Reconstructing the integral of a distribution of radioactivity along a line from Compton camera data could be used to produce a parallel projection, or to perform tomosynthesis, or to reconstruct the whole three-dimensional distribution itself. Analytic methods for reconstructing an integral of radioactivity along a line are presented here. The sets of data that allow an integral along a line and a parallel projection of the distribution to be reconstructed are determined here by interpreting these methods from a geometric viewpoint. These methods and the sets of data depend upon which of the two models is assumed for the data. Both of these models have been previously proposed by other researchers. In addition, a new camera design is proposed here that makes it possible to measure all these sets of data. In this design, a first detector element has to be seen from the second detector in only a "semicircle of direction." Also, two techniques for increasing the sensitivity of the new camera design are proposed here. Computer simulations are performed to illustrate these reconstruction methods.