The Shape of Space: Evidence for Spontaneous but Flexible Use of Polar Coordinates in Visuospatial Representations

2021 ◽  
pp. 095679762097237
Author(s):  
Sami R. Yousif ◽  
Frank C. Keil
Keyword(s):  
Spatial Representation ◽  
Spatial Scales ◽  
Human Mind ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Coordinate Systems ◽  
Spatial Tasks ◽  
Polar Coordinate ◽  
Visual Matching ◽  
2D Space

What is the format of spatial representation? In mathematics, we often conceive of two primary ways of representing 2D space, Cartesian coordinates, which capture horizontal and vertical relations, and polar coordinates, which capture angle and distance relations. Do either of these two coordinate systems play a representational role in the human mind? Six experiments, using a simple visual-matching paradigm, show that (a) representational format is recoverable from the errors that observers make in simple spatial tasks, (b) human-made errors spontaneously favor a polar coordinate system of representation, and (c) observers are capable of using other coordinate systems when acting in highly structured spaces (e.g., grids). We discuss these findings in relation to classic work on dimension independence as well as work on spatial representation at other spatial scales.

scholarly journals The shape of space: Evidence for spontaneous but flexible use of polar coordinates in visuospatial representations

2021 ◽  
Author(s):  
Sami Ryan Yousif ◽  
Frank Keil
Keyword(s):  
Spatial Representation ◽  
Dimensional Space ◽  
Spatial Scales ◽  
Human Mind ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Coordinate Systems ◽  
Spatial Tasks ◽  
Polar Coordinate ◽  
Visual Matching

What is the format of spatial representation? In mathematics, we often conceive of two primary ways of representing two-dimensional space, Cartesian coordinates, which capture horizontal and vertical relations, and polar coordinates, which captures angle and distance relations. Do either of these two coordinate systems play a representational role in the human mind? Six experiments utilizing a simple ‘visual matching’ paradigm show that (1) representational format is recoverable from the errors observers make in simple spatial tasks; (2) human-made errors spontaneously favor a polar coordinate system of representation; and (3) observers are capable of using other coordinate systems when acting in highly structured spaces (e.g., grids). We discuss these findings in relation to classic work on dimension independence, as well as work on spatial representation at other spatial scales.

Outlier detection by the EM algorithm for laser scanning in rectangular and polar coordinate systems

2015 ◽  
Vol 9 (3) ◽  
Author(s):  
Karl-Rudolf Koch ◽  
Boris Kargoll
Keyword(s):  
Em Algorithm ◽  
Coordinate System ◽  
Linear Model ◽  
Laser Scanning ◽  
Local Coordinate System ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Coordinate Systems ◽  
The Em Algorithm ◽  
Polar Coordinate

AbstractTo visualize the surface of an object, laser scanners determine the rectangular coordinates of points of a grid on the surface of the object in a local coordinate system. Vertical angles, horizontal angles and distances of a polar coordinate system are measured with the scanning. Outliers generally occur as gross errors in the distances. It is therefore investigated here whether rectangular or polar coordinates are better suited for the detection of outliers. The parameters of a surface represented by a polynomial are estimated in the nonlinear Gauss Helmert (GH) model and in a linear model. Rectangular and polar coordinates are used, and it is shown that the results for both coordinate systems are identical. It turns out that the linear model is sufficient to estimate the parameters of the polynomial surface. Outliers are therefore identified in the linear model by the expectation maximization (EM) algorithm for the variance-inflation model and are confirmed by the EM algorithm for the mean-shift model. Again, rectangular and polar coordinates are used. The same outliers are identified in both coordinate systems.

scholarly journals Polar Transforms for ITK

2006 ◽  
Author(s):  
Jakub Bican
Keyword(s):  
Image Processing ◽  
Coordinate System ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Cartesian Coordinates ◽  
Polar Coordinate ◽  
2D Space

Polar transform is a geometric transform, that transforms points form cartesian coordinates to so-called polar coordinates. In case of 2D space, a point in polar coordinate system is addressed by radius and angle. In image processing, polar transform is usually used to convert rotations around the origin of polar coordinate system to translations.This submission contains the implementation of forward and inverse polar transforms in two very simple classes for the Insight Toolkit.

