Mar 29, 2002
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- Introduction to splines
	- piecewise polynomials that are 
	  "as smooth as possible"

- Applications and contexts
	- draftsmen in 1800s (plastic and wood strips)
	- free-form curves in Microsoft Word
	- surface splines (e.g., automobile surfaces)
	- computer animation (in-betweening)
	- aesthetics (e.g., reversing a car)

- Advantages of being piecewise
	- allows you to be local in your modeling
	- an interpolating polynomial oscillates widely
	  as degree increases
	  - recall f(x) = max(0,1-x)

- Advantages of being a polynomial
	- have continuous derivatives of all orders!

- Examples of piecewise polynomials
	- piecewise constant
	- piecewise linear
	- piecewise quadratic
	- piecewise cubic

- Types of continuity
	- values only: C^0
	- values and first derivatives: C^1
	- " + second derivatives: C^2

- Definition of spline
	- piecewise polynomial of degree n
	- joined together at break points (called knots)
	- with (n-1) continuous derivatives

- Why only (n-1)?

- Usage of word spline varies widely from books
	- your book: first two conditions above are enough
	- other books: a C^2 piecewise cubic (very specific)
		- because human eye can't notice jumps
		  in third derivative
	- we will use the above definition

- Two types of splines
	- interpolating spline
	- approximating spline

- Evaluating piecewise polynomials
	- linear search through entries
	- binary search
	- shifted power form (around left knot) is advantageous

- Finding splines: piecewise linear
	- 2k unknowns (coefficients)
	- 2k conditions

- Finding splines: piecewise quadratic
	- 3k coefficients
	- 2k + k - 1 = 3k-1 conditions
	  (impose one more condition yourself,
	   e.g., S'(t_0) = 0)

- Started on Problem 7.1.7 from textbook

Apr 1, 2002
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- Revisiting the three conditons for splines
	- why are they necessary?

- Completed Problem 7.1.7

- Cubic splines
	- two missing conditions here
	- many ways of cooking these up

- Natural cubic splines: one specific way of cooking up
	- S''(t_0) = S''(t_n) = 0

Apr 3, 2002
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- Algorithm for natural cubic splines

- Tridiagonal system solution
	- formulating the system around second derivatives
	- solving it
	- finding piecewise polynomials from previous

- Parametric splines

Apr 5, 2002
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- More working out problems from book
	- Example 2
	- Problem 7.2.32
	- Problem 7.2.26

- B-Splines
	- need for a basis

Apr 8, 2002
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- Introduction to B-Splines
	- B^0 splines
	- B^1 splines

- Finding coefficients for B^0 and B^1 splines
	- just read off the values!

- Recursive definition of B^1 splines
	- linear combination of two B^0 splines

- Extending the analogy

- Curves and surfaces using B-splines