New Techniques for Seed Shape Description in Silene Species

Seed shape in Silene species is often described by means of adjectives such as reniform, globose, and orbicular, but the application of seed shape for species classification requires quantification. A method for the description and quantification of seed shape consists in the comparison with geometric models. Geometric models based on mathematical equations were applied to characterize the general morphology of the seeds in 21 species of Silene. In addition to the previously described four models (M1 is the cardioid, and M2 to M4 are figures derived from it), we present four new geometric models (model 5–8). Models 5 and 6 are open cardioids that resemble M3, quite different from the flat models, M2 and M4. Models 7 and 8 were applied to those species not covered by models 2 to 6. Morphological measures were obtained to describe and characterize the dorsal view of the seeds. The analyses done on dorsal views revealed a notable morphological diversity and four groups were identified. A correlation was found between roundness of dorsal view and the geometric models based on lateral views, such that some of the groups defined by seed roundness are also characterized by the similarity to particular models. The usefulness of new morphological tools of seed morphology to taxonomy is discussed.

Geometric Models for Seed Shape Description and Quantification in the Cactaceae

Seed shape in species of the Cactaceae is described by comparison with geometric models. Three new groups of models are presented, two for symmetric seeds, and a third group for asymmetric seeds. The first two groups correspond, respectively, to superellipses and the combined equations of two semi-ellipses. The third group contains models derived from the representation of polar equations of Archimedean spirals that define the shape of asymmetric seeds in genera of different subfamilies. Some of the new models are geometric curves, while others are composed with a part resulting from the average silhouettes of seeds. The application of models to seed shape quantification permits the analysis of variation in seed populations, as well as the comparison of shape between species. The embryos of the Cactaceae are of the peripheral type, strongly curved and in contact with the inner surface of the seed coat. A relationship is found between seed elongation and the models, in which the genera with elongated seeds are represented by models with longer trajectories of the spiral. The analysis of seed shape opens new opportunities for taxonomy and allows quantification of seed shape in species of the Cactaceae.

The Method of Constructing a Multidimensional Model for Finding the Values of the Parameters of the Incremental Sheet Forming Process

The object of research is geometric models of the process of incremental forming of parts from sheet material. The subject of research is a graphical model for finding the optimal values of the parameters of the forming process based on multidimensional descriptive geometry. The author of the article discusses the main optimizing factors and process parameters. Particular attention was paid to the problems of constructing geometric models for determining the optimizing factors for incremental forming. The research method is a way of constructing a graphical optimization model of the process using the projection drawing of Radishchev for multidimensional space. Mathematical modeling was also applied to check the correctness of the obtained optimal parameters.