While eggs shapes and sizes have been subject of many studies, we still know little about factors affecting these characteristics of birds' eggs. We revealed that shapes of pelvis and egg correlated less than their respective sizes. Egg measurements (length or diameter) scaled with negative allometry against pelvis size, that is, eggs become relatively larger with decreasing pelvis size. Studied birds with altricial developmental mode had on average the smallest pelvic dimensions and the largest relative size of eggs. However, this is due to the effect of small pelvis size (and body as a whole) of most altricials. At the similar size of the pelvis, birds with altricial developmental mode had a smaller relative size of eggs than their precocial counterparts. Correlation between the shape of egg and pelvis is affected by habitat. Narrow pelvis with an elongated postacetabular region correlated with elongated eggs in diving waterfowl. In raptorial birds, the relatively wide pelvis with the shortened postacetabular region correlated with the nearly rounded shape of eggs.

The original technique of description of avian eggs on the basis of the geometry of asymmetrical oval (ovoid) is suggested. Specific properties of this figure allow to create a system of 80 basic ovoid standards, each given an appropriate name, digital and letter coding, and distinct quantitative characteristics. Combining infundibular zones (blunt poles) of basic ovoids in pairs gives 80 standards of symmetric pseudo-ovoids, 44 of which are found in birds. The same procedure applied to different ovoids produces 375 standards of asymmetrical pseudo-ovoids. This totality can be divided into six groups. Use of such system of standards enables us to identify real shapes of avian eggs, to analyze relation of morphometric parameters to incubatory properties of eggs, and also to carry out comparisons and generalizations of other authors' data. Each standard is quantitatively characterized by means of indexes (namely, indices of infundibular, cloacal, and lateral zones; index of asymmetry, elongation index, complementarity index, interporal index, arc radiuses, length and diameter).

We performed comparative analysis of curvature characteristics of bird eggs and used ovoid profiles from various authors, our own geometric profiles, and archive bird egg profiles from our database. We suggested that the possible ovoids arise by changing the curvature radii of the polar and lateral zones. We compared the constructed curves and curves presented in oological literature with the pictures of the real bird egg profiles. The volume of actual material includes 16,490 eggs from 800 species of 20 bird orders. Specially designed computer software had calculated the radius of curvature of real bird eggs from photos and drew out the half-profiles presented in our article. We supposed that the asymmetrical ovoid is the most widespread geometric egg-shaped figure, which can easily be obtained by combination of circles. We also calculated that if the ovoid diameter were taken as a unit, then the radius obtuse (infundibular) pole would be equal to its half-length, lateral (side arcs) is equal two lengths, and the cloacal arch is 1–√2/2. From this suggestion, we concluded that the length of the ovoid is 2–√2/2 and its cloacal radius is equal to the difference between the length of ovoid and its diameter (rc = L–D). We analyzed the geometry of this asymmetrical oval and came to the conclusion that this is the basic form able to generate the derivative forms. We discovered that the centres of the arcs which form it lie on one basic/supporting circle, the diameter of which is equal to the ovoid diameter. All other ovoids, which have radius equal to radius of the infundibular zone we called the real ovoids. The changes in the radii of the lateral arcs are caused by the movement of their centres along a horizontal line passing through the centre of the base circle. This causes the prolongation or shortening of the ovoids. Sizes of cloacal arcs depended on the lateral, and their centres are placed along the vertical axis. Variability of the sizes of the abovementioned arcs caused the variability of form of the bird eggs. For their classification, we have proposed to divide them into five groups: sphere-like, circle, obtuse, droplet, and cone-shaped. Further, we separated some more groups: short, shortened, normal, lengthened, and long; according to the sizes of cloacal arcs – large radius, middle radius, and small radius ovoids. As a result, we have 80 forms of real ovoids – standards which describe the varieties of bird eggs. Each of them has its own number, name, geometrical figure, polynomic equation, and correlations of ovoid parameters. This set of ovoids is sufficient to describe the specifics of any form of bird eggs.

We evaluated the variation in absolute size and shape of birds’ eggs and the effects of developmental mode and phylogenetic relatedness on these traits. Eggs were characterized by length, diameter, and three indices of egg shape. Indices of egg shape were calculated as the ratio of radii that described the curvature of pointed end (cloacal zone), blunt end (infundibular zone), and lateral zone to egg diameter. We found that eggs shape was less variable than the absolute size of eggs. Index of the cloacal zone was the most changeable and index of the infundibular zone was very conservative. Size and shape of eggs could be better explained with phylogenetic relatedness than developmental mode.

The general principles of ovoid shapes and their mathematical interpretation were considered concerning previous data and experience. Previously, bird egg description was carried out using the composite ovoid model. According to this model, an ovoid is considered as a set of arcs with a smooth transition between them. The studied group of eggs was named true ovoid. They differ from other forms in size of their infundibular zone radius (thick end) that is almost equal to half of the diameter (0.5D ± 0.01 ˂ ri = 0.5D).We suggested that this commonality, a priori, implies the presence of an abstract geometric model, which is a satisfactory solution and logical approach for analyzing the diversity of natural ovoids. Such a model is a system of circles passing into each other. This allows, within a single system, to assign a vendor code to each form, which involves the name, geometric shape, and quantitative parameters that can be implemented in bird taxonomy.Early, 0.01 D was chosen as the model difference value and the ratio of symmetrical eggs in the analyzed database was 1.1%. In this research, we extended the difference value to 0.05 D and this covered 6.0% of the egg shapes. This is the maximum interval at which the curvature of the polar zones does not visually differ. We revealed that the variability in the egg shapes depends on the radii of curvature of the lateral and polar arcs. The larger the radius of the lateral arches, the greater the degree of freedom for variation of the lateral arches. We supposed that our data could associate any form of bird egg with its biological content. In turn, many ovoid features relevant to other natural objects can be used in bird taxonomical study.