Ultrastructural aspects of spermatogonesis in Metamicrocotyla macracantha (Alexander, 1954) Koratha, 1955 (Monogenea; Microcotylidae)

Ultrastructural aspects of spermatogenesis, spermiogenesis and of the mature spermatozoon of a microcotylid monogenean Metamicrocotyla macracantha parasite from Mugil liza, are described. The irregularly-shaped spermatogonia divides by successive mitoses, forming the primary spermatocytes, identified by the presence of synaptonemal complexes in their nuclei. The spermatids formed by meiotic cell divisions of the secondary spermatocytes, differentiate into a mature spermatozoon. Cross sections of the head and the middle region of mature spermatozoa show the nucleus with strong condensed chromatin, the mitochondria with short cristae, peripheral microtubules and two axonemes with a 9+1 pattern, confirming the characteristics of this genus.

Algorithm to determine the intersection curves between bezier surfaces by the solution of multivariable polynomial system and the differential marching method

The determination of the intersection curve between Bézier Surfaces may be seen as the composition of two separated problems: determining initial points and tracing the intersection curve from these points. The Bézier Surface is represented by a parametric function (polynomial with two variables) that maps a point in the tridimensional space from the bidimensional parametric space. In this article, it is proposed an algorithm to determine the initial points of the intersection curve of Bézier Surfaces, based on the solution of polynomial systems with the Projected Polyhedral Method, followed by a method for tracing the intersection curves (Marching Method with differential equations). In order to allow the use of the Projected Polyhedral Method, the equations of the system must be represented in terms of the Bernstein basis, and towards this goal it is proposed a robust and reliable algorithm to exactly transform a multivariable polynomial in terms of power basis to a polynomial written in terms of Bernstein basis .