*Here is the summary of section 3 of Chapter 6; Robotics and Automatic Geometric Theorem Proving of the book {Ideals, Varieties, and Algorithms} By David A. Cox , John Little and Donal O’Shea *

The problem is:

Given c \in \mathcal{C} , can we determine one or all the j \in \mathcal{J} such that f(j) = c ?

Indeed we wish to determine j whether it is possible to place the hand of the robot at that point with that orientation. And if it is possible, we wish to find all combinations of joint settings that will accomplish this.

In other words, we want to determine the **image** of the mapping f: \mathcal{J} \to \mathcal{C} for this robot; for each c in the image of f , we want to determine the ** inverse image** f^{-1}(c) .