FreeForm Curves and Surfaces

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Free-Form Curves and Surfaces
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How a scene is created
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Geometric Modeling

• Objective Develop methods and algorithms to
  mathematically model shape of real world objects
• Most virtual scenes are designed using modeling
  tools
• Maya3D, XSI, 3DSMax, Autocad, .
• Need methods for prescribing smooth and
  continuous shapes.
• The most common way today is using parametric
  curves and surfaces
• We will cover mostly curves, for the rest CAGD
  course
• We will cover subdivision curves and surfaces

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Freeform Representation

• Explicit form z z(x, y)
• Implicit form f(x, y, z) 0
• Parametric form S(u, v) x(u, v), y(u, v),
  z(u, v)
• Example origin centered sphere of radius R

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A representation of choice

• What makes a good representation for solid
  modeling?
• Ease/Intuitive shape manipulation
• Class of representable models
• Ease of display and evaluation
• Ease of query answering such as inside/outside,
  subdivision, interference
• Robustness and accuracy
• Closure under operations such as Booleans.

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Splines Free Form Curves

• To allow control of curves need description via
  set of basis functions coefficients
• Require geometric meaning of coefficients (base)
• Approximate/interpolate set of positions,
  derivatives, etc..
• Typically parametric

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Polynomial Bases

• Polynomials are easy to analyze, derivatives
  remain polynomial, etc.
• Monomial basis 1, x, x2, x3,
• Coefficients are geometrically
• meaningless
• Manipulation is not robust
• Number of coefficients
• polynomial rank
• We seek coefficients with geometrically intuitive
  meanings
• Other polynomial bases (with better geometric
  intuition)
• Lagrange (Interpolation scheme)
• Hermite (Interpolation scheme)
• Bezier (Approximation scheme)
• B-Spline (Approximation scheme)

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Mathematical Continuity

• C1(t) C2(t), t ? 0,1 - parametric curves
• Level of continuity of the curves at C1(1) and
  C2(0) is
• C-1C1(1) ? C2(0) (discontinuous)
• C0 C1(1) C2(0) (positional continuity)
• Ck, k gt 0 continuous up to kth derivative
• Continuity of single curve inside its parameter
  domain is similarly defined - for polynomial
  bases it is C?

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Geometric Continuity

• Mathematical continuity is too strong a
  requirement
• May be relaxed to geometric continuity
• Gk, k ? 0 Same as Ck
• Gk, k 1 C'1(1) ? C'2(0)
• Gk, k? 0 There is a reparameterization of C1(t)
  C2(t), where the two are Ck
• E.g.
• C1(t)cos(t),sin(t), t???/2,0
• C2(t)cos(t),sin(t), t?0,?/2
• C3(t)cos(2t),sin(2t), t?0,?/4
• C1(t) C2(t) are C1 ( G1) continuous
• C1(t) C3(t) are G1 continuous (not C1)

C2, C3
C1
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More on Continuity

• Consider a set of points Pii0,n
• A piecewise constant interpolant will be C-1
• A piecewise linear interpolant will be C0
• Polyhedra are usually piecewise linear
  approximation of freeform polynomials
• Polyhedra are C0 continuous
• Higher order polynomials posses better continuity

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Cubic Hermite Basis

• Set of polynomials of degree k is linear vector
  space of degree k1
• The canonical, monomial basis for polynomials is
• 1, x, x2, x3,
• Define geometrically-oriented basis for cubic
  polynomials
• hi,j(t) i, j 0,1, t?0,1
• Has to
• satisfy

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Hermite Cubic Basis (contd)

• The four cubics which satisfy these conditions
  are
• Obtained by solving four linear equations in four
  unknowns for each basis function
• Prove Hermite cubic
• polynomials are linearly
• independent and form a
• basis for cubics

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Hermite Cubic Basis (contd)

• Lets solve for h00(t) as an example.
• h00(t) a x3 b x2 c x d
• must satisfy the following four
• constraints
• Four equations and four unknown to solve for.

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Hermite Cubic Basis (contd)

• Let C(t) be a cubic polynomial defined as the
  linear combination
• What are C(0), C(1), C(0), C(1) ?

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Parametric Splines

• So far we solve for a single explicit function
  C(t)
• Cannot represent closed curves such as a circle
• Why ?
• Solution, let Pi (xi, yi) and Ti (tix, tiy)
  be vector functions
• Solve for each individual
• axis independently
• Here is an example of
• four Hermite cubis that
• approximate a circle

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Cubic Splines

• Standard spline input set of points Pii0, n
• No derivatives specified as input
• Can we prescribe derivatives values in order to
  achieve C2?
• Interpolate by n cubic segments (4n DOF)
• Derive Tii0 , n from C2 continuity constraints
• Solve 4n equations

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Cubic Splines

• Have two degrees of freedom left (to reach 4n
  DOF)
• Options
• Natural end conditions C1''(0) 0, Cn''(1) 0
• Complete end conditions C1'(0) 0, Cn'(1) 0
• Prescribed end conditions (derivatives available
  at the ends)
• C1'(0) T0, Cn'(1) Tn
• Quadratic end conditions
• C1''(0) C1''(1), Cn''(0) Cn''(1),
• Periodic end conditions
• C1'(0) Cn'(1), C1''(0) Cn''(1),
• Question What parts of C(t) are affected as a
  result of a change in Pi ?

