A bicriteria optimal path simultaneously satisfies two bounds on two measures of path quality. The complexity of finding such a path depends on the particular choices of path quality. This thesis studies bicriteria path problems in a geometric setting using several pairs of path quality, including: path length measured according to different norms ($Lp$ and $Lq$); Euclidean length within two or more classes of regions; total turn and Euclidean length; total turn and number of links; and Euclidean length and number of links.

For several cases, finding the bicriteria optimal path is shown to be NP-hard. These NP-hard cases include minimizing path length in two different norms, minimizing travel through two regions, and minimizing length and total turn. In the last case, an $O(En2N2)$ pseudo-polynomial time algorithm to find an approximate answer is presented. In contrast, when the two measures of path quality are total turn and number of links, an $O(E3n\log 2n)$ exact algorithm is given.

A main result of this thesis examines minimizing the Euclidean length and number of links of a path. When the geometric setting of this problem is a polygon without holes, this thesis presents an $O(n3k3\log (Nk/ϵ1/k))$ algorithm to find a $k$-link path with Euclidean length at most $1+ϵ$ times the length of the shortest $k$-link path. A faster algorithm for a relaxed case, when the output path is allowed to have $2k$ links, is presented for a polygon with or without holes.

Finally, some approximation algorithms are outlined for finding a minimum link path among polyhedral obstacles.