Time geography provides a powerful analytic framework to identify alternative space-time paths as well as space-time opportunities based on the concept of a space-time prism. Both of these are important to activity scheduling, which is a potentially popular traveler assistance service. Spatiotemporal analysis of alterna-tive space-time paths between origin-destination pairs helps identify time-dependent critical links involving joint participants. In real activity scheduling sit-uations, space-time opportunities can be highly dependent on time-dependent critical links. This chapter presents a time-dependent prism-based approach to identifying critical links and opportunities for scheduling joint participation of multiple individuals. The first section introduces the generation of time-varying network-based prisms. In this part, a tracking dataset of 12,325 taxis in Wuhan, China, over one week was used to capture travel speed and time-dependent con-gestion of each link in the road network. This prism was identified based on time-dependent travel speeds. Then the chapter presents the identification of time-dependent critical links and opportunities between each origin-destination pair. Spatiotemporal analysis of alternative space-time paths between origin-destination pairs is considered in the evaluation of the use priority of links in real taxi services. This analysis supports the identification of time-dependent critical links involving joint participants. Finally, this chapter introduces a multi-objective approach to scheduling joint participation of multiple individuals by incorporating the previous two approaches. Five objectives are used here: (i) minimizing travel distance; (ii) minimizing travel time; (iii) maximizing expected participation activi-ty time; (iv) minimizing the added travel time over alternative space-time paths; and (v) maximizing the utility index of the selected critical activity opportunity. A scenario involving joint participation among four people is designed and imple-mented to demonstrate the feasibility of this approach.