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When dealing with the multiplication of three-dimensional vectors, there are actually two separate operations that can be performed. The first of these yields a scalar (nonvector) value, and is called the dot product. To compute the dot product, you multiply the x of the first vector by the x of the second one, and then add that to the product of the two y values and the product of the two z values. To see how this works, we ll simply show you our JavaScript implementation in Listing 10-3, which takes two arguments, each of which is assumed to be a three-element array representing a vector. Listing 10-3. Function for Calculating a Dot Product function dotProduct(a, b) { return (a[0] * b[0]) + (a[1] * b[1]) + (a[2] * b[2]); } The other type of vector multiplication returns another vector as its result. This is called the cross product, and the resulting vector has the geometric property of being perpendicular to the two initial vectors. Our function for calculating the cross product is shown in Listing 10-4. Listing 10-4. Function for Calculating the Cross Product function crossProduct(a, b) { return [(a[1] * b[2]) - (a[2] * b[1]), (a[2] * b[0]) - (a[0] * b[2]), (a[0] * b[1]) - (a[1] * b[0])]; } Now we can assemble a final solver for a given angle, as in Listing 10-5. Listing 10-5. Function for the Angle Between Three Points on a Sphere function spherePointAngle(A, B, C) { // returns angle at B return Math.atan2(dotProduct(crossProduct(C, B), A), dotProduct(crossProduct(B, A), crossProduct(B, C))); } And now we have all the pieces to solve for the area of a spherical triangle. Before we get to work on that, though, there s an important thing you should know.

o far, we have focused on individual aspects of Flex and Spring by defining many integration points involving security, ORM solutions, project definition, and database connectivity. Much of programming reflects one of my favorite analogies: crawl, walk, then run. You learn the individual pieces, then begin to put them together, and in the end, you have a complete software solution. What better way to learn the complete solution than to build one from scratch The remainder of this book will involve taking what you have learned and building an enterprise Flex application on Spring. I will be sharing what went into building the AF Client Manager application in detail, so you will have a complete understanding of large-scale RIA development. I know that if you wanted to learn a single platform or single language, you would not be reading this book. You are reading it to find out how to integrate Flex with Spring and its solid architecture. For Flex, we will use Cairngorm as the developer framework to deliver the user interface. In Spring, a DAO will be integrated with a service implementation. In 2, we defined the project work breakdown structure. This chapter will define the three projects that we need to support Flex and Spring development. I will describe everything you need to develop the application, including the SWCs, JARs, and RIA artifacts. RIA artifacts are located in a common repository where any Flex application modules such as reusable components, local images, local shared CSS, and compiled modules can access them. To get started, we will refine the project s business needs and how to deliver those needs with Flex and Spring.

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After reading this chapter, you should be able to Understand how users are represented internally in Drupal Understand how to store information associated with a user in several ways Hook into the user registration process to obtain more information from a registering user Hook into the user login process to run your own code at user login time Understand how the two different kinds of external user authentication work Implement your own external authentication module

As it turns out, the triangle formula that we showed at the beginning of this is actually just the n=3 case of a general formula for shapes traced on spheres:

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