Curves

1
Chapter 11

• Curves

2
Modeling

• Define an object or group of objects in terms of
  its form or shape
• surface modeling
• solid modeling
• particle system modeling

3
Surface Model

• No interior information
• Surface only
• Visible Man Fly Thru Bill Lorenson, GE

4
Solid Model

• Interior information is preserved.
• Requires huge amounts of space.

5
Particle-system Model

• Model objects consisting of many of the same
  shape.
• Water, fire, trees, dandelion seeds, bubbles and
  grapes might be modeled by particle systems.

6
Polygonal Modeling
z

• 0 dimension
• point
• 1 dimensions, length (x)
• line
• 2 dimensions, length (x) and height (y)
• plane
• 3 dimensions, length (x), height (y) and width
  (z)
• cube

x
y
7
Polygonal Modeling

• Vertex
• Polygon
• Polygonal Approximation

8
Polygon Reduction (or Polygon Culling or Polygon
Thinning)

• Specifying maximum number of polygons
• Specify minimum angle between polygons

9
Another Global Polygon Adjustments

• Polygon expansion
• the opposite of polygon reduction

10
Local Operations

• Restricting polygon expansion to the areas of
  high detail
• efficient use of polygons

11
Linear Approximation of a Curve

• 2 or 3 dimensions
• linear approximation (polyline)
• simple concept
• awkward to edit
• many points necessary for good approximation of
  curve
• never smooth

12
Parameterized Curves

• Known as spline curves
• have direction
• beginning point
• ending point
• may be non planar

13
Explicit Functions

• y f(x)
• x g(y)
• y mx b
• Problems -- with vertical lines and circles
• y sqrt(r2 - x2)
• y -sqrt(r2 - x2)
• Only if 0ltxltr
• Problems compound with 3D
• z f(x,y) cant work because a given x,y
  generates multiple points on a sphere.

14
Curves

• Developed in the CAD industry
• aviation
• automobiles
• Hermite Curve
• Bezier Curve

15
Splines

• Shipbuilders forced wood splines around ducks
• Curve shaped by
• control points
• control vertices
• Types of Splines used in Computer Graphics
• natural spline
• B-Splines
• NURBs

16
Continuity

• C0 - connectedness
• C1 - smoothness (no tangent breaks)
• C2 - curvature

17
Hermite Curve

• Used for interpolation of keyframe data
• Use
• hermite basis functions
• points p1 and p2 and tangent vectors t1 and t2

t1
p1
p2
t2
18
Bezier Curve

• Developed at Renault by Pierre Bezier
• Pair of endpoints and control points (not on the
  curve)

Control Point
Control Point
Endpoint
Endpoint
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B Spline

• B stands for basis or blending function
• Control points do not interpolate the curve.
• Difficult to edit because control points must be
  moved significantly to see change in curve shape.
• Can only be cut at a knot (between curve segments
  -- difficult to ascertain since control points
  are not on the curve)

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NURBSNon-Uniform Rational B-Spline

• There is no NURB, only NURBS
• a type of B-spline
• each control point can have a weight
• can cut anywhere on the length of the curve

21
Spline Patches

• Original direction of curve, u
• Moved through space along a second curve with
  direction v

22
Homework for next week

• Work Day Wednesday 4/11. No class.
• Read Lasseter, John. Principles of Traditional
  Animation Applied to 3D Computer Animation,
  Proceedings of SIGGraph 1987.