Computational Quantification of Map Projection Distortion by Fractal Dimension of Coastlines

Maps, essential tools for portraying the Earth’s surface, inherently introduce distortions to geographical features. While various quantification methods exist for assessing these distortions, they often fall short when evaluating actual geographic features. In our study, we took a novel approach by analyzing map projection distortion from a geometric perspective. We computed the fractal dimensions of different stretches of coastline before and after projection using the divide-and-conquer algorithm and image processing. Our findings revealed that map projections, even when preserving basic shapes, inevitably stretch and compress coastlines in diverse directions. This analysis method provides a more realistic and practical way to measure map-induced distortions, with significant implications for cartography, geographic information systems (GIS), and geomorphology. By bridging the gap between theoretical analysis and real-world features, this method greatly enhances accuracy and practicality when evaluating map projections.

Approximately Conformal,Equivalent and Equidistant Map Projections

If geodetic coordinates from an ellipsoid are included in the equations of a projection for mapping a sphere instead of geographical coordinates,the result will be a projection of the ellipsoid into a plane.This will slightly change the distortion distribution of the initial map projection.The question is to what extent the replacement of geographical with geodetic coordinates will affect this change.In this paper,we deal with conformal,equal-area and equidistant projections of the sphere,which we modify by using geodetic coordinates instead of geographical ones.The result will be an approximately conformal,approximately equal-area and approximately equidistant projection.It is shown that in this case the maximum distortion of the angles in approximately conformal projections will be approximately 23.09′,the maximum distortion of the area in approximately equal-area projections less than 0.7% and the maximum distortion of lengths in approximately equidistant projections less than 0.7%,therefore on the maps imperceptible.

Seven possible states of geospatial data with respect to map projection and definition: a novel pedagogical device for GIS education

Conceptually,the theory and implementation of“map projection”in geographic information system(GIS)technology is difficult to comprehend for most introductory students and novice users.Compounding this difficulty is the concept of a“map projection file”that defines map projection parameters of geo-spatial data.The problem of the“missing projection file”appears ubiquitous for all users,especially in practice where data is widely shared.Another common problem is inadvertent misapplication of the“Define Projection”tool that can result in a GIS dataset with an incorrectly defined map projection file.GIS education should provide more guidance in differentiating the concepts of map projection versus projection files by increasing understanding and minimizing common errors.A novel pedagogical device is introduced in this paper:the seven possible states of GIS data with respect to map projection and definition.The seven possible states are:(1)a projected coordinate system(PCS)that is correctly defined,(2)a PCS that is incorrectly defined,(3)a PCS that is undefined,(4)a geographic coordinate system(GCS)that is correctly defined,(5)a GCS that is incorrectly defined,(6)a GCS that is undefined,and(7)a non-GCS.Recently created automated troubleshooting tools to determine a missing map projection file are discussed.

Local linear scale factors in map projections in the direction of coordinate axes

This paper explains that the terms“horizontal and vertical scales”are not appropriate in map projections theory.Instead,the authors suggest using the term“scales in the direction of coordinate axes.”Since it is not possible to read a local linear scale factor in the direction of a coordinate axis immediately from the definition of a local linear scale factor,this paper considers the derivation of new formulae that enable local linear scale factors in the direction of coordinate x and y axes to be calculated.The formula for computing the local linear scale factor in any direction defined by dx and dy is also derived.Furthermore,the position and magnitude of the extreme values of the local linear scale factor are considered and new formulas derived.

Design of Authoring Tool for Static and Dynamic Projection Mapping

This study introduces the design details of a tool to create interactive projection-mapping content in a convenient manner.For the proposed tool design,a homography-based camera–projector calibration method was applied with the use of red–green–blue-depth images from a Kinect V2 sensor that did not require accurate camera calibration prerequisites.In addition,the proposed tool simultaneously achieved static projection mapping that projected the image content onto a fixed object,and dynamic projection mapping that projected the image content onto a user’s body,by tracing the moving user.To verify the effectiveness of the proposed content-creation tool,users with no programming capabilities were employed to create contents that were projected onto various objects in fixed positions and a user’s body in various poses,thereby analyzing the tool’s completeness.Moreover,the projection accuracy was analyzed at different depth positions,and the projection-mapping accuracy was verified with the use of the proposed method.

f-Harmonic maps of doubly warped product manifolds

In this paper,we study f-harmonicity of some special maps from or into a doubly warped product manifold.First we recall some properties of doubly twisted product manifolds.After showing that the inclusion maps from Riemannian manifolds M and N into the doubly warped product manifold M ×（μ,λ） N can not be proper f-harmonic maps,we use projection maps and product maps to construct nontrivial f-harmonic maps.Thus we obtain some similar results given in [21],such as the conditions for f-harmonicity of projection maps and some characterizations for non-trivial f-harmonicity of the special product maps.Furthermore,we investigate non-trivial f-harmonicity of the product of two harmonic maps.

Inverse full state hybrid projective synchronization for chaotic maps with different dimensions

A new synchronization scheme for chaotic（hyperchaotic） maps with different dimensions is presented.Specifically,given a drive system map with dimension n and a response system with dimension m,the proposed approach enables each drive system state to be synchronized with a linear response combination of the response system states.The method,based on the Lyapunov stability theory and the pole placement technique,presents some useful features：（i） it enables synchronization to be achieved for both cases of n 〈 m and n 〉 m;（ii） it is rigorous,being based on theorems;（iii） it can be readily applied to any chaotic（hyperchaotic） maps defined to date.Finally,the capability of the approach is illustrated by synchronization examples between the two-dimensional H′enon map（as the drive system） and the three-dimensional hyperchaotic Wang map（as the response system）,and the three-dimensional H′enon-like map（as the drive system） and the two-dimensional Lorenz discrete-time system（as the response system）.

