Curves and Surfaces

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Curves and Surfaces
CS 230
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Curves

• Curves are one dimensional entities where the
  function is nonlinear

Curves
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Curves

• Modeling computer graphics element with curves
  requires specialized handles that manage the look
  of a curve in an intuitive way
• In addition, seaming curves to build models is
  also important, for example changing complexity
  in the middle of a curve (Maya makes it trivial)

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Representations for curves

• Parametric representation
• x x(u), y y(u) or simply p(u) where p
• More robust and general than other forms
• Gives better control over curves and surfaces
• Notation
• p(u) c0 c1u c2u2
• where each c is a vector, ci

x y
cix ciy
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Representations for curves

• Example, quadric parametric curve
• p(u) c0 c1u c2u2
• Like curve
• x 3u2
• y 2u 3
• for u -1,1
• c0 c1 c2

0 3
0 2
3 0
Note, more coefficients than a quadratic
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Representations for curves

• Parametric curve, p(u) can as easily represent a
  curve in 3D (x,y,z)
• Simply
• x fx(u)
• y fy(u)
• z fz(u)
• Quadric coefs become
• c0 c1 c2

cox coy coz
c2x c2y c2z
c1x c1y c1z
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Parametric Cubic (pc) curves

• Extension of quadrics to cubics
• Represented as
• Also called Hermite curves after
• the 17th century mathematician

p(u) c0 c1u c2u2 c3u3
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Parametric Cubic (pc) curves

• Algebraic form
• (Note,unless otherwise specified u goes
  from 0,1)
• Not very intuitive, 12 values of c
• Instead want a better specify the curve
• More intuitive control by start point p(0) and
  ending point p(1) and their derivatives

p(u) c0 c1u c2u2 c3u3
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Parametric Cubic (pc) curves

• From
• Build a geometric form in order to specify a
  curves by their end points and tangents

p(u) c0 c1u c2u2 c3u3
p(1)
p(1)
p(0)
p(0)
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Parametric Cubic curves

• This leads to the Geometric Form
• where
• F1(u) 2u3 3u2 1
• F2(u) -2u3 3u2
• F3(u) u3 2u2 u
• F4(u) u3 u2

p(u) F1(u)p(0) F2(u)p(1) F3(u)p(0)
F4(u)p(1)
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Hermite curves basis

• F curves

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Joining Multiple Segments
use p p3 p4 p5 p6T
use p p0 p1 p2 p3T
Get continuity at join points but not continuity
of derivatives C0, C1, C2 continuity
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Bezier Curves

• Family of curves developed in the 1970's by
  Bezier, a engineer for Renault, car manufacturer
• Bezier's curves are only guaranteed to pass
  through the end points, but other control points
  controlled the derivative at the end points
• Specifically, the tangent was controlled by the
  next control point in, the 2nd derivative by the
  second control point in, and the nth derivative
  by the nth and so on...

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Bezier Curves
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Bezier Curves

• The general form of the Bezier curve is
• Vertices p control the curve and blending
  functions, fi(u), that satisfy the "derivative"
  condition
• Bernstein polynomials were a family of functions
  that were chosen by Bezier to satisfy his needs,
  these are not the only functions that could be
  used though

n
p(u) S pi fi(u) 0 lt u lt 1
i0
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Convex Hull Property

• The properties of the Bernstein polynomials
  ensure that all Bezier curves lie in the convex
  hull of their control points
• Hence, even though we do not reach all the input
  data, we cannot be too far away

p1
p2
convex hull
Bezier curve
p3
p0
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Bernstein Polynomials

• The blending functions are a special case of the
  Bernstein polynomials
• These polynomials give the blending polynomials
  for any degree Bezier form
• All zeros at 0 and 1
• For any degree they all sum to 1
• They are all between 0 and 1 inside (0,1)

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Bezier Matrix

• For a curve where n 3, p is defined as
• p(u) (1-u)2p0 2u(1-u)p1 u2p2

p(u) uTMBp b(u)Tp
blending functions
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Bezier Curves

• p(u) (1-u)2p0 2u(1-u)p1 u2p2

p2
p1
p0
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n 3
n 5
n 4
n 6
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Bezier Curves
Bezier curves have intuitive control, are nicely
formed, convex hull of all points
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B Splines

• One problem with the curves we have looked at is
  that changing a single control point affects the
  whole curve (this is called global propogation,)
  
• Also, depends on of control pts
• B-Splines offer an alternative, to only affect
  the local region if a single control point is
  modified (i.e. local propogation)

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B Splines

• B-splines are also called Basis Splines
• They have a form similar to Bezier, B-splines are
  defined as
• where n 1 is the number of control points and
  k controls the degree of the blending (or basis)
  functions

n
p(u) S pi fi,k(u) 0 lt u lt n 2 - k
i0
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B Splines

