In some ray casting algorithm I want to integrate the following integral:
$\int_{0}^{Z} \frac{c^x}{\|(\vec{o}+x \vec{d})-\vec{l}\|^3} dx$
- $Z$ is a constant until which I want to integrate
- $\vec{o}$ is the origin of the ray.
- $\vec{d}$ is the direction of the ray.
- $\vec{o} + x \vec{d}$ is the position of the ray at time $x$.
- $\vec{l}$ is the position of a light.
- $\|(\vec{o}+x \vec{d})-\vec{l}\|$ is the distance between the current position and the light.
- $c^x$ is a weighting factor which I want to apply. When I move further away from the camera by increasing x, I want the weight to decrease ($c$ < 1).
- $c$ is a constant value (like 0.9).
Solving it via a Riemann sum is not efficient enough on the Xbox. Therefore, I want to solve it analytically. However, as there is a norm of vectors involved together with some other ugly parts, I don't really know where to start solving this.
- Is it possible to solve this integral analytically? (If yes, how?)
- Is there a way to approximate this integral by using non-uniform step-sizes using some Riemann-style integration to increase efficiency of my current implementation?
- Maybe some other solutions? It doesn't have to be exact; as long as it is not instable for small changes to $Z$ and it somehow approximates the integral, it should be fine since it is used in a game I am working on (where importance of performance is bigger than importance of correctness).