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# Date Archive: 2009 January

## Splines Come to Mathematica

One of the areas I contributed to Mathematica 7 was support for splines. The word “spline” originated from the term used by ship builders referring to thin wood pieces. Over the last 40 years, splines have become very popular in computer graphics, computer animation and computer-aided design fields. From containers for household goods to state-of-the-art airplanes, it is hard to find any industrial product without spline surfaces. Also, they are widely used in other mathematical studies, such as interpolation and approximation. Through its integration of numerics, symbolics and graphics, Mathematica has the opportunity to go much further with splines than has ever been possible before. Mathematica has had basic spline packages for a long time. But in Mathematica 7 we decided to make highly general spline support a core feature of the system. Splines give another way to represent classes of functions. For decades, mathematicians had been using polynomials for numerical analysis. In the early 20th century, with advances in approximation theory, spline functions were beginning to emerge. The basic idea is simple. In essence, they consist of piecewise polynomials with local supports. Since Version 5.1, Mathematica has offered general support for piecewise functions, both numerically and symbolically. In Mathematica 7, the B-spline functions can be expanded using PiecewiseExpand. For example, a uniform cubic B-spline basis function can be expanded to the following.
Announcements & Events

## Quick-Starting Mathematica with Palettes

I have taught collegiate mathematics for more than 20 years and have used Mathematica for 15 or so of these years to explore, learn and teach. For the last eight years Mathematica has been my primary tool to write all of my exams, handouts, letters, reports, papers, presentations and even a complete electronic textbook. New features introduced recently have been revolutionary in the teaching and learning environment and make possible the creation of materials that integrate text, typeset mathematics and interactive figures, which can be created efficiently and used effectively in ways not possible with other software tools. For faculty and students to benefit from using Mathematica in the teaching and learning process, they must be able to use Mathematica sufficiently well to remain focused on course concepts and not become frustrated by the technology. Without question, the main challenge I face teaching new users how to use Mathematica is helping them master the task of creating syntactically correct commands, followed closely by the challenge of teaching how to use Mathematica to write rich documents that combine text, typeset mathematics and figures. When the use of technology gets in the way of the teaching, learning and writing about content, which should remain the focus of academic learning, then all involved in the teaching and learning process experience frustration! If enough example commands are provided, if the ways of Mathematica are carefully explained, and if patient help is readily available, then some new users are able to work their way up the learning curve and reach a point where they can focus on the subject matter and are able to comfortably use Mathematica to explore, learn, teach and write about the concepts. Members of this group are often able to independently deepen their understanding and use of Mathematica by relying on the Wolfram Mathematica Documentation Center and other resources; but not enough new users reach this level of Mathematica knowledge and thus do not experience firsthand the marvelous capabilities of Mathematica to explore, investigate, learn, teach and write about interesting ideas!
Computation & Analysis

## Visualizing Weather Patterns in Mathematica 7

Weather visualizations are very interesting—there are television channels that thrive by showing nothing else. Online, there are several sources for specific maps of current weather conditions. Generally these are produced and maintained by government agencies or other large organizations. But with Mathematica 7, you can easily produce completely customizable weather visualizations on your own computer. As usual, this is made possible by Mathematica’s tight integration of several areas of functionality. Two new features that enable this particular application are powerful new vector visualization functions and built-in weather data. Vector visualization has been present in Mathematica since Version 2. In Mathematica 7 it has been dramatically improved, adding modern techniques in vector data visualization and new algorithms developed at Wolfram Research. Traditional arrow-based vector plots, new methods based on automatic streamline placement, support for vector glyphs and high-resolution images produced using line integral convolutions are all now supported.
Computation & Analysis