Aspects of Geometric Design

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Aspects of Geometric Design

• Jean Gallier
• CIS Department
• University of Pennsylvania

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Thanks, Dianna, Marcelo, Gary
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WhatIs Computer-Aided Geometric Design?

• Computer Aided Geometric Design CAGD
• Techniques for modelling curved shapes
• Using curves and surfaces
• Computational Geometry
• Scientific Visualization
• Computer Vision
• Architecture
• Computer Animation (movies)

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What is CAGD?

• Product Design and Manufacturing
• (cars, planes, ships, etc.)
• Medical Imaging (brain, liver, lungs, )
• Robotics Motion interpolation
• Biology and Computational Chemistry
• Art, Archaeology

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Some History

• Even before the 19th century, ship hulls were
  designed using long flexible wooden strips called
  splines.
• In the forties, engineers at BOEING experimented
  with CAGD methods for designing airplanes.
• Various types of surfaces were created.

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More History (1)

• In the sixties, independently, two French
  Engineers working for Citroen and Renault
  pioneered the use of CAGD in car design.
• Pierre Bezier and Paul de Casteljau proposed
  schemes for specifying curves and surfaces using
  control points (as opposed to explicit coeffs.).

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More History (2)

• In the early 70s, Carl de Boor and Richard
  Riesenfeld came up with
• B-splines and the de Boor algorithm.
• In 1978, Catmull and Clark proposed subdivision
  surfaces. Catmull is now one of the vps of
  Pixar!
• Subdivision surfaces heavily used by Pixar to
  make animated movies.

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Some Problems Considered

• Approximating a curved shape (accuracy not so
  crucial)
• Interpolating a curved shape. Here some data
  points must be on the surface.
• Drawing (rendering) smooth surfaces (or curves).

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Mathematical Representations of Curves and
Surfaces

• System of equations
• Parametric form
• Implicit form

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Parametrization
A 3D point is assigned a unique pair of values
(u,v) in the plane by means of projection
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Parametric Curves

• Cubic
• Helix

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Wiggling Effect

• Example
• Four data points
• Third degree polynomial
• Might look something like
• This is the ONLY third degree polynomial which
  fits the data
• Wiggling gets much worse with higher degree

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Problems with Polynomial Curves

• To interpolate many points, high degree is
  required
• Too expansive to draw curves of high degree
• Excessive wiggling
• Precision issues
• Impossible to make local changes (lack of local
  control)
• Lack of modularity
• Lack of incrementality

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A solution Splines

• Use piecewise polynomial curves
• The complete curve consists of several simple
  polynomial pieces
• All pieces are of low order
• Third order (cubic) is the most common
• Pieces join smoothly (typically,C2)

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Parametric Continuity

• common endpoint
• tangent vectors also agree
• change in tangents also agree
• 0 through kth derivatives match

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Bézier Curves

• A polynomial curve of degree is uniquely
  determined by control points,
  , forming a
• Control polygon

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Bézier Curves Examples

• Changing the position of a control point will
  change the shape of the curve

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The de Casteljau algorithm (1)

• A method for computing a point on the curve,
  F(t), using affine interpolation
• If , then F(t) is in the convex
  hull of the control polygon
• This method yields a recursive algorithm to draw
  a curve segment

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The de Casteljau algorithm (2)
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Parametric Surfaces

• Generalizing from curves to surfaces by using two
  parameters u and v
• Parametric surfaces can be either rectangular or
  triangular, depending on how the parameter plane
  is divided

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Bézier Surfaces (rectangular case)

• Defined in terms of a two dimensional control net

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B-spline Surfaces local control

• Local control is one of the most desirable
  properties of B-splines
• Modification of a control point only affects a
  small neighborhood

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The Difficulties with Triangular Splines

• Rectangular spline surfaces are constructed as a
  cross product of spline curves in two directions
• The B-spline curve framework does not carry over
  to triangular spline surfaces
• Lack simple and intuitive algorithms that offer
  local control

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Why Triangular Splines?

• Arbitrary topology
• how to represent a sphere?
• More generally, any closed surface?
• Many related areas generate triangular meshes
• laser scanning
• mesh construction and simplification
• Generality

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Motivations for triangular spline surfaces

• Design and/or interpolation of complex 3D shapes
• Such shapes may have holes or sharp corners
• Incremental algorithm
• Local flexibility

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Polar Forms (blossoms)

• Polynomials can be linearized in terms of
  symmetric multiaffine maps
• The specification of (polynomial) curves and
  surfaces in terms of control points is best
  explained by polar forms
• Specifying Ck -continuity is greatly simplified
  when polar forms are used

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Multiaffine Maps

• A map is affine if
• for all , and all
• A map is multiaffine if it
  is affine in each of its arguments
• A map is symmetric if it
  does not depend on the order of its arguments

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Example Ennepers Surface

• Parametric forms
• Polar forms

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Polar Forms of Polynomial Surfaces

• Given a surface , (degree m)
  there exists a unique symmetric multiaffine map,
  
  
• such that
• is also known as the polar form of the
  polynomial surface

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Example of a Triangular Control Net

• Control Net 0, 0, 0, 2, 0, 2, 4, 0, 2,
  6, 0, 0, 1, 2, 2, 3, 2, 5, 5, 2, 2, 2,
  4, 2, 4, 4, 2, 3, 6, 0
• The surface patch associated is approximated with
  the subdivision version of the de Casteljau
  algorithm

