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Ultimately, what you want is to generate a second set of texture coordinates that indexes into the right place in the environment map based on the eye position and on the surface normal. If the surface were a perfect mirror, what would you see To find out, you could cast a ray from the eye to the surface and then bounce it off the surface taking the normal into account. This new vector is called a reflection vector (see Figure 11-5).

(5.35)

As the illustration shows, network access is exactly that: the collection of technologies that support connectivity between the customer and the transport resources of the network. At its most common level, access is the local loop the (typically) two-wire circuit that connects a customer s telephone to the local switch, which provides telephony service to that customer. As the network has become more data-aware, other solutions have emerged that provide greater bandwidth as well as multiservice capability. ISDN, which uses the two-wire local loop, provides greater bandwidth than the traditional analog local loop through digitization and time-division multiplexing (both explained shortly). Digital subscriber line (DSL) is also a local loop-based service, but offers even more diverse service than ISDN in the areas where it is available. Cable

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The deformation of path theorem (see Complex Variables Demysti ed ) tells us that we can shrink or expand the contour used for the integration in Eq. (5.35) and leave the integral invariant. Since movement along the radial distance in the complex plane corresponds to time translation, this tells us that the Virasoro operators are invariant under a time translation, in other words the integrals [Eq. (5.35)] are related to conserved charges.

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Mathematically, a reflection vector is computed with the following formula: E is the unit eye vector, and N is the unit normal vector. The incident vector is the vector from the eye position to the point on the surface of the model. Because the incident vector changes for each point on the mesh, it needs to be recomputed for each vertex. Also, since the reflection vector depends on the eye position, it will change as the camera moves around the object (or the object changes position relative to the camera), so you need to recompute it every frame. Once you have this reflection vector, you can apply a formula to index into your environment map. This is the standard formula for spherical environment mapping:

1

To gain insight into the physics of the problem, we will begin by calculating propagators for the theory. In the text we will derive the closed string propagator, you can try the open string case in the chapter quiz. We begin by considering the propagator or Wick expansion for X ( , ) for the closed string. This is done by calculating X ( , ), X ( , ) = T ( X ( , ), X ( , )) : X ( , ) X ( , ) : This is also called the two-point function. There are two notations you should be aware of in this expression. The T indicates that the expression is time ordered. It is shorthand for

That s quite a bit of work to do for each vertex, especially with those square roots on the iPhone CPU. Fortunately, you can make some approximations again that will let you speed things up significantly. The key observation is that you could fake the reflection vector by simply using the normal vector at each vertex. The reflection wouldn t be physically accurate, but it would reflect something, which is good enough. That by itself isn t enough, though: if you use only the normal vector, the reflection wouldn t change as the camera moves around the object or the object changes position. To fix that, you need to rotate the vertex normal by the camera transform. Finally, you discard the z component of the normal and scale and bias the x and y components so they are between 0 and 1. The results are the texture coordinates corresponding to the environment mapping as seen from the correct viewpoint. This technique is sometimes referred to as normal environment mapping, because it uses the normals of the model instead of the true reflection vectors.

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