MATHEMATICAL BASIS FOR LARGE SCALE GIS AND TERRESTRIAL DIGITAL PRODUCTS

The problem of spatial mathematical basis has been encountered by both large scale GIS and spatial digital products theoretically and practically.It is also a basic problem in the development of the whole geo_information science.After analyzing the status quo and the limitations of the space mathematical base of GIS,this paper points out definitely that the geodetic coordinate system is uniform,which can show the location of any point of the global exactly and uniquely in form of ( B,L,H ) and is the most proper reference system of large_scale GIS and Digital Earth.Moreover,this paper also puts forward a set of practical model of the standard "map projection".Finally,this paper introduces a DRG system based on this model.

Real-time raster projection for web maps

The virtual globe is the default visualization for Digital Earth applications,but it can only show one half of the Earth.This article introduces user-adjustable,on-the-fly projection of georeferenced raster images for web mapping and web GIS applications.This technique allows users to center a map on arbitrary locations,while still seeing the entire Earth surface.Modern web mapping libraries can apply map projection transformations to vector data in the browser,but they do not currently support the projection of raster images at interactive frame rates.We use the cross-platform WebGL(Web Graphics Library)graphics pipeline for hardware-accelerated projection of raster data in web browsers.Two algorithmic techniques–inverse per-fragment projection,and combined forward per-triangle and inverse per-fragment projection–for georeferenced raster imagery are documented.The resulting raster maps combine the ease of use and flexibility of interactive virtual globes with the ability to show the entire Earth.The adjustable projection of raster imagery allows users to observe global phenomena that are difficult to visualize on virtual globes or traditional maps with static projections.

Adaptive stray-light compensation in dynamic multi-projection mapping

Projection-based mixed reality is an effective tool to create immersive visualizations on real-world objects. Its wide range of applications includes art installations, education, stage shows, and advertising. In this work, we enhance a multi-projector system for dynamic projection mapping by handling various physical stray-light effects: interreflection,projector black-level, and environmental light in real time for dynamic scenes. We show how all these effects can be efficiently simulated and accounted for at runtime, resulting in significantly improved projection mapping results. By adding a global optimization step,we can further increase the dynamic range of the projection.

含有非线性扰动的正半Fredholm映射方程的逼近可解性

本文利用A—proper理论研究A是半Fredholm映射,N是某种非线性扰动时的算子方程Ax+Nx=f,得到了一些逼近可解性结果。

一类非紧算子方程 Ax=x 在楔中的近似可解性

设 T={X_n,P_n}是实 Banach 空间 X 的投影完备格式,F(?)X 是一楔形(Wedoe),它满足条件:存在eo∈F-{θ}使得-eo(?)F.D 是 X 中的有界开集,D∩F≠(?),AL(?)∈→F 连续、有界并使得 A-tl 关于 T={X_n,P_n)A-固有,t∈[1,1+k]或 t∈[(1+k′)^(-1) ,1+k],则 A 在 F 中不动点存在且 Ax=x 是弱(强)近似可解的,这些结果是许多作者的锥的拉仲与压缩不动点存在定理及其近似的改进和推广.

High-Throughput Chemotyping of Cannabis and Hemp Extracts Using an Ultraviolet Microplate Reader and Multivariate Classifiers

As the use of Cannabis products as natural medicines burgeons,it is also appearing as a food ingredient.It is important to screen Cannabis samples as ingredients by profiling their chemical compositions,which is referred to as chemotyping.Two sets of botanical extracts were studied.The first set is referred to as Cannabis contained plant materials from 15 samples of the sativa,indica,and hybrids of the two species.The second set contained 20 extracts from the variety of Cannabis sativa with low tetrahydrocannabinol(THC)concentrations,i.e.,below 0.3%,and,henceforth,will be referred to as hemp.An ultraviolet(UV)microplate reader provides a cost-effective and high-throughput method for identifying chemotypes of plant extracts by their spectra.The microplate reader affords rapid measurements of small volumes,e.g.,50μL,which demonstrates a potential to significantly reduce the analysis time and cost for Cannabis and hemp chemotyping or chemi-cal profiling.Replicate samples were measured on different days to demonstrate the robustness of the method.Projected difference resolution(PDR)maps were used to visualize the separations among the classes.Five multivariate classifiers,fuzzy rule-building expert system(FuRES),super partial least squares-discriminant analysis(sPLS-DA),support vector machine(SVM),and two tree-based support vector machines(SVMtreeG and SVMtreeH)were evaluated.The classifiers were validated with ten bootstrapped Latin partitions(BLPs).For the Cannabis extracts,the SVMtreeG yielded the best performance and the classification accuracy was 99.1±0.4%for spectra collected in the nonlinear absorbance range.For the hemp extracts,the SVM classifier performed the best with a 97.4±0.6%classification accuracy.These results demonstrate that the UV microplate reader coupled with multivariate classifiers can be used as a high-throughput and cost-effective approach for chemotyping Cannabis.