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Parameterization

• The assumption t ? 0,1 (uniform
  parameterization) is arbitrary
• Implicitly implies same curve length for each
  segment
• Not natural if points are not equally spaced
• One alternative - chord-length parameterization
• Question Can chord-length parameterization be
  used for cubic Hermite splines?

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Parameterization
Question Why do we need arc/chord length curves
for?
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Bezier Curves

• Bezier curve is an approximation of given control
  points
• Denote by ? (t) t?0,1
• Bezier curve of degree n is defined over n1
  control points Pii0, n
• Have two
• formulations
• Constructive
• Algebraic

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Algebraic Form of Bezier Curves

• Bezier curve for set of control points Pii0, n
  
• where Bin(t)i0, n Bernstein basis of
  polynomial of
• degree n
• Here is a
• Cubic case
• example

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Algebraic Form of Bezier Curves
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Algebraic Form of Bezier Curves

• why?
• Curve is linear combination of basis functions
• Curve is affine combination of control points

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De Casteljau Construction
Select t?0,1 value. Then,
Bezier
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Properties of Bezier Curves

• ? (t) is polynomial of degree n
• ? (t) ? CH(P0,,Pn)
• ? (0) P0 and ? (1) Pn
• ? '(0) n(P1?P0) and ? '(1) n(Pn?Pn-1)
• ? (t) is affine invariant and variation
  diminishing
• ? (t) is intuitive to control via Pi and it
  follows the general shape of the control polygon
• ? '(t) is a Bezier curve of one degree less
• Questions
• What is the shape of Bezier curves whose control
  points lie on one line?
• How many line Bezier segments between two given
  points exist of degree 2? Degree 3? Higher?
• How can one connect two Bezier curves with c0
  continuity? c1 ? c2 ?

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Drawbacks of Bezier Curves

• Degree corresponds to number of control points
• Global support change in one control point
  affects the entire curve
• For large sets of points curve deviates far
  from the points
• Cannot represent conics exactly. Most noticeably
  circles
• Can be resolved by introducing a more powerful
  representation of rational curves.
• As an example, a 90 degrees arc as a rational
  Bezier curve

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B-Spline Curves

• Idea Generate basis where functions are
  continuous across the domains while presenting
  local support
• For each parameter value only a finite set of
  basis functions is none zero
• The parametric domain is subdivided into sections
  at parameter values denoted knots, ?i .
• The B-spline functions are then defined over the
  knots
• The knots are denoted uniform knots if ?i - ?i-1
  c, constant. W.l.o.g.,
• assume c 1.

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Uniform Cubic B-Spline Curves

• Definition (uniform knot sequence, ?i - ?i-1 1)

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Uniform Cubic B-Spline Curves

• For any t ? 3, n
• (prove it!)
• For any t ? 3, n at most four basis functions
  are non zero
• Any point on a cubic B-Spline is an affine
  combination of at most four control points

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Boundary Conditions for Cubic B-Spline Curves

• In general, B-Splines do not interpolate control
  points
• in particular, the uniform cubic B-spline curves
  do not interpolate the end points of the curve.
• Why is the end points interpolation important?
• Two ways are common to force endpoint
  interpolation
• Let P0 P1 P2 (same for other end)
• Add a new control point (same for other end) P-1
  2P0 P1 and a new basis function N-13(t).
• Question
• What is the shape of the curve at the end points
  if the first method is used?
• What is the derivative vector of the curve at the
  end points if the first method is used?

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Local Control of B-spline Curves

• Control point Pi
• affects ? (t) only for
• t?(?i, ?i4)

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Properties of B-Spline Curves

• For n control points, ? (t) is a piecewise
  polynomial of degree 3, defined over t?3, n)
• and for specific
  t t0?
• ?(t) is affine invariant and variation
  diminishing
• ?(t) follows the general shape of the control
  polygon and it is intuitive and ease to control
  its shape
• Questions
• What is ? (?i) equal to?
• What is ? '(?i) equal to?
• What is the continuity of ? (t) ? Prove it!

B-spline
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From Curves to Surfaces

• A curve is expressed as inner product of
  coefficients Pi and basis functions
• One why of extending curves into surfaces is to
  treat surface as a curves of curves. This
  approach is also known as tensor product surfaces
• Assume Pi is not constant, but are functions of a
  second, new parameter v

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From Curves to Surfaces (contd)

• Then
• or
• Question What is the difference between u and v?

BezPatch