Diffuse Reflection Diameter in Simple Polygons

We prove a conjecture of Aanjaneya, Bishnu, and Pal that the minimum number of diffuse reflections sufficient to illuminate the interior of any simple polygon with $n$ walls from any interior point light source is $\lfloor n/2 \rfloor - 1$. Light reflecting diffusely leaves a surface in all directions, rather than at an identical angle as with specular reflections.Comment: To appear in Discrete Applied Mathematic

The opaque square

The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916. We call such a set {\em opaque} or a {\em barrier} for the square. The shortest known barrier has length $\sqrt{2}+ \frac{\sqrt{6}}{2}= 2.6389\ldots$. The current best lower bound for the length of a (not necessarily connected) barrier is $2$, as established by Jones about 50 years ago. No better lower bound is known even if the barrier is restricted to lie in the square or in its close vicinity. Under a suitable locality assumption, we replace this lower bound by $2+10^{-12}$, which represents the first, albeit small, step in a long time toward finding the length of the shortest barrier. A sharper bound is obtained for interior barriers: the length of any interior barrier for the unit square is at least $2 + 10^{-5}$. Two of the key elements in our proofs are: (i) formulas established by Sylvester for the measure of all lines that meet two disjoint planar convex bodies, and (ii) a procedure for detecting lines that are witness to the invalidity of a short bogus barrier for the square.Comment: 23 pages, 8 figure

Abrasion of flat rotating shapes

We report on the erosion of flat linoleum "pebbles" under steady rotation in a slurry of abrasive grit. To quantify shape as a function of time, we develop a general method in which the pebble is photographed from multiple angles with respect to the grid of pixels in a digital camera. This reduces digitization noise, and allows the local curvature of the contour to be computed with a controllable degree of uncertainty. Several shape descriptors are then employed to follow the evolution of different initial shapes toward a circle, where abrasion halts. The results are in good quantitative agreement with a simple model, where we propose that points along the contour move radially inward in proportion to the product of the radius and the derivative of radius with respect to angle

The Penetration of Solar Radiation into Carbon Dioxide Ice

Icy surfaces behave differently to rocky or regolith‐covered surfaces in response to irradiation. A key factor is the ability of visible light to penetrate partially into the subsurface. This results in the Solid‐State Greenhouse Effect (SSGE), as ices can be transparent or translucent to visible and shorter wavelengths, whilst opaque in the infrared. This can lead to significant differences in shallow sub‐surface temperature profiles when compared to rocky surfaces. Of particular significance for modelling the SSGE is the e‐folding scale, otherwise known as the absorption scale length, or penetration depth, of the ice. Whilst there have been measurements for water ice and snow, pure and with mixtures, to date there have been no such measurements published for carbon dioxide ice. After an extensive series of measurements we are able to constrain the e‐folding scale of CO2 ice for the cumulative wavelength range 300 nm to 1100 nm, which is a vital parameter in heat transfer models for the Martian surface, enabling us to better understand surface‐atmosphere interactions at Mars’ polar caps

3DFlow: Continuous Summarization of Mesh Editing Workflows

Mesh editing software is continually improving allowing more detailed meshes to be create efficiently by skilled artists. Many of these are interested in sharing not only the final mesh, but also their whole workflows both for creating tutorials as well as for showcasing the artist\u27s talent, style, and expertise. Unfortunately, while creating meshes is improving quickly, sharing editing workflows remains cumbersome since time-lapsed or sped-up videos remain the most common medium. In this paper, we present 3DFlow, an algorithm that computes continuous summarizations of mesh editing workflows. 3DFlow takes as input a sequence of meshes and outputs a visualization of the workflow summarized at any level of detail. The output is enhanced by highlighting edited regions and, if provided, overlaying visual annotations to indicated the artist\u27s work, e.g. summarizing brush strokes in sculpting. We tested 3DFlow with a large set of inputs using a variety of mesh editing techniques, from digital sculpting to low-poly modeling, and found 3DFlow performed well for all. Furthermore, 3DFlow is independent of the modeling software used since it requires only mesh snapshots, using additional information only for optional overlays. We open source 3DFlow for artists to showcase their work and release all our datasets so other researchers can improve upon our work

Minimizing Visible Edges in Polyhedra

We prove that, given a polyhedron $\mathcal P$ in $\mathbb{R}^3$, every point in $\mathbb R^3$ that does not see any vertex of $\mathcal P$ must see eight or more edges of $\mathcal P$, and this bound is tight. More generally, this remains true if $\mathcal P$ is any finite arrangement of internally disjoint polygons in $\mathbb{R}^3$. We also prove that every point in $\mathbb{R}^3$ can see six or more edges of $\mathcal{P}$ (possibly only the endpoints of some these edges) and every point in the interior of $\mathcal{P}$ can see a positive portion of at least six edges of $\mathcal{P}$. These bounds are also tight.Comment: 19 pages, 9 figure