In general, GA and its variants like PGA are good at calculating tangents and offsets for points, lines, planes, spheres, and other parametric surfaces. The key benefit is that the exact same algorithm will work in 2D, 3D, 4D, or any higher number of dimensions. This could in principle allow very elegant formulas for tasks such as "find the biggest thing that won't hit this other thing at any point in time" or the same thing but with a 0.1 mm gap. Similarly, it's possible to reuse the same code and "human intuition" in very high dimensional spaces, such phase space used in more complex motion planning or engineering simulation scenarios.
I actually have that book in my "to be read" pile, underneath some math texts which I believe will make it accessible to me --- will have to get to studying....
In general, GA and its variants like PGA are good at calculating tangents and offsets for points, lines, planes, spheres, and other parametric surfaces. The key benefit is that the exact same algorithm will work in 2D, 3D, 4D, or any higher number of dimensions. This could in principle allow very elegant formulas for tasks such as "find the biggest thing that won't hit this other thing at any point in time" or the same thing but with a 0.1 mm gap. Similarly, it's possible to reuse the same code and "human intuition" in very high dimensional spaces, such phase space used in more complex motion planning or engineering simulation scenarios.
Even just the pictures on the cover might give you an idea: https://www.amazon.com.au/Projective-Geometric-Algebra-Illum...