[Computer Animation] Natural Cubic Spline

Natural Cubic Spline

Natural Cubic Splines

• $C^{n-1}$-continuity can be achieved from splines of degree $n$.
• Cubic Splines can have $C^{2}$-continuity.

 

 

• We have $4n$ unknowns.
  • $n$ cubic segments (4 coefficients for each segment)

 

• We have $4n$ unknowns
  • $n$ cubic segments (4 coefficients for each segment)
• We have $(4n-2)$ equations
  • $2n$ equations for end point interpolation
  • $(n-1)$ equations for tangential continuity
  • $(n-1)$ equations for second derivative continuity

 

• Two more equations are required!
  • Natural Spline Boundary Condition
  • Closed Boundary Condition

Natural Spline Boundary Condition	Closed Boundary Condition

 

$4n$ Equations

 

• We have $4n$ unknowns and $4n$ equations.
• We can solve the linear system.

 

Natural Cubic Spline Properties

• Variation Diminishing
  • the curve in 2D space does not oscillate about any straight line more often than the control point polygon.
• $C^{2}$-continuity
• No local controllability

 

Question

• Find the natural cubic spline interpolating the four key points.
• $p(t) = L_{0}(t)p_{0} + L_{1}(t)p_{1} + L_{2}(t)p_{2} + L_{3}(t)p_{3}$
  • $p_{0} = (0, 0)$
  • $p_{1} = (10, 10)$
  • $p_{2} = (15, 5)$
  • $p_{3} = (20, 15)$