The Relation Between Offset and Conchoid Constructions

The one-sided offset surface Fd of a given surface F is, roughly speaking, obtained by shifting the tangent planes of F in direction of its oriented normal vector. The conchoid surface Gd of a given surface G is roughly speaking obtained by increasing the distance of G to a fixed reference point O by d. Whereas the offset operation is well known and implemented in most CAD-software systems, the conchoid operation is less known, although already mentioned by the ancient Greeks, and recently studied by some authors. These two operations are algebraic and create new objects from given input objects. There is a surprisingly simple relation between the offset and the conchoid operation. As derived there exists a rational bijective quadratic map which transforms a given surface F and its offset surfaces Fd to a surface G and its conchoidal surface Gd, and vice versa. Geometric properties of this map are studied and illustrated at hand of some complete examples. Furthermore rational universal parameterizations for offsets and conchoid surfaces are provided

Conchoid surfaces of spheres

The conchoid of a surface $F$ with respect to given fixed point $O$ is roughly speaking the surface obtained by increasing the radius function with respect to $O$ by a constant. This paper studies {\it conchoid surfaces of spheres} and shows that these surfaces admit rational parameterizations. Explicit parameterizations of these surfaces are constructed using the relations to pencils of quadrics in $\R^3$ and $\R^4$. Moreover we point to remarkable geometric properties of these surfaces and their construction

Quantification of synmagmatic flow structures: a tool for Rock Quality Evaluation

The result of the analysis using methods based on fractal geometry provides information about magmatic flux directions and mineral equilibrium processes in the crystallizing magma chamber. Additionally, the used methods may also provide important information for the rock industry, because they allow a fast and automatically evaluation of rock quality.Fundação para a Ciência e a Tecnologia (FCT

Quantification of synmagmatic flow structures of the Vila Pouca de Aguiar Pluton, NE Portugal

Methods based on fractal geometry offer the opportunity to quantify complex rock patterns (Kruhl. 2013), which provide information about the pattern forming processes. Mineral distribution patterns of Variscan post-tectonic granites from NE of Portugal (Vila Pouca de Aguiar Pluton) were analyzed with the MORFA and Map-­Counting software (Peternell, 2011). The result of the analysis provides information about pattern inhomogeneity and anisotropy, i.e. magmatic flux directions and mineral equilibrium processes in the crystallizing magma chamber.Fundação para a Ciência e a Tecnologia (FCT

Cissoid constructions of augmented rational ruled surfaces

J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683)Given two real affine rational surfaces we derive a criterion for deciding the rationality of their cissoid. Furthermore, when one of the surfaces is augmented ruled and the other is either an augmented ruled or an augmented Steiner surface, we prove that the cissoid is rational. Furthermore, given rational parametrizations of the surfaces, we provide a rational parametrization of the cissoid.Ministerio de Economía y CompetitividadEuropean Regional Development Fun

Interpretative model of shearband boudins internal evolution in HT ductile shear zones

The internal structure of a shearband boudin resulting from an original igneous, hydrothermal or metamorphic segregation tabular rigid body is a subject of scientific interest. It allows understanding the deformation mechanisms acting on homogeneous quartz aggregate activated during simple shear progressive deformation. This work is focused on the characterization of the internal evolution of shearband boudins, using microtextural analysis, fluid inclusions studies, fractal and OCP analysis. The proposed interpretative model shows the several structural stages that can be well established during the process of the internal evolution of shearband boudin

P-T Path of a Variscan Shear Zone recorded on Quartz-Aluminous Shearband Boudins

The work is focused on the P-T path recorded on internal shearband boudin microstructures, developed during simple shear progressive deformation (Malpica-Lamego Ductile Shear Zone – MLDSZ, NW Portugal). In the studied area, MLDSZ is a NW-SE striking Variscan crustal shear zone with a subvertical and west-dipping foliation and a sub-horizontal stretching lineation; it is recorded as a heterogeneous simple shear zone with bulk left-lateral kinematics (Pamplona and Rodrigues, 2011). The deformation zone is marked by a generalized foliation (Sn) defined by Bt+Ms±Sil and a stretching mineral lineation marked by sillimanite fibres. Microstructural analysis, fluid inclusions studies, Raman spectroscopy, crystallographic preferred orientation on quartz grains and fractal geometry based analysis were applied to the boudins.Fundação para a Ciência e a Tecnologia (FCT

Offsets, Conchoids and Pedal Surfaces

We discuss three geometric constructions and their relations, namely the offset, the conchoid and the pedal construction. The offset surface F d of a given surface F is the set of points at fixed normal distance d of F. The conchoid surface G d of a given surface G is obtained by increasing the radius function by d with respect to a given reference point O. There is a nice relation between offsets and conchoids: The pedal surfaces of a family of offset surfaces are a family of conchoid surfaces. Since this relation is birational, a family of rational offset surfaces corresponds to a family of rational conchoid surfaces and vice versa. We present theoretical principles of this mapping and apply it to ruled surfaces and quadrics. Since these surfaces have rational offsets and conchoids, their pedal and inverse pedal surfaces are new classes of rational conchoid surfaces and rational offset surfaces

Quantification of quartz microstructures

Quartz microstructure analysis based on classical optical microscopy, is appointing to the identification of features related with dominial quartz studies. Microstructural features defined in this study include the concepts of microstructures, microstructural elements, microstructural features, fabric elements, elementary textures, textural elements and textural features (e.g., Vernon, 1976 and Kosaka, 1980). Petrographic analysis was focused on the identification of quartz microstructures, such as intra-granular microstructures, inter-granular boundary, and their relation to the deformation mechanisms activated during a simple shear progressive deformation event. These structures were quantified following the methodology developed by Kosaka (1980). To ensure a random sampling, measurement points were picked from a standard grid that covers the total thin section area. A statistically significant number of points were analysed.Fundação para a Ciência e a Tecnologia (FCT