Related papers: Crocheting Mathematics
We describe a method for constructing $n$-orthogonal coordinate systems in constant curvature spaces. The construction proposed is a modification of Krichever's method for producing orthogonal curvilinear coordinate systems in the…
Additive manufacturing builds physical objects by accumulating layers upon layers of solidified material. This process is typically done with horizontal planar layers. However, fused filament printers have the capability to extrude material…
Complexity is an interdisciplinary concept which, first of all, addresses the question of how order emerges out of randomness. For many reasons matrices provide a very practical and powerful tool in approaching and quantifying the related…
Visualization of implicit surfaces is an actively researched topic. While raytracing can produce high quality images, it is not well suited for creating a quick preview of the surface. Indirect algorithms (e.g. Marching Cubes) create an…
Cylindrical Algebraic Decomposition (CAD) is a key tool in computational algebraic geometry, best known as a procedure to enable Quantifier Elimination over real-closed fields. However, it has a worst case complexity doubly exponential in…
We study collections of curves in generic position on a closed surface whose complement consists of one disk only, up to orientation-preserving homeomorphism of the surface. We define a surgery operation on the set of such collections and…
Spherical $t$-designs are finite point sets on the unit sphere that enable exact integration of polynomials of degree at most $t$ via equal-weight quadrature. This concept has recently been extended to spherical $t$-design curves by the use…
A contractible simplicial complex is constructed that parametrizes different ways of representing a fixed one-dimensional homology class in a closed orientable surface by isotopy classes of systems of disjoint oriented simple closed curves.…
From the pragmatic to the symbolic, textiles play a prominent role in some of the most demanding yet ubiquitous scenarios, such as covering the complex and dynamic geometries of the human body. Textiles are made by repeated manipulations of…
We consider collections of disjoint simple closed curves in a compact orientable surface which decompose the surface into pairs of pants. The isotopy classes of such curve systems form the vertices of a 2-complex, whose edges correspond to…
A new deterministic floating-point arithmetic called precision arithmetic is developed to track precision for arithmetic calculations. It uses a novel rounding scheme to avoid excessive rounding error propagation of conventional…
This study starts from the counter-intuitive question of how we can render a conventional stiff, non-stretchable and even brittle material conformable so that it can fully wrap around a curved surface, such as a sphere, without failure.…
I-nteract is a cyber-physical system that enables real-time interaction with both virtual and real artifacts to design 3D models for additive manufacturing by leveraging on mixed reality technologies. This paper presents novel advances in…
Casting is a manufacturing process where liquid material is poured into a mold having the shape of a desired product. After the material solidifies, the product is removed from the mold. We study the case where the mold is made of a single…
Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as…
The square root velocity transformation provides a convenient and numerically efficient approach to functional and shape data analysis of curves. We study fundamental geometric properties of curves under this transformation. Moreover,…
Mathematically representing the shape of an object is a key ingredient for solving inverse rendering problems. Explicit representations like meshes are efficient to render in a differentiable fashion but have difficulties handling topology…
An elliptical invisible cloak is proposed using a coordinate transformation in the elliptical-cylindrical coordinate system, which crushes the cloaked object to a line segment instead of a point. The elliptical cloak is reduced to a…
Generating images from hand-drawings is a crucial and fundamental task in content creation. The translation is difficult as there exist infinite possibilities and the different users usually expect different outcomes. Therefore, we propose…
This paper builds a novel bridge between algebraic coding theory and mathematical knot theory, with applications in both directions. We give methods to construct error-correcting codes starting from the colorings of a knot, describing…