K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

We show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

Optimal Intersection Curves for Surfaces

In this article, an algorithm has been established to approximate parametric-parametric, explicit-implicit, and explicit-explicit surface intersection. Foremost, it extracts the characteristic points (boundary and turning points) from the sequence of intersection points and fits an optimal cubic spline curve to these points. Moreover, this paper utilizes genetic algorithm (GA) for optimization of shape parameters in the portrayal of cubic spline so that the error is minimal. The proposed algorithm is demonstrated with different types of surfaces to analyze its robustness and proficiency. In the end, all illustrations show the effectiveness of the algorithm which makes it more influential to resolve all complexities arises during intersection with a minimal error.

The Impact of Slicing Softwares on the Mechanical Properties of 3D Printed Parts

“Slicing tool” or “Slicing Software” computes the intersection curves of models and slicing planes. They improve the quality of the model being printed when given in the form of STL file. Upon analyzing a specimen that has been printed using two different slicing tools, there was a drastic variation on account of the mechanical properties of the specimen. The ultimate tensile strength and the surface roughness of the material vary from one tool to another. This paper reports an investigation and analysis of the variation in the ultimate tensile strength and the surface roughness of the specimen, given that the 3D printer and the model being printed is the same, with a variation of usage of slicing software. This analysis includes ReplicatorG, Flashprint as the two different slicing tools that are used for slicing of the model. The variation in the ultimate tensile strength and the surface roughness are measured and represented statistically through graphs. An appropriate decisive conclusion was drawn on the basis of the observations and analysis of the experiment on relevance to the behavior and mechanical properties of the specimen.

Area-Construction Algorithms Using Tangent Circles

Computer aided geometric design employs mathematical and computational methods for describing geometric objects, such as curves, areas in two dimensions (2D) and surfaces, and solids in 3D. An area can be represented using its boundary curves, and a solid can be represented using its boundary surfaces with intersection curves among these boundary surfaces. In addition, other methods, such as the medial-axis transform, can also be used to represent an area. Although most researchers have presented algorithms that find the medial-axis transform from an area, a algorithm using the contrasting approach is proposed; i.e., it finds an area using a medial-axis transform. The medial-axis transform is constructed using discrete points on a curve and referred to as the skeleton of the area. Subsequently, using the aforementioned discrete points, medial-axis circles are generated and referred to as the muscles of the area. Finally, these medial-axis circles are blended and referred to as the blended boundary curves skin of the area; consequently, the boundary of the area generated is smooth.

Mathematical Method to Unfolding Couplings

The purpose of this paper is to establish the intersection curves between cylinders, using Mathematica program. The equations curves which are inferred by mathematical methods are introduced in this program. This paper takes into discussion the case of four cylinders.