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Joining Triangular Patches with Cn Continuity
s
A
B
p
q
A
B
r

• The patches FA and FB (degree m) join with Cn
• continuity along (r, s) iff their polar forms
  satisfy
• 
  

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C1 Constraints along an Edge
s
B
p
q
A
B
r

• Along the edge ,
• Every diamond must be
  the image of

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Dianna Xus dissertation

• Incremental Algorithm for the Design of
  Triangular-Based Spline Surfaces (2002)

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C1 Constraints around a Vertex special case

• There are only 5 significant cases given
  equilateral triangles
• There are 3 degrees of freedom among star points
  in all cases when equations are examined

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C1 Constraints around a Vertex general case

• The star control points must observe the same
  affine relations given by the vertices of the
  template triangles
• There are again only 3 degrees of freedom in all
  cases

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Algorithm for Choosing Prescribed Control Points

• Systematically prescribe a set of control points
• The rest of the control points are computed
  efficiently via propagation
• The resulting surface has guaranteed continuity
• We look at both equilateral and irregular
  triangulations

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Prescription around a Vertex
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Prescription along the Edges
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Prescription 6 patches
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Irregular Triangulations

• Algorithm adapts with minimal modifications
• Complexity remains the same
• Any three non-collinear points are picked at the
  corners
• We have similar quadrilaterals instead of
  parallelograms along the edges

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Closed Surfaces

• The method generalizes to one that is based on a
  triangulated polyhedron
• Algebraic topology offers some insights

• Example
• Triangulation of a sphere

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Geometric Continuity

• Fitting a parametrically continuous surface over
  a triangulated polyhedron cannot always be done
• Two surface patches F and G are said to be
• -continuous at the joining point a if and
  only if there exist two reparameterizations
• and
  such that
• and are
  -continuous at a

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Icosahedron

• The simplest local reparameterization maps were
  chosen.
• The experimental results are surprisingly good.

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Examples
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And now, cheese and dessert!
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Marcelo Siqueiras work

• Ph.D dissertation in progress
• Generating provably good 2D and 3D meshes from
  biomedical imaging data

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Mesh Generation (1)

• Given a polygonal region (resp. polyhedron) ? in
  R2, a mesh of ? is a collection T of triangles
  (resp.. tetrahedra) such that
• the union of all triangles (resp. tetrahedra) in
  T is equal to ? and,
• for any two t1 and t2 in T, the intersection of
  t1 and t2 is either empty or a common vertex,
  edge, or face (if t1 and t2 are tetrahedra) of t1
  and t2.

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Mesh Generation (2)

• The mesh generation problem finding a mesh for a
  given polygonal region (polyhedron) ?.
• We can add more constraints to the above problem.
  For instance
• We may ask for a mesh T such that T has mostly
  well-shaped triangles (resp. tetrahedra).
• We can formally define what we mean by
  welll-shaped. For instance, we can assume that
  a triangle is well-shaped if it has no angle
  smaller than 30o.
• We can also ask for a mesh T in which the area of
  the triangles in a given region of ? does not
  exceed a given upper bound.

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Mesh Generation (3)

• The problem of generating a mesh for a polygonal
  region such that mesh triangles are mostly
  well-shaped is well understood. There are very
  good solutions available.
• However, the problem of generating a mesh for a
  polyhedron such that mesh tetrahedra are mostly
  well-shaped is not so well understood.
• There are several open questions related to this
  problem. You can find some of these questions in
• http//www.ics.uci.edu/eppstein/280g/open.html

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Mesh Generation (4)

• Why is the mesh generation problem important?
• The availability of a mesh for a domain is an
  essential prerequisite for the use of powerful
  numerical methods to solve PDEs, such as the
  Finite Element Method (FEM).
• The accuracy of the solution provided by such
  numerical methods is highly dependent on a
  variety of mesh parameters, one of which is the
  shape and the number of mesh triangles (resp.
  tetrahedra).

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Mesh Generation from Imaging Data (1)

• We are interested in generating meshes from 2D
  and 3D digital images.
• A digital image can be seen as a map that assigns
  a real value (color) to points of a 2D or 3D
  grid of integer points.
• Since a digital image is a map rather than a
  polygonal region or a polyhedron, what do we mean
  by generating a mesh from a 2D or a 3D digital
  image?

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Mesh Generation from Imaging Data (2)

• We perceive the integer points of the image
  domain as points in R2 (resp. R3) with integer
  coordinates, and consider the polygonal region
  (resp. polyhedron) defined by the pixels (resp.
  voxels) corresponding to the image points.

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Mesh Generation from Imaging Data (3)

• We want our meshes to respect the boundaries
  between polygonal regions (polyhedra) defined by
  pixels (resp. voxels) whose points are assigned
  the same color.

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Mesh Generation from Imaging Data (4)

• Our motivation for generating meshes from images
  is to enable the use of numerical methods such as
  the FEM on imaging data.
• Images adds new challenges to the mesh generation
  problem
• The polygonal regions (resp. polyhedra) obtained
  from a 2D (resp. 3D) digital image tend to have
  too many vertices, edges, and faces, which causes
  meshing algorithms to generate an excessive
  number of triangles (tetrahedra) near the
  boundary of the polygonal regions (resp.
  polyhedra).
• In our work, we solve the above problem using
  some of the CAGD tools we saw before.

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Thanks! For more challenges, come and
talk to me.