B-curves and B-surfaces

1
B-curves and B-surfaces

• - parametrically defined
• (parameterization example 0lttlt1)
• - piecewise polynomial
• - rational or non-rational
• (weight wgt0 and default to 1)
• - Non-Uniform Rational B-spline (NURB)

2
B-curves and B-surfaces

• Explicit (B-spline data, splining, lofting etc.,)
• Automatic (created during modeling)
• B-surfaces
• Boolean
• Sweeping and Spinning

3
Surface forms

• Analytic surfaces
• Blends
• B-surfaces
• Offset surfaces
• Foreign Geometry Surfaces

4
Implicit Surfaces

• Solutions to implicit equation
• F(x) 0
• Divide space into two regions
• F(x) gt 0 (positive) and F(x) lt 0 (negative)
• Boundary between two regions
• Sphere (closed), Plane, cylinder (infinite)

5
B-surfaces

• Defined over a finite region
• Bounded
• B-curves are bounded
• Swept surfaces infinite in swept direction
• Spun surfaces closed in the spin direction

6
Creation of B-geometry

• B-spline control points
• Piecewise data ( Bezier, Hermine etc.,)
• Sweeping, Spinning B-curves
• Covert line/circle/ellipse
• Splining a set of points
• Lofting B-curves
• Constant parameter curves (extraction)

7
B-spline-data

• B-spline control points
• Knot vectors
• Continuity conditions
• Rational or non-rational
• Periodic or non-periodic
• Consider U and V directions
• From piecewise data

8
B-spline properties

• Spline function of degree n
• Continuity at the joins (position derivatives)
• (n-1) derivative continuity
• Linear sum of basis functions
• Basis functions bi, weights wi, vertices vi,
  knot vector t, degree n

9
Knot Vectors

• Knots are values
• Ordered vector of knot values (increasing)
• May be coincident (multiplicity of gt 1)
• Knot multiplicity lt degree (Interior)
• Knot multiplicity lt degree1 (first or last)

10
B-geometry

• Bell shaped basis functions (non-zero over the
  span of degree1 intervals)
• N-knots n-vertices degree 1

11
Closed vs. Periodic

• Periodic and closed are both required to have
  positional continuity
• Periodic has additional requirements ( tangent
  continuity)

12
Properties

• Convex hull property (bspline)
• For a given t, only degree1 basis functions are
  non-zero ( local modifications )
• P(t) lies within convex hull of (degree 1)
  adjacent vertices ( figure 24-1)