Fabrics are three-dimensional and geometrically complex. They can be resolved into fabric elements, the simplest constituents of compound fabrics. The fabric elements can range from geometrically simple structures with families of planes or lines to complex structures with families of more or less regularly curved, congruent or incongruent surfaces or lines. The fabric elements are understood more easily if one projects derived families of parallel lines (axes) with well-defined geometric relations to the fabric itself into them. As fabric elements become more complex, these axes can degenerate into discrete lines or curves with a definite position. The purely geometrical character of Sander's fold axis β is confirmed, and the stacking axis α is defined. These axes may coexist in certain fabric elements and together define a family of parallel planes, the axial plane. Individual members of the penetrative family of axial planes are singled out as axial surfaces, such as the axial hinge surface. Other discrete axial surfaces result from the coexistence of discrete axial lines or curves with a penetrative axis. One can extrapolate along axes, axial planes, and axial surfaces. They are, therefore, important for the geologist whose knowledge by observation is always restricted to the small accessible part of his object. Examples demonstrate application of geometry to specific geological problems, ranging from current bedding to refolded folds and the position of oil traps.