + si gi(ti), then,

F(u0, v0)/u = Ri(0, t0)/u and 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 footprintParameter 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).

G1 continuity

Contouring of three and four-sided patch configuration.Matched loftsacross curves andat vertices.

Higher order continuityLetF(u) = S i(u)3Qi(u)

where i(u) = wi(u)/Sj wj(u) and

Qi(u) = (1-si)2 fi(ti) + 2 (1-si) si gi(ti) + si 2 hi(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) .

G2 continuityLet Qi(s,t) = (1-s)2 fi(t) + 2(1-s)s gi (t) + s2 hi(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 in which QLi(s,t) and QRi(s,t) meet with G2 continuity on the boundary curves of L(u) and R(u), then L(u) and R(u) are G2 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.

G2 ContinuitySet of 2, 3, 4 and 5-sided patchesIsophote showing curvature continuity

G2 ContinuityHighly reflective, aesthetic surfaces

EditabilityMulti-sided patches:Car with 2-, 3- 4- and 5-sided patches

A-pillar a single 7-sided patch

EditabilityMinimal curve input for high expressive content

EditabilityEditting cuves across interior, arbitrary parameter position.

Editability auto footprintFootprint space inferred from the shape of the patch.Automobile A-pillar:7-sided and lengthsand flattening!

Editability demo

Floating edgesUnattached footprint maps to unattached attribute curve;surface interpolates floating curve

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

Floating edgesFootprints are orthogonal projections of (red) attributecurves. Appalachian mountain trimmed to square (arbitrary, no polygon!)

Template parameter spacesCylindrical footprint space. Circular footprints.Distance is is vertical height from point to circle.udistance

Template parameter spacesSimilar shapes. Cylindrical footprint space reapplied to daffodil- twice.udistance

Interpolation to fractalsProperly defined lofts yields slope of ridgebrown = fi(ti), red = gi(ti).

Two-sided attributes, single patchFloating edge with two lofts. Switch loft on footprint C0 continuity

glefti(ti)

grighti(ti)

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

Two-sided attributes, curves7 Bezier curves with fractal noise is total data base

Floating edge with two attribute functions. Switch functions again C-1 continuity!

flefti(ti)

frighti(ti)Two-sided attributes, single patch

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

Closure of Parametric CurvesAttributes include any parametric curve, whichallows adding trigonometric noise, for example:

finew(t) = fiold(t) + [0, 0.3*sin(5*pi*t), 0.2*sin(4*pi*t)],

Closure of Parametric Curves Trim curvesTrimming from footprint (parameter) space to object space is traditional (5-sided, noise)

Closure of Parametric Curves Watertight merging of two surfacesUsing trim curve as an attribute for 3-sided

Closure of Parametric Curves Watertight merging of two surfaceMoving 3-sided and trim curve

Closure of Parametric Curves Watertight merging of two surfaceEmblem on shieldand floating curve

Operations degree of surfaceG1 surface with cubic attributes:

F(u) = S i(u)2Li(u):

For ith term:Cubic in t (parameter of attribute)Loft is linear in s (distance parameter)Weight is order (N-1)2 over (N-1)2 in s, where N is number of sides.

For examples, if N=2, 3, 4, and 5then degree in s is 2/1, 5/4, 10/9 and 16/15

(s and t are affine maps of u and v.)

Operations - order of computationLimit singularity, wi(u) is large number:

F(u) = i fi(u) wi(u) / i wi(u).

Computation is linear with number of attributes

Remove singularity:F(u) = i ji [1/ wj (u)] fi(ui) / i P ji [1/ wj (u)].

Recall wj (u) is reciprocal distance

Computation is quadratic with number of attributes

Economy of data/inputObjectPatches CurvesLofts NoisesAuto body 28 25 25 0Cartoon bird 7 21 21 0Morning glory 2 3 2 2Appalachia 1 6 0 0Pikes Peak 1 7 14 1Daffodil 4 5 4 2A-pillar 1 7 7 0

Economy of data/input Palo Duro Canyon captured with 128 edges

Thank you!

It is impossible to be a mathematician without being a poet in soul.Sofia Kovalevskaya

Poetry is as exact a science as geometry Flaubert

Geometry is as sublime an art as poetry - AR

See also www.fredesign3d.com

Leifs questionVar. MeshAB surfacePiecewise linearAlgebraically exactNumerical issuesAnalyticMesh database(104-6) Small database(102-3)Vertex/feature editingAttribute editingPDE solutions (flow) None -yetLinear time in verticesLinear in attributesLOD re-initializeLOD add attributesResolution re-initializeresampleNoneFootprint issues

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Feature Based Modeling and Design