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## Surfaces Comp 575 Fall 2008

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### Trimming Curves. Sometimes we want 'holes' in the surface. Can define them using trimming curves. Define a NURBS curve on the NURBS surface ... – PowerPoint PPT presentation

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Title: Surfaces Comp 575 Fall 2008

1
Surfaces Comp 575 - Fall 2008
2
Parametric Bicubic Surfaces
• An extension to curves by adding another dimension

t 1.0
t 0.8
t 0.6
t 0.4
t 0.2
t 0.0
s 0.6
s 0.8
s 0.0
s 0.2
s 0.4
s 1.0
3
Curve Equation
• Remember, the equation of the curve is
• Or, equivalently
• Where G is the geometry vector, and is a constant

4
The Surface Equations
• If we allow G to be a function, we get
• i.e., the geometry can now change, based on t

5
The Surface Equations
• For a fixed t1, Q(s, t1) is a curve, since G(t1)
is a constant.
• Taking a new t2 that is near in value to t1 will
give another curve which is slightly different
from the first.
• Repeating the process for some number of
parameters t, with 0 t lt 1 will give a group of
curves that define a surface.

6
The Surface Equations
• Each of the Gi(t) functions are cubics, and can
be represented as
• where

7
Surface Equations
• Taking the transpose of this gives us
• If we substitute this back into our original
equation, and expand to include all four geometry
terms, we get

8
Surface Equations
• So,
• With being the geometry matrix and M being
the basis matrix.

9
Bezier Surfaces
• The ends of each patch require 4 control points
in the s direction.
• The t direction gives rise to 4 control points
also
• Thus, there are 4x4 or 16 control points required
for each patch.

10
Bezier Surface Example
11
Joining Bezier Surfaces
• Multiple Bezier surfaces can be joined with C0
continuity by
• Making the four control points at the join common
between the two patches

12
Joining Bezier Surfaces
• To join with C1 continuity we must also enforce
the following stipulations
• The control points on either side of the join
must be collinear
• All of the pairs of line segments joining the
three collinear control points must have lengths
that have the same ratios

13
Joining Bezier Surfaces
• For this example the following ratios must be
equal

14
Displaying Surfaces
• Surfaces are displayed in a manner similar to
curves.
• Iteratively evaluate the surface equation at s
and t intervals, then draw polygons for those
patches
• Subdivide the surface until the patch size is
small enough

15
NURBS Surfaces
• NURBS
• Non Uniform
• - knots can have any spacing desired
• Rational
• the blending functions are the ratios of two
polynomials
• B-Spline
• the surface type is B-spline
• Very flexible and powerful
• Also somewhat complex
• Can represent conics exactly
• Used extensively, particularly in CAD

16
NURBS Examples
http//www.geomagic.com
17
http//gallery.mcneel.com/
18
http//gallery.mcneel.com/
19
http//gallery.mcneel.com/
20
NURBS in OpenGL
• gluNurbsSurface(GLUnurbs nurb,
• Glint sKnotCount,
• GLfloat sKnots,
• Glint tKnotCount,
• GLfloat tKnots,
• Glint sStride,
• Glint tStride,
• GLfloat control,
• Glint sOrder,
• Glint tOrder,
• GLenum type)

21
NURBS in OpenGL
gluNurbsSurface(GLUnurbs nurb, Glint
sKnotCount, GLfloat sKnots, Glint
tKnotCount, GLfloat tKnots,
Glint sStride, Glint
tStride, GLfloat control,
Glint sOrder, Glint
tOrder, GLenum type)
• nurb specifies the NURBS object created with
gluNewNurbsRenderer
• sKnotCount number of knots in the s direction
• sKnots the array of s knots
• tKnotCount number of knots in the t direction
• tKnots the array of t knots
• sStride offset between control points in the s
direction
• tStride offset between control points in the t
direction
• control array of control points
• sOrder order of the NURBS in s
• tOrder order of the NURBS in t
• type type of surface

22
Trimming Curves
• Sometimes we want holes in the surface
• Can define them using trimming curves
• Define a NURBS curve on the NURBS surface
• Draw the surface everywhere except inside the
curve

23
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24
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25
Subdivision Surfaces
• One problem with the surfaces we have discussed
is the difficulty in changing resolution for a
portion of the surface
• If we want more detail at one part of the patch,
we have to introduce a whole new patch
• Subdivision surfaces allow local refinement of
the control mesh
• This gives more flexibility in the objects to be
modeled

26
Subdivision Surfaces
• Idea recursively subdivide the patch to finer
and finer resolution

27
Standard Subdivision
• Given initial control points, recursively
subdivide until desired smoothness is reached

28
Adaptive Subdivision
• Generally some areas of the surface have higher
curvature, and thus should be subdivided further
than other areas
• Can apply an adaptive subdivision scheme to
subdivide more where we want finer control of the
surface

29
Adaptive Subdivision Surface Example
After one refinement
Original Mesh
After two refinements
The limit infinite refinement
from http//grail.cs.washington.edu/projects/subdi
vision/
30
Subdivision Surfaces
Geris Game (1997) Pixar Animation Studios
31
Subdivision Surface Example
http//mrl.nyu.edu/dzorin/sig98course/multires/sl
d005.htm
32
Allowing for Sharp Edges and Creases
• Often we want to permit sharp edges
• How can we smooth some of the surface, but not
all?
• Tag edges as sharp or non-sharp
• If an edge is sharp, apply sharp subdivision
rules
• Otherwise apply normal subdivision rules

33
T-splines
• Introduced by Dr. Sederberg in 2003
• Allow T-junctions in the surfaces

34
T-Splines
35
T-spline Hole Filling
36
T-spline example
37
Summary
• Surfaces allow for a higher level of realism
• Used in all CAD packages
• NURBS surfaces are the most popular type
• Subdivision surfaces allow for increased surface
detail, and for local adaptation
• T-splines allow for even more flexibility
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