I find the distinction between algebraic and geometric approaches somewhat overstated and misleading. Both these representations are in fact algebraic representations. At any rate, past algebraic approaches to the intersection of quadric surfaces have made use of general implicit algebraic representations of the two surfaces. As the author notes, these methods require the solution of fourth-degree equations and may not be numerically robust. In contrast, the author's approach is to develop a special-case solution for each of the possible pairs of natural quadrics.
Performance is significantly better than that of the general algebraic method.
The material in this paper is well presented and should be of value to implementors. The Digital Library is published by the Association for Computing Machinery.