CS 430/585 Computer Graphics I 3D Surfaces Week 9, Lecture 18

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CS 430/585Computer Graphics I3D Surfaces Week
9, Lecture 18

• David Breen, William Regli and Maxim Peysakhov
• Geometric and Intelligent Computing Laboratory
• Department of Computer Science
• Drexel University
• http//gicl.cs.drexel.edu

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Overview

• Quadratic and super-quadratic surfaces
• Bicubic surfaces
• Bezier surfaces
• Normals to surfaces
• Surface rendering
• Blobby objects

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Quadratic Surfaces

• Sphere
• Ellipsoid
• Torus
• General form

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Superellipsoid Surfaces

• Obtain from ordinary ellipsoids
• Control parameters s1 and s2
• If s1 s2 1 then regular ellipsoids
• Has an implicit and parametric form

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CSG with Superquadrics
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CSG with Superellipsoids
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Subdivision Surfaces

• Coarse Mesh Subdivision Rule
• Define smooth surface as limit of sequence of
  refinements

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Biparametric Surfaces

• Biparametric surfaces
• A generalization of parametric curves
• 2 parameters s, t (or u, v)
• Two parametric functions

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Bicubic Surfaces

• Recall the 2D curve
• G Geometry Matrix
• M Basis Matrix
• S Polynomial Terms s3 s2 s 1
• For 3D, we allow the points in G to vary in 3D
  along t as well

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Observations About Bicubic Surfaces

• For a fixed t1, is a curve
• Gradually incrementing t1 to t2, we get a new
  curve
• The combination of these curves is a surface
• are 3D curves

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Bicubic Surfaces

• Each is , where
• Transposing , we get

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Bicubic Surfaces

• Substituting into ,
  we get Q(s, t)
• The g11, etc. are the control points for the
  Bicubic surface patch

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Bicubic Surfaces

• Writing outgives

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Plotting Isolines
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Plotting Isolines
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Bézier Surfaces

• Bézier Surfaces(similar definition)

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Bézier Surfaces

• C0 and G0 continuity can be achieved between two
  patches by setting the 4 boundary control points
  to be equal
• G1 continuity achieved when cross-wise CPs are
  co-linear

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Bézier Surfaces Example

• Utah Teapot modeled by 32 Bézier Patches with G1
  continuity

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Faceting
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Faceting
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B-spline Surfaces

• Representation for B-spline patches
• C2 continuity across boundaries is automatic with
  B-splines

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Normals to Surfaces

• Normals used for
• Shading
• Interference detection in robotics
• Calculating offsets for numerically controlled
  machining

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Computing the Normals to Surfaces

• For a bicubic surface, first, compute the s
  tangent vector

t
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Computing the Normals to Surfaces

• Next, compute the t tangent vector

t
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Computing the Normals to Surfaces

• Since s and t are tangent to the surface, their
  cross product is the normal vector to the
  surface!
• xs - x component of s tangent
• ys - y component of s tangent
• zs - z component of s tangent

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NURBS Surfaces

• Similar to B-spline patches

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Drawing Parametric Surfaces

• Usually done patch by patch
• Two choices
• Draw/render directly from the parametric
  description
• Approximate the surface with a polygon mesh, then
  draw/render the mesh

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Direct Rendering

• Use a scan-line algorithm
• Evaluate pixel by pixel
• Problem How to go from (x,y) screen space to
  point on the 3D patch
• Easy for a planar polygon where we know max/min
  y, equations for edges, screen depth
• Not as easy for parametric surfaces

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Issues for Direct Rendering

• Max/Min y coords may not lie on boundaries
• Silhouette edges result from patch bulges
• Need to track both silhouettes and boundaries
• What if they intersect?
• Note patch edges need not be monotonic in x or y
• Idea Scan convert patch plane-by-plane, using
  scan planes instead of scan lines

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Direct Scan Conversion of Patches

• Basic idea
• Find intersection of patch with XZ plane
• Producing a planar curve
• Draw the curve
• De Boor, DCasteljeau
• Note if doing rendering, one can compute
  pixel-by-pixel color values this way

Patch xX(u,v), yY(u,v), zZ(u,v)
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Direct Scan Conversion of Patches Algorithm
Outline

• Patch xX(u,v), yY(u,v), zZ(u,v)
• u,v range from 0 to 1 this defines 4 bounds
• Intersection of scan line Ys with boundary

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Patch to Polygon Conversion

• Two methods
• Object Space Conversion
• Techniques
• Uniform subdivision
• Non-uniform subdivision
• Resolution depends on object space
• Image Space Conversion
• Resolution depends on pixels and screen

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Object Space Conversion Uniform Subdivision

• Basic Procedure
• Cut parameter space into equal parts
• Find new points on the surface
• Recurse/Repeat until done
• Split squares into triangles
• Render

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Object Space Conversion Non-Uniform Subdivision

• Basic idea
• Facet more in areas of high curvature
• Use change in normals to surface to assess
  curvature
• More derivatives
• Break patch into sub-patches based on curvature
  changes

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Image Space Conversion

• Idea control subdivision based on screen
  criteria
• Minimum pixel area
• Stop when patch is basically one pixel
• Screen flatness
• Stop when patch converges to a polygon
• Screen flatness of silhouette edges
• Stop when edge is straight or size of pixel

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How do I know if Ive found a silhouette edge?

• If the ray that Im casting is tangent to the
  surface at the point it hits the surface!
• N L 0
• Where N is the normal at the point where L, the
  line of sight, hits the surface

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Blobby Objects

• Do not maintain shape, topology
• Water drops
• Molecules
• Force fields
• But can maintain other properties, like volume

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Gaussian Bumps

• Model object as a sum of Gaussian bumps/blobs
• Where and T is a
  threshold.

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Metaballs (Blinn Blobbies)
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Ray-traced Metaballs