scholarly journals PLANAR IMMERSIONS WITH PRESCRIBED CURL AND JACOBIAN DETERMINANT ARE UNIQUE

2021 ◽  
pp. 1-6
Author(s):  
ANTHONY GRUBER
Keyword(s):  
Computer Graphics ◽  
Grid Generation ◽  
Jacobian Determinant ◽  
Two Dimensional ◽  
Boundary Values ◽  
Planar Domains ◽  
Dimensional Version

Abstract We prove that immersions of planar domains are uniquely specified by their Jacobian determinant, curl function and boundary values. This settles the two-dimensional version of an outstanding conjecture related to a particular grid generation method in computer graphics.

scholarly journals Geometry of the 3D Pythagoras' Theorem

2016 ◽  
Vol 8 (6) ◽  
pp. 78 ◽  
Author(s):  
Luis Teia
Keyword(s):  
Three Dimensional ◽  
Transformation Process ◽  
Two Dimensional ◽  
Natural Evolution ◽  
Dimensional Version ◽  
Pythagoras Theorem

This paper explains step-by-step how to construct the 3D Pythagoras' theorem by geometric manipulation of the two dimensional version. In it is shown how $x+y=z$ (1D Pythagoras' theorem) transforms into $x^2+y^2=z^2$ (2D Pythagoras' theorem) via two steps: a 90-degree rotation, and a perpendicular extrusion. Similarly, the 2D Pythagoras' theorem transforms into 3D using the same steps. Octahedrons emerge naturally during this transformation process. Hence, each of the two dimensional elements has a direct three dimensional equivalent. Just like squares govern the 2D, octahedrons are the basic elements that govern the geometry of the 3D Pythagoras' theorem. As a conclusion, the geometry of the 3D Pythagoras' theorem is a natural evolution of the 1D and 2D. This interdimensional evolution begs the question -- Is there a bigger theorem at play that encompasses all three?

Computer Animation of Gait

1978 ◽  
Vol 7 (2) ◽  
pp. 111-113 ◽  
Author(s):  
D. R. Broome
Keyword(s):  
Computer Graphics ◽  
Computer Animation ◽  
Fourier Coefficients ◽  
Two Dimensional ◽  
Ankle Angle ◽  
Hip Knee Ankle Angle ◽  
Displacement Information ◽  
Leg Movements

A program has been written which enables two dimensional visualization of leg movements on a computer graphics display. Hip, knee, ankle angle, and pelvic displacement information can be input and processed to obtain Fourier coefficients characterizing these motions. The thigh, shank and foot are then displayed at, for example, two hundred points within each walking cycle, at natural speed or as slowly as desired.

Solutions to the Tipping Point Problem

2012 ◽  
Vol 106 (1) ◽  
pp. 60-63
Keyword(s):  
Ice Cream ◽  
Tipping Point ◽  
Rectangular Prism ◽  
Point Problem ◽  
Two Dimensional ◽  
Dimensional Version ◽  
Fixed Area

The problem posed in MT August 2011 (vol. 105, no. 1, pp. 62-66) asked readers to consider the two-dimensional version of tipping a bowl (assumed to be a rectangular prism) to spoon out the last little bit of melted ice cream. Here is the essence of the problem: Given a fluid region of fixed area A contained in a rectangle whose width is W, find a formula for the fluid depth D when the container is tilted through a known angle T that is measured from horizontal.

ON TWO-DIMENSIONAL DYNAMICAL SYSTEMS ASSOCIATED WITH MEMBRANE KINETICS UNDERLYING CARDIAC ARRHYTHMIAS

2009 ◽  
Vol 19 (05) ◽  
pp. 1709-1732 ◽  
Author(s):  
B. M. BAKER ◽  
M. E. KIDWELL ◽  
R. P. KLINE ◽  
I. POPOVICI
Keyword(s):  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Two Dimensional ◽  
Stimulus Period ◽  
Smooth Maps ◽  
Periodic Stimulation ◽  
Piecewise Smooth Maps ◽  
Dimensional Version ◽  
Bifurcation Law ◽  
The One

We study the orbits, stability and coexistence of orbits in the two-dimensional dynamical system introduced by Kline and Baker to model cardiac rhythmic response to periodic stimulation — as a function of (a) kinetic parameters (two amplitudes, two rate constants) and (b) stimulus period. The original paper focused mostly on the one-dimensional version of this model (one amplitude, one rate constant), whose orbits, stability properties, and bifurcations were analyzed via the theory of skew-tent (hence unimodal) maps; the principal family of orbits were so-called "n-escalators", with n a positive integer. The two-dimensional analog (motivated by experimental results) has led to the current study of continuous, piecewise smooth maps of a polygonal planar region into itself, whose dynamical behavior includes the coexistence of stable orbits. Our principal results show (1) how the amplitude parameters control which escalators can come into existence, (2) escalator bifurcation behavior as the stimulus period is lowered — leading to a "1/n bifurcation law", and (3) the existence of basins of attraction via the coexistence of three orbits (two of them stable, one unstable) at the first (largest stimulus period) bifurcation. We consider the latter result our most important, as it is conjectured to be connected with arrhythmia.

Microstrip Gas Counters

2014 ◽  
pp. 31-51
Keyword(s):  
Glass Surface ◽  
Two Dimensional ◽  
Large Area ◽  
Proportional Chamber ◽  
Microelectronic Technology ◽  
Multiwire Proportional Chamber ◽  
Dimensional Version ◽  
A Chain ◽  
New Generation ◽  
First Time

In this chapter, the first micropattern gaseous detector, the microstrip gas counter, invented in 1988 by A. Oed, is presented. It consists of alternating anode and cathode strips with a pitch of less than 1 mm created on a glass surface. It can be considered a two-dimensional version of a multiwire proportional chamber. This was the first time microelectronic technology was applied to manufacturing of gaseous detectors. This pioneering work offers new possibilities for large area planar detectors with small gaps between the anode and the cathode electrodes (less than 0.1 mm). Initially, this detector suffered from several serious problems, such as charging up of the substrate, discharges which destroyed the thin anode strips, etc. However, by efforts of the international RD28 collaboration hosted by CERN, most of them were solved. Although nowadays this detector has very limited applications, its importance was that it triggered a chain of similar developments made by various groups, and these collective efforts finally led to the creation of a new generation of gaseous detectors-micropattern detectors.

Hybrid grid generation for two-dimensional high-Reynolds flows

2009 ◽  
Vol 38 (10) ◽  
pp. 1863-1875 ◽  
Author(s):  
D. Dussin ◽  
M. Fossati ◽  
A. Guardone ◽  
L. Vigevano
Keyword(s):  
Grid Generation ◽  
Two Dimensional ◽  
Hybrid Grid ◽  
High Reynolds
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