Probing Convex Polygons with a Wedge

Minimizing the number of probes is one of the main challenges in reconstructing geometric objects with probing devices. In this paper, we investigate the problem of using an $\omega$-wedge probing tool to determine the exact shape and orientation of a convex polygon. An $\omega$-wedge consists of two rays emanating from a point called the apex of the wedge and the two rays forming an angle $\omega$. To probe with an $\omega$-wedge, we set the direction that the apex of the probe has to follow, the line $\overrightarrow L$, and the initial orientation of the two rays. A valid $\omega$-probe of a convex polygon $O$ contains $O$ within the $\omega$-wedge and its outcome consists of the coordinates of the apex, the orientation of both rays and the coordinates of the closest (to the apex) points of contact between $O$ and each of the rays. We present algorithms minimizing the number of probes and prove their optimality. In particular, we show how to reconstruct a convex $n$-gon (with all internal angles of size larger than $\omega$) using $2n-2$ $\omega$-probes; if $\omega = \pi/2$, the reconstruction uses $2n-3$ $\omega$-probes. We show that both results are optimal. Let $N_B$ be the number of vertices of $O$ whose internal angle is at most $\omega$, (we show that $0 \leq N_B \leq 3$). We determine the shape and orientation of a general convex $n$-gon with $N_B=1$ (respectively $N_B=2$, $N_B=3$) using $2n-1$ (respectively $2n+3$, $2n+5$) $\omega$-probes. We prove optimality for the first case. Assuming the algorithm knows the value of $N_B$ in advance, the reconstruction of $O$ with $N_B=2$ or $N_B=3$ can be achieved with $2n+2$ probes,- which is optimal.Comment: 31 pages, 27 figure

Periglacial landforms and processes in the southern Kenai Mountains, Alaska

The distribution and characteristics of periglacial landforms in the southern Kenai Mountains, Alaska, were investigated during the summer of 1979. The principal area of study was a 1300-metre high mountain mass which stood as a nunatak during the last general glaciation. Periglacial features in the area include gelifluction lobes, nivation hollows, cryoplanation terraces, tors, a string bog, and various for111S of patterned ground such as sorted circles, sorted polygons, earth hummocks, sorted steps, sorted stripes, and small ice-wedge polygons. Ground temperature measurements indicate that permafrost recently existed in the area but is no longer present. The sorted polygons, cryoplsnation terraces, and nivation hollows are relic features which have been inactive for a considerable time. The turf-banked sorted steps and large gelifluction lobes probably were active until the very recent thawing of permafrost. Cryofraction and frost sorting still are vigorous active processes. Finely jointed bedrock, a previous colder environment, and long exposure in the absence of glacial ice has allowed periglacial processes to be the dominant surface agents both in the principal study area and in similar areas along the western side of the Kenai Mountains

An Investigation of Sixth-grade Students\u27 Conceptualization of Angle and Angle Measure: A Retrospective Analysis of Design Research Study of a Real-world Context

A strong foundation in students’ understanding of the multifaceted nature of the angle concept is of paramount significance in understanding trigonometry and other advanced mathematics courses involving angles. Research has shown that sixth-grade students struggle understanding the multifaceted nature of the angle concept (Keiser, 2004). Building on existing work on students’ understanding of angle and angle measure and instructional supports, this study asks: How do sixth-grade students conceptualize angle and angle measure before, during, and after learning through a geometry unit of instruction set in a miniature golf context? What instructional supports contribute to sixth-grade students’ conceptualization of angle and angle measure in such a context? I conducted a retrospective analysis of existing data generated using design-based research methodology and guided by Realistic Mathematics Education (RME) theory. Using Cobb and Yackel’s (1996) Emergent Perspective as an interpretive framework, I analyzed transcripts of video and audio recordings from nine days of lessons in a collaborative teaching experiment (CTE), focusing on two pairs of students in sixth-grade mathematics classes. I also analyzed transcripts of pre-interviews before instruction, midway interviews during instruction, and post-interviews after instruction with each student in the two pairs. To answer research question one, I developed codes from data guided by the existing literature. For research question two, I used Anghileri’s (2006) levels of supports framework. Overall, the findings revealed that sixth-grade students conceptualized an angle as a static geometric figure defined by two rays meeting at a common point, and conceptualized angle measure through their body turns. In addition, Anghileri’s three levels of supports, such as the use of structured tasks, teacher’s use of probing questions, generation of conceptual discourse were evident in contributing to students’ conceptualization of angle and angle measure during the miniature golf geometry unit of instruction. The findings of this study have implications for the school mathematics curriculum, and how to teach and to prepare teachers to teach angle and angle measure. This study emphasizes the need to redefine the angle concept in the curriculum documents, the need to increase activities involving body turns and the use of Anghileri’s (2006) levels of supports in the teaching and learning of angle and angle measure in a real-world context. Further research is needed to identify instructional supports, in particular activities that can support students’ conceptualization of slopes and turns as angles in a real-world context

