Abstract Morphing Using the Hausdorff Distance and Voronoi Diagrams

This paper introduces two new abstract morphs for two 2-dimensional shapes. The intermediate shapes gradually reduce the Hausdorff distance to the goal shape and increase the Hausdorff distance to the initial shape. The morphs are conceptually simple and apply to shapes with multiple components and/or holes. We prove some basic properties relating to continuity, containment, and area. Then we give an experimental analysis that includes the two new morphs and a recently introduced abstract morph that is also based on the Hausdorff distance [Van Kreveld et al., 2022]. We show results on the area and perimeter development throughout the morph, and also the number of components and holes. A visual comparison shows that one of the new morphs appears most attractive

Between Shapes, Using the Hausdorff Distance

Given two shapes A and B in the plane with Hausdorff distance 1, is there a shape S with Hausdorff distance 1/2 to and from A and B? The answer is always yes, and depending on convexity of A and/or B, S may be convex, connected, or disconnected. We show a generalization of this result on Hausdorff distances and middle shapes, and show some related properties. We also show that a generalization of such middle shapes implies a morph with a bounded rate of change. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two sets and show how to approximate or compute it

Between shapes, using the Hausdorff distance

Given two shapes A and B in the plane with Hausdorff distance 1, is there a shape S with Hausdorff distance 1/2 to and from A and B? The answer is always yes, and depending on convexity of A and/or B, S may be convex, connected, or disconnected. We show that our result can be generalized to give an interpolated shape between A and B for any interpolation variable α between 0 and 1, and prove that the resulting morph has a bounded rate of change with respect to α. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two input sets. We show how to approximate or compute this middle shape, and that the properties relating to the connectedness of the Hausdorff middle extend from the case with two input sets. We also give bounds on the Hausdorff distance between the middle set and the input

Towards Characterizing Graphs with a Sliceable Rectangular Dual

\u3cp\u3eLet G be a plane triangulated graph. A rectangular dual of G is a partition of a rectangle R into a set R of interior-disjoint rectangles, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge. A rectangular dual is sliceable if it can be recursively subdivided along horizontal or vertical lines. A graph is rectangular if it has a rectangular dual and sliceable if it has a sliceable rectangular dual. There is a clear characterization of rectangular graphs. However, a full characterization of sliceable graphs is still lacking. The currently best result (Yeap and Sarrafzadeh, 1995) proves that all rectangular graphs without a separating 4-cycle are slice- able. In this paper we introduce a recursively defined class of graphs and prove that these graphs are precisely the nonsliceable graphs with exactly one separating 4-cycle.\u3c/p\u3

Multidecadal ocean variability and NW European ice sheet surges during the last deglaciation

A multiproxy paleoceanographic record from the Atlantic margin off the British Isles reveals in unprecedented detail discharges of icebergs and meltwater in response to sea surface temperature increases across the last deglaciation. We observe the earliest signal of deglaciation as a moderate elevation of sea surface temperatures that commenced with a weakly developed thermocline and the presence of highly ventilated intermediate waters in the Rockall Trough. This warming pulse triggered a series of multidecadal ice-rafted debris peaks that culminated with a major meltwater discharge at 17,500 years before present related to ice sheet disintegration across the NW European region. The impact of meltwater caused a progressive reduction in deep water ventilation and a sea surface cooling phase that preceded the collapse of the Laurentide Ice Sheet during Heinrich event 1 by 500-1000 years. A similar sequence of rapid ocean-ice sheet interaction across the European continental margin is identified during the Bølling-Allerød to Younger Dryas transition. The strategic location of our sediment core suggests a sensitive and rapid response of ice sheets in NW Europe to transient increases in thermohaline heat transport

Warm Atlantic surface water inflow to the Nordic seas 34–10 calibrated ka B.P.

Author Posting. © American Geophysical Union, 2008. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Paleoceanography 23 (2008): PA1201, doi:10.1029/2007PA001453.A number of short-lasting warm periods (interstadials) interrupted the otherwise cold climate of the last glacial period. These events are supposedly linked to the inflow of the warm Atlantic surface water to the Nordic seas. However, previous investigations of planktonic foraminifera from the Nordic seas have not been able to resolve any significant difference between the interstadials and intervening cold stadials, as the faunas are continuously dominated by the polar species Neogloboquadrina pachyderma s. Here we examine the planktonic foraminifera assemblages from a high-resolution core, LINK17, taken at 1500 m water depth off northern Scotland below the warmest part of the inflowing Atlantic water. The core comprises the time period 34–10 calibrated ka B.P., the coldest period of the last glaciation and the deglaciation. The results reveal a hitherto unknown faunistic variability indicating significant fluctuations in both surface water inflow and in summer sea surface temperatures. During the interstadials, relatively warm Atlantic surface water (4–7°C) flowed north into the eastern Norwegian Sea. During the stadials and Heinrich events the surface inflow stopped and the temperatures in the study area dropped to <2°C. The Last Glacial Maximum was nearly as warm as the interstadials, but the inflow was much more unstable. The data reveal two previously unrecognized warming events each lasting more than 1600 years and preceding Heinrich events HE3 and HE2, respectively. By destabilizing the ice sheets on the shelves the warmings may have played a crucial role for the development of Heinrich events HE2 and HE3.The study of LINK17 was financed by UNIS as a part of the ESF-EuroClimate Program Resolution (grant 04-ECLIM-FP33)