Additional Separated-Variable Solutions of the Biharmonic Equation in Polar Coordinates

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
I. H. Stampouloglou ◽  
E. E. Theotokoglou
Keyword(s):  
Fourth Order ◽  
Biharmonic Equation ◽  
Direct Determination ◽  
Additional Term ◽  
Elastic Solid ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Radial Coordinate ◽  
Displacement Fields ◽  
Polar Coordinate

From the biharmonic equation of the plane problem in the polar coordinate system and taking into account the variable-separable form of the partial solutions, a homogeneous ordinary differential equation (ODE) of the fourth order is deduced. Our study is based on the investigation of the behavior of the coefficients of the above fourth order ODE, which are functions of the radial coordinate r. According to the proposed investigation additional terms, φ¯−m(r,θ)(1≤m≤n) other than the usually tabulated in the Michell solution (1899, “On the Direct Determination of Stress in an Elastic Solid, With Application to the Theory of Plates,” Proc. Lond. Math. Soc., 31, pp. 100–124) are found. Finally the stress and the displacement fields due to each one additional term of φ¯−m(r,θ) are determined.

scholarly journals Cubic Spline Iterative Method for Poisson’s Equation in Cylindrical Polar Coordinates

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
R. K. Mohanty ◽  
Rajive Kumar ◽  
Vijay Dahiya
Keyword(s):  
Iterative Method ◽  
Spline Approximation ◽  
Cubic Spline ◽  
Poisson’S Equation ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Poisson's Equation ◽  
Polar Coordinate ◽  
Cylindrical Polar Coordinates ◽  
Diffusion Convection

Using nonpolynomial cubic spline approximation in x- and finite difference in y-direction, we discuss a numerical approximation of O(k2+h4) for the solutions of diffusion-convection equation, where k>0 and h>0 are grid sizes in y- and x-coordinates, respectively. We also extend our technique to polar coordinate system and obtain high-order numerical scheme for Poisson’s equation in cylindrical polar coordinates. Iterative method of the proposed method is discussed, and numerical examples are given in support of the theoretical results.

scholarly journals Determination of Ship Approach Parameters in the Polar Coordinates System

2014 ◽  
Vol 96 (1) ◽  
pp. 1-8
Author(s):  
Andrzej Banachowicz ◽  
Adam Wolski
Keyword(s):  
Coordinate System ◽  
Target Position ◽  
Geometric Interpretation ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Polar Coordinate ◽  
Radar Measurements ◽  
Cartesian Coordinate ◽  
Radar Antenna

Abstract An essential aspect of the safety of navigation is avoiding collisions with other vessels and natural or man made navigational obstructions. To solve this kind of problem the navigator relies on automatic anti-collision ARPA systems, or uses a geometric method and makes radar plots. In both cases radar measurements are made: bearing (or relative bearing) on the target position and distance, both naturally expressed in the polar coordinates system originating at the radar antenna. We first convert original measurements to an ortho-Cartesian coordinate system. Then we solve collision avoiding problems in rectangular planar coordinates, and the results are transformed to the polar coordinate system. This article presents a method for an analysis of a collision situation at sea performed directly in the polar coordinate system. This approach enables a simpler geometric interpretation of a collision situation

scholarly journals The Symplectic Method in Polar Coordinates for Linear Viscoelastic Materials

2015 ◽  
Vol 9 (1) ◽  
pp. 130-134
Author(s):  
W.X. Zhang ◽  
Y. Bai
Keyword(s):  
Analytical Form ◽  
Fundamental Solutions ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Symplectic Method ◽  
Polar Coordinate ◽  
Linear Viscoelastic ◽  
Annular Sector ◽  
Particular Solution ◽  
Variable Substitution

In this study, the symplectic method is applied to a two-dimensional annular-sector viscoelastic domain under the polar coordinate system. By applying variable separation approach, all fundamental solutions are derived in analytical form. Furthermore, using the method of variable substitution, lateral conditions are transformed into finding a particular solution for the governing equations, and this particular solution is derived with the use of eigensolution expansion. In the numerical example, the boundary condition problem is discussed in detail to analyze the stress response of viscoelastic solids.

Extreme precipitations measured by radar

2021 ◽  
Author(s):  
András Bárdossy ◽  
Geoff Pegram
Keyword(s):  
Reduction Factor ◽  
Polar Coordinates ◽  
Polar Coordinate System ◽  
Large Area ◽  
Area Reduction ◽  
Polar Coordinate ◽  
Radar Measurements ◽  
Stationary Conditions ◽  
The Mean ◽  
Block Sizes

<p>Radar measurements provide information on precipitation in space and time. They do not measure precipitation but reflectivity. The transformation to precipitation is not straightforward.  The result is that different, partly random, partly systematic errors may occur.  Radar precipitation pixels are usually considered to measure the mean over a large area of 500 x 500 m. However the measurement itself is represented in polar coordinates and is subsequently transformed to a Cartesian system. As the measurements in the polar coordinate system deliver areal averages corresponding to different block sizes this is likely to have an effect on the estimates of the true precipitation values. This particularly applies to extremes. In the outer circles of the radar scan the blocks are bigger, and thus the measurements deliver areal extremes where a kind of area reduction factor is the result of the resolution. In order to investigate the influence of this on the extremes two numerical simulation examples were considered. Results from a long high resolution simulation using the String of Beads model applied with data from South Africa, and of a copula based direct simulation, are analysed and presented. The results show that the extremes towards the outer ring of the radar observations may, under stationary conditions, be reduced by up to 20 %.</p>

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