• B-splines' blending functions are defined
  recursively
• fi,1(u) 1 if ti lt u lt ti1
• 0 otherwise
• and
• fi,m(u)
• m goes from 2 to k
• ti's are knot points relating u to control
  points, pi

(u - ti) fi,m-1(u)
(u - ti m) fi1,m-1(u)
_
tim-1 - ti
tim - ti1
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B Splines

• Knot points,ti's, follow along like this
• ti 0 if i lt k
• ti i - k 1 if k lt i lt n
• ti n- k 2 if i gt n

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B Splines

• For 6 control pts k 1,
• n 5 and 0 lt u lt 6
• We get the degenerate case
• p(u) p0 0 lt u lt 1
• p(u) p1 1 lt u lt 2
• p(u) p2 2 lt u lt 3
• p(u) p3 3 lt u lt 4
• p(u) p4 4 lt u lt 5
• p(u) p5 5 lt u lt 6

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B Splines
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B Splines

• For 6 control pts k 2,
• n 5 and 0 lt u lt 5
• We get a linear average of neighbors
• p(u) (1 - u)p0 u p1 0 lt u lt 1
• p(u) (2 - u)p1 (u - 1)p2 1 lt u lt 2
• p(u) (3 - u)p2 (u - 2)p3 2 lt u lt 3
• p(u) (4 - u)p3 (u - 3)p4 3 lt u lt 4
• p(u) (5 - u)p4 (u - 4)p5 4 lt u lt 5

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B Splines
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B Splines

• For 6 control pts k 3,
• n 5 and 0 lt u lt 4
• We get
• for 0 lt u lt 1
• p1(u) (1 - u)2p0 .5u(4 - 3u) p1 .5
  u2p2
• for 1 lt u lt 2
• p2(u) .5(2 - u)2p1 .5(-2u2 6u - 3) p2
  .5(u - 1)2p3
• for 2 lt u lt 3
• p3(u) .5(3 - u)2p2 .5(-2u2 10u - 11) p3
  .5(u - 2)2p4
• for 3 lt u lt 4
• p4(u) .5(4 - u)2p3 .5(-3u2 20u - 32) p4
  (u - 3)2p5

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B Splines

• Note, influence grows with degree

k 2 k 3 k 4
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B Splines

• Thus, local influence of control points on
• curve as

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Cubic B-spline
p(u) uTMSp b(u)Tp
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Curves vs. Surfaces

• Curves are one dimensional entities where the
  function is nonlinear
• Surfaces are formed from two-dimensional
  functions
• Linear functions give planes and polygons

y
x
Curves
z
Surfaces
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Curves vs. Surfaces

• Parametric curve, p(u) can as easily represent a
  curve in 3D (x,y,z)
• x fx(u)
• y fy(u)
• z fz(u)

u 1
u -1
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Curves vs. Surfaces

• Parametric surface is very similar
• p(u,v)
• Simply
• x f(u, v)
• y f(u, v)
• z f(u, v)

(Just, more difficult to comprehend in a 2D
representation, we must go behind the image
plane)
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Parametric Surfaces

• Surfaces require 2 parameters
• xx(u,v)
• yy(u,v)
• zz(u,v)
• p(u,v) x(u,v), y(u,v), z(u,v)T
• Want same properties as curves
• Smoothness
• Differentiability
• Ease of evaluation

y
p(u,1)
p(0,v)
p(1,v)
x
p(u,0)
z
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Bezier Patches
Using same data array Ppij as with
interpolating form
Patch lies in convex hull
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B-Spline Patches
defined region
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Rendering Curves

• How do we draw the curve, given p(u)?
• void evaluateCurve(u,pixelX,pixelY)

p(1)
p(1)
tangent
tangent
p(0)
p(0)
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Rendering Curves

• Polyline approximation

p(1)
piecewise linear discrete approximation
p(0)
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Splitting a Cubic Bezier
p0, p1 , p2 , p3 determine a cubic Bezier
polynomial and its convex hull
Consider left half l(u) and right half r(u)
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l(u) and r(u)
Since l(u) and r(u) are Bezier curves, we should
be able to find two sets of control points l0,
l1, l2, l3 and r0, r1, r2, r3 that determine
them
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Efficient Form
l0 p0 r3 p3 l1 ½(p0 p1) r1 ½(p2
p3) l2 ½(l1 ½( p1 p2)) r1 ½(r2 ½( p1
p2)) l3 r0 ½(l2 r1)
Requires only shifts and adds!
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Surfaces

• Can apply the recursive method to surfaces if we
  recall that for a Bezier patch curves of constant
  u (or v) are Bezier curves in u (or v)
• First subdivide in u
• Process creates new points
• Some of the original points are discarded

original and discarded
original and kept
new
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Second Subdivision
16 final points for 1 of 4 patches created
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Normals

• We can differentiate with respect to u and v to
  obtain the normal at any point p

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Utah Teapot

• Most famous data set in computer graphics
• Widely available as a list of 306 3D vertices and
  the indices that define 32 Bezier patches