Spatial variability of aircraft-measured surface energy fluxes in permafrost landscapes

Arctic ecosystems are undergoing a very rapid change due to global warming and their response to climate change has important implications for the global energy budget. Therefore, it is crucial to understand how energy fluxes in the Arctic will respond to any changes in climate related parameters. However, attribution of these responses is challenging because measured fluxes are the sum of multiple processes that respond differently to environmental factors. Here, we present the potential of environmental response functions for quantitatively linking energy flux observations over high latitude permafrost wetlands to environmental drivers in the flux footprints. We used the research aircraft POLAR 5 equipped with a turbulence probe and fast temperature and humidity sensors to measure turbulent energy fluxes along flight tracks across the Alaskan North Slope with the aim to extrapolate the airborne eddy covariance flux measurements from their specific footprint to the entire North Slope. After thorough data pre-processing, wavelet transforms are used to improve spatial discretization of flux observations in order to relate them to biophysically relevant surface properties in the flux footprint. Boosted regression trees are then employed to extract and quantify the functional relationships between the energy fluxes and environmental drivers. Finally, the resulting environmental response functions are used to extrapolate the sensible heat and water vapor exchange over spatio-temporally explicit grids of the Alaskan North Slope. Additionally, simulations from the Weather Research and Forecasting (WRF) model were used to explore the dynamics of the atmospheric boundary layer and to examine results of our extrapolation

Surficial geology and morphology of the Alaskan central Arctic coastal plain

Thesis (Ph.D.) University of Alaska Fairbanks, 1990Mapping and analyses have defined the distribution, morphology, character, and age of marine, fluvial, glacial, eolian, and lacustrine sediments of the late Cenozoic Gubik Formation in approximately 12,000 km\sp2 of the Alaskan central Arctic Coastal Plain, and allowed interpretations of the depositional, climatic, and tectonic histories. Amino-acid analysis of wood and some shell materials has defined broad age groups: young, middle and old. The old group has been abandoned because of probable leaching of acids or other modification. These groups are the basis for correlation of deposits between areas and have been assigned minimum relative ages. The young group is at least Sangamonian and the middle group is probably at least middle Pleistocene. Notable among interpretations of the surficial geology and morphology are: (1) Transgression of early Wisconsinan and perhaps Sangamonian seas as far as 9 km inland from the present coast. (2) Tertiary glacial advances as far north as uplands near Kavik airstrip and perhaps the headwaters of the Kachemach and Miluveach Rivers. (3) Three marine terraces as old as middle to late Pliocene and three late Pleistocene alluvial terraces east of the Colville River. (4) Middle Pleistocene minimum age for the Ugnuravik gravel is indicated by wood of the middle amino-acid group. (5) Coexistence of coniferous and nonconiferous wood on the Coastal Plain in middle to early Pleistocene time is possibly explained by greater accumulation of summer warmth associated with a continental climate resulting from greater exposure of the continental shelf. (6) Late Pliocene through Pleistocene outwash and alluvium and Holocene alluvium compose the Canning gravel. (7) Folding of the Coastal Plain in western ANWR and up to 95 m of uplift in the Sadlerochit Mountains since latest Pliocene time. (8) Late middle through late Wisconsinan age for the Beechey sand. (9) Late Wisconsinan through early Holocene age for thaw lakes in which broad-based mounds formed. While other findings and interpretations may be less significant, collectively they have allowed a start toward definition of the surficial geology