This four-hour session will discuss the basics of splines and their most common and useful representation scheme, the B-spline series.
The first two hours will cover the basic theory, including polynomials, piecewise polynomials, continuity, breakpoints, knots, knot multiplicity, B-splines, and recurrence relations. In addition, some of the basic approximation theory for this interesting class of functions will be discussed, including positivity, unisolvency, and sufficient conditions for shape control.
The second two hours will focus on applications and algorithms for applications, including finite element methods, contouring, root finding, and surface intersections, all with a view toward providing the basic spline technology for a solid modeling system. Mass property calculations will be discussed if time permits
Tom Grandine is a specialist in geometric modeling and numerical analysis in the Mathematics and Engineering Analysis group in Boeing's Phantom Works. His areas of expertise include curve and surface modeling, numerical approximation, splines, and multidisciplinary design optimization. He has extensive experience in computational methods for both design and manufacturing applications. Tom has a Ph.D. and M.Sc. degree in Computer Science and a Master of Arts degree in Mathematics from the University of Wisconsin.
Tom Hogan is a mathematician at The Boeing Company. His background is in approximation theory and numerical analysis. He has an MS in Computer Science and a PhD in Mathematics from the University of Wisconsin-Madison. He has been Assistant Professor of Mathematics at University of Alberta, Vanderbilt University and Ohio University where his research focused primarily on wavelets, refinable functions and shift-invariant spaces. Tom's current position with The Math Group at Boeing is in geometric modeling---developing algorithms for curve and surface approximation, shape-preserving interpolation and splines that are especially well-suited for multidisciplinary design optimization. Tom is also Editorial Assistant for Journal of Approximation Theory.