1. Splines
By: Marina Uchenik

2. History and Dilemma
Early designers of ship hulls or air crafts were faced with the problem of how to create curvatures
They began by bending large strips of wood or metal and tracing them with chalk
Then facing another problem if they needed one part to be modeled in one direction while the other part needed to be in a different direction

3. Solution
Isaac Schoenberg introduced the idea of using mathematical splines
He found that for small deflections, the shape of the physical spline was a piece-wise cubic polynomial.
The development of parametric spaces arose

4. What is a Spline?
An approximated curve through previously defined points
Many cubic functions pieced together to create a free-form shape

5. How do we approximate one?
Cubic Splines
Given n + 1 data points P0 , P1 , ..., Pn we want to construct a curve which passes through them.
Gobal control: moving one of the control points affects the entire curve.
Do not require geometric constraints, such as tangent directions or control points, it can be derived as a set of scalar functions Si(x).

6. Approximation Cont. (1) B-Splines
Most basic method which uses cubic splines.
Only approximate the position of a set of points.
Local control: moving one data point only changes a portion of the curve.
Useful in geometric design and modeling.

7. Approximation Cont. (2) Hermite Curve
Defined by two end points P1 and P2 and their respective tangent directions T1 and T2.

8. Approximation Cont. (3) ∑ Bézier Curves
The Bézier curve is given by the parametric function:
Bézier Curves
Defined for any polynomial of degree n, given n+1 control points P1 , P2 , ..., Pn n ∑
B(t) =
Bn,k (t) Pk
k = 0
where the blending functions Bn,k are the Bernstein polynomials defined by:
Bn,k(t) = (
) tk (i -t)(n-k) n k

9. Approximation Cont. (4) Non-Uniform B-Splines Uniform B-Splines
Uses parametric cubic polynomials on a uniform knot sequence of successive integers
Non-Uniform B-Splines
Most widely used
Involves the use of irregularly spaced knot sequences

10. Who uses them? CAD Any type of computer aided design
Their initial problem was to be able to completely define a shape
Before splines, designers could only use existing shapes pushed together to create other shapes; but there are only so many shapes that could be constructed by using cubes, cylinders, and cones

11. Who else uses them? Animated and 3D films
Simulated surgery and other educational programs
Analysis of particle movement
Paths of projectiles

12. How do they make my life better?
Aerodynamic sports cars
Traveling safely in a perfectly designed plane
Avoiding frog disections