The 3-D Global Spatial Data Model

Trigonometry (Summary of Mathematical Concepts) (The 3-D Global Spatial Data Model)

Trigonometry is the study of triangles and relationships between the various sides and angles. The trigonometric functions are defined as ratios of the sides of a right triangle. Once an angle is identified, the defining trigonometric ratio for angle θ are as given in Figure 3.5 (opp = opposite, adj = adjacent, hyp = hypotenuse). […]

Spherical Trigonometry (Summary of Mathematical Concepts) (The 3-D Global Spatial Data Model)

The rules of spherical trigonometry can be used to solve for the great circle arc distance between latitude/longitude points on the Earth or to solve triangles on the celestial sphere when determining the astronomical azimuth of a line on the ground. Two important considerations are as follows: • The Earth is slightly flattened at the […]

Calculus (Summary of Mathematical Concepts) (The 3-D Global Spatial Data Model)

Calculus is a valuable mathematical tool that can be described as a study of rates of change or, said differently, cause and effect. Calculus has an undeserved reputation of being difficult to learn. That may be true for some, but consider that most people who receive a paycheck for wages are already experts at calculus. […]

Probability and Statistics (Summary of Mathematical Concepts) (The 3-D Global Spatial Data Model)

Introduction The fields of probability and statistics are distinct disciplines each deserving more coverage than given here. Since knowledge of underlying mathematical principles is essential to understanding the importance of each discipline’s contribution to spatial data, the reader is referred to a variety of sources for more information. Some topics are written purposefully with a […]

Models (Summary of Mathematical Concepts) (The 3-D Global Spatial Data Model)

Models provide a connection between abstract concepts and human experience. In the context of spatial data, models give relevance and meaning to the concepts of location and geometrical relationships. Two kinds of models are used in this topic: functional models and stochastic models. Functional Models Functional models consist of physical, geometrical, mechanical, electrical, and other […]

Error Propagation (Summary of Mathematical Concepts) (The 3-D Global Spatial Data Model)

The theory of error propagation is derived in topics such as Mikhail (1976) and Wolf and Ghilani (1997), and is presented concisely in matrix form as where Error propagation involves calculus and is used to answer the question “If something is computed on the basis of a measurement and the measurement contains uncertainty, how is […]

Error Ellipses (Summary of Mathematical Concepts) (The 3-D Global Spatial Data Model)

Error ellipses are a graphical tool used to illustrate the pair-wise correlation that exists between computed values. Using 2-D plane coordinates as an example, if the correlation between the computed coordinates is zero, then the orientation of the error ellipse corresponds to that of the host coordinate system. Special case: if the correlation is zero […]

Least Squares (Summary of Mathematