Review of Spline Concepts

1
Review of Spline Concepts

• Sections 10-6 to 10-13 in Hearn Baker
• Splines can be 2D or 3D
• Control points
• Interpolation vs. approximation
• Convex hull
• Continuity conditions
• Local vs. global control
• Three methods of specifying splines
• Types of splines
• Cubic
• Bézier
• B-splines
• Other

2
Spline Basics (1/3)

• Definitions
• Textbook calls a spline curve any composite
  curve formed with polynomial sections satisfying
  specified continuity conditions at the boundary
  of the pieces - can be 2D or 3D
• Spline surfaces formed from two sets of
  orthogonal spline curves
• From now on spline refers to spline curve
• Control points
• Used to specify spline curves
• Two ways to fit splines
• Interpolation - curve passes through each control
  point
• Approximation - curve does not necessarily pass
  through any control point
• Convex hull
• minimum convex polygon boundary that encloses set
  of control points

3
Spline Basics (2/3)

• Splines are specified using parametric equations
• P(u) x(u) y(u) z(u) - row vector
• P(u) Au3 Bu2 Cu D - each of A, B, C, and
  D are row vectors
• From now on, we treat P as scalar
• P(u) au3 bu2 cu d
• Unless otherwise specified, all derivatives are
  with respect to parameter u
• Continuity conditions (between two curve
  sections)
• parametric
• C0 - same point
• C1 - tangent lines (1st derivatives) equal
• C2 - 1st and 2nd derivatives equal
• etc
• geometric
• G0 C0
• parametric derivatives proportional (in same
  direction) but not necessarily equal
• G1, G2, etc
• note
• there exists a case where you can have G1
  continuity without C1 continuity
• Local vs. global control

4
Spline Basics (3/3)

• Recall from previous slide
• P(u) au3 bu2 cu d
• We can write as
• P(u) U ? Co U u3 u2 u 1 Co a b c d
  T
• U ? Mspline ? Mgeom
• B ? Mgeom
• U is the 1?4 parameter vector
• Co is the 4?1 coefficient matrix
• Three methods of spline specification
• Boundary conditions
• Description given in English
• Actual values placed in 4?1 geometry vector Mgeom
• Basis matrix
• Mspline is the 4?4 basis matrix
• This is constant for a given type of spline
• Co Mspline ? Mgeom
• Blending functions
• B is a 1?4 matrix representing the blending
  functions
• B U ? Mspline F0(u) F1(u) F2(u) F3(u)

5
Types of Splines

• Why cubic?
• Higher order less stable, more complex to
  calculate
• Lower order dont look as good
• Cubic is smallest that specifies endpoints and
  derivatives, and which is nonplanar in 3D
• Cubic splines (interpolation)
• Natural
• Hermite
• Cardinal
• Kochanek-Bartels
• Bézier (approximation)
• B-splines (approximation)
• Other

6
Review of Cubic Splines

• Features
• Cubic splines interpolate - passes through every
  control point
• We fit one curve section at a time between each
  pair of successive control points (u0 to u1)
• We have 4 coefficients (cubic) so for n curve
  sections need 4n boundary conditions
• Natural
• Boundary conditions endpoints on control
  points, 1st and 2nd parametric derivatives match
  between adjacent curve sections
• Bad exhibits global control
• Hermite
• Boundary conditions endpoints on control
  points, parametric slope specified at control
  points
• Exhibits local control, but may be inconvenient
  to specify slopes
• Cardinal
• Same as Hermite, but we compute tangents from
  neighboring control points
• Has tension parameter t
• Kochanek-Bartels
• Same as cardinal but adds two additional
  parameters, bias and continuity
• Good for modelling abrupt changes

7
Bézier Curves - Features

• Approximation
• Global control
• Number of control points determines degree of
  Bézier curve
• Easy to implement and calculate recursively
• Always passes through first and last control
  points
• Lies within convex hull
• Easy to connect sections together

8
Note from 27 October lecture

• In class we went over how to derive the following
  boundary conditions from the definition of Bézier
  curve
• P(0) p0 P(1) pn
• P'(0) -np0 np1 P'(1) -npn-1 npn
• Dr. Chawla points out that there is a simpler
  (though equivalent) way to view our derivation
• For u 0, we see that every term except the
  first has a u and therefore goes to 0.
• For u 1, we see that every term except the last
  has a (1-u) and therefore goes to 0. So we have
• Taking the parametric derivative, we get