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This paper presents methods to obtain analytical solutions to a class of continuous traffic equilibrium problems, where continuously distributed customers from a bounded two-dimensional service region seek service from one of several discretely located facilities via the least congested travel path. We show that under certain conditions, the traffic flux at equilibrium, which is governed by a set of partial differential equations, can be decomposed with respect to each facility and solved analytically. This finding paves the foundation for an efficient solution scheme. Closed-form solution to the equilibrium problem can be obtained readily when the service region has a certain regular shape, or through an additional conformal mapping if the service region has an arbitrary simply connected shape. These results shed light on some interesting properties of traffic equilibrium in a continuous space. This paper also discusses how service facility locations can be easily optimized by incorporating analytical formulas for the total generalized cost of spatially distributed customers under congestion. Examples of application contexts include gates or booths for pedestrian traffic, as well as launching sites for air vehicles. Numerical examples are used to show the superiority of the proposed optimization framework, in terms of both solution quality and computation time, as compared with traditional approaches based on discrete mathematical programming and partial differential equation solution methods. An example with the metro station entrances at the Beijing Railway Station is also presented to illustrate the usefulness of the proposed traffic equilibrium and location design models.