Feature Based Modeling and Design

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Feature Based Modeling and Design
Alyn Rockwood Kun Gao KAUST
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Greetings from Saudi Arabia

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

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Modeling with rectangular patchesProblems

• 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

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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

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What is a Feature?

• Boundaries, G1 discontinuities, creases, high
  curvature regions, ridges, peaks

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Modeling with features

• A model that more closely reflects the artists
  conception

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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.

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Desiderata
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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

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A New Approach

• In parameter space,
• find ui, the closest point on the ith footprint
  to given point u

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A New Approach

• In parameter space,
• find di, the distance to closest point on the
  ith footprint

d1
d3
d2
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A New Approach

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

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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
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A New Approach

• F(u) x

F(u) x
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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

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Example
x1
x
x3
Move u See x move
u1
u
u3
u2
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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
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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
•   ?i xi(u,v) wi(u,v) x ?i wi(u,v).
•   implies
•   x ?i xi(u,v) wi(u,v) / ?i wi(u,v).
•  
• Putting it together
•  

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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

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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.

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Minimal energy soap film effect
Three lines and a sine curve no connection
needed
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Higher order continuity

• Let
• F(u) S Wi(u)2Li(u)
• where Wi(u) wi(u)/Sj wj(u) and the loft
• Li(u) (1-si) fi(ti) si gi(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).

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Interpolation to derivatives

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

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Interpolation to derivatives

• Five sided and slopes,
• linear lofts

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Interpolation to derivatives

• Five sided and slopes,
• linear lofts

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G1 continuity

• Theorem 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 sisi(u,v) is the distance to the ith
  footprint
• Parameter titi(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).

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G1 continuity

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

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Higher order continuity

• Let
• F(u) S Wi(u)3Qi(u)
• where Wi(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) .

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G2 continuity

• Let 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.

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G2 Continuity

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

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G2 Continuity

• Highly reflective, aesthetic surfaces

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Editability

• Multi-sided patches
• Car with 2-, 3- 4- and 5-sided patches
• A-pillar a single 7-sided patch

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Editability

• Minimal curve input for high expressive content

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Editability

• Editting cuves across interior, arbitrary
  parameter position.

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Editability

• Editting cuves across interior, arbitrary
  parameter position.

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Editability auto footprint

• Footprint space inferred from the shape of the
  patch.
• Automobile A-pillar
• 7-sided and lengths
• and flattening!

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Editability demo
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Floating edges

• Unattached footprint maps to unattached attribute
  curve
• surface interpolates floating curve

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Floating edges topology?

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

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Floating edges

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

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Template parameter spaces

• Cylindrical footprint space. Circular
  footprints.
• Distance is is vertical height from point to
  circle.

distance
u
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Template parameter spaces

• Similar shapes. Cylindrical footprint space
  reapplied to daffodil- twice.

distance
u
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Interpolation to fractals

• Properly defined lofts yields slope of ridge

brown fi(ti), red gi(ti).
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Two-sided attributes, single patch

• Floating edge with two lofts.
• Switch loft on footprint C0 continuity
• glefti(ti)
• grighti(ti)

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Two-sided attributes, single patch

• Several floating edges (7) with paired lofts.
• Switch lofts on footprint C0 continuity

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Two-sided attributes, curves

• 7 Bezier curves with fractal noise is total data
  base

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Two-sided attributes, single patch

• Floating edge with two attribute functions.
• Switch functions again C-1 continuity!
• flefti(ti)
• frighti(ti)

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Two-sided attributes, single patch

• Several floating edges (3) with paired lofts.
• Antelope Island. Cliff

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Closure of Parametric Curves

• Attributes include any parametric curve, which
• allows adding trigonometric noise, for example
• finew(t) fiold(t) 0, 0.3sin(5pit),
  0.2sin(4pit),

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Closure of Parametric Curves Trim curves

• Trimming from footprint (parameter) space to
  object space is traditional (5-sided, noise)

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Closure of Parametric Curves Watertight merging
of two surfaces

• Using trim curve as an attribute for 3-sided

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Closure of Parametric Curves Watertight merging
of two surface

• Moving 3-sided and trim curve

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Closure of Parametric Curves Watertight merging
of two surface

• Emblem on shield
• and floating curve

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Operations degree of surface

• G1 surface with cubic attributes
• F(u) S Wi(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 N2, 3, 4, and 5
• then degree in s is 2/1, 5/4, 10/9 and 16/15
• (s and t are affine maps of u and v.)

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Operations - order of computation

• Limit 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 ?j?i 1/ wj (u) fi(ui) / ?i P j?i 1/
  wj (u).
• Recall wj (u) is reciprocal distance
• Computation is quadratic with number of
  attributes

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Economy of data/input

• Object Patches Curves Lofts Noises
• Auto body 28 25 25 0
• Cartoon bird 7 21 21 0
• Morning glory 2 3 2 2
• Appalachia 1 6 0 0
• Pikes Peak 1 7 14 1
• Daffodil 4 5 4 2
• A-pillar 1 7 7 0

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Economy of data/input Palo Duro Canyon captured
with 128 edges
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Thank you!It is impossible to be a
mathematician without being a poet in soul.Sofia
KovalevskayaPoetry is as exact a science as
geometry FlaubertGeometry is as sublime an art
as poetry - ARSee also www.fredesign3d.com
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Leifs question

• Var. Mesh AB surface
• Piecewise linear Algebraically exact
• Numerical issues Analytic
• Mesh database(104-6) Small database(102-3)
• Vertex/feature editing Attribute editing
• PDE solutions (flow) None -yet
• Linear time in vertices Linear in attributes
• LOD re-initialize LOD add attributes
• Resolution re-initialize resample
• None Footprint issues