Geometric probes in combination with systematic-random sampling are used to estimate length and surface area. If test geometric probes make random intersections with the biological objects, the unknown length and surface area of the objects is proportional to the number of intersections with the test probe. Geometric probes with two dimensions (planes) and one dimension (lines) are used to estimate parameters of length (L) and surface area (S) on tissue sections, respectively. In practice, we calculate local estimators of length density (L_V) and surface density (S_V), then scale these quantities to the total reference volume using either the two-stage method or fractionator approach.

As shown in the figures below, estimating the length of linear features of biological interest can be accomplished using the 2-D surface of a sphere. Because the surface of a sphere is isotropic (equal orientation in all directions), the probability of an intersection is proportional to the length of the linear feature. This probability, which is based on Buffon's needle problem from the 18th century, allows for the estimation of total length for thin linear objects on tissue sections.

Counting the intersections between the surface of the sphere (the surface of a circle on 2-D) and the linear fibers of interest allows the investigator to make estimates of total length.

This probe for length estimation is called a virtual probe (for more informaiton, see Mouton et al., J. Microscopy, 206, 54-64, 2002). A virtual probe for unbiased estimation of surface area is called the virtual cycloid (VC). The approach, which derives from the vertical section approach of Baddeley et al. (1986), specifies the vertical axis as the direction of sectioning (i.e., the direction perpendicular to the tissue section), then orients computer-generated cycloids (virtual cycloids) with their minor axis parallel to the vertical axis. At each x-y location the number of surface-cycloid intersections counted on focal planes scanned through the z-axis is proportional to the surface area in the tissue, with no further assumptions about the shape or isotropy of the surface of interest.