The geometric aspects of Burmester Theory, as used in planar four-bar linkage synthesis, are examined to define a general procedure which is applied to the generation of the joint loci of spatial dyads. For a given dyad type a figure is drawn showing the joints of the dyad in their assumed positions, subject to the motion constraints of the dyad. A standard approach is used to geometrically relate the joints to the screw axes of the prescribed motion, by means of a screw triangle. The screw triangle relates the geometry between any three related screws. The geometric relationships are typically separated into several geometric constraints. Each geometric constraint is considered separately to generate a loci of lines or points representing joints which satisfy the constraint. The intersection of all of the loci produces a single loci of all the possible fixed or moving joints. The geometric approach is shown to have several advantages over numerical and pure analytical techniques, especially in relating the characteristics of the loci to the physical linkage and its required motion. The cylindrical-cylindrical dyad is considered in detail as a general case for dyads involving joints with axes. The angular and positional constraints are considered separately as independent constraints. Families of quadric cones are generated which correspond to the families of circles for three precision positions in the planar case. The intersection of the families of quadric cones produces the cubic screw cone which degenerates to Burmester's curve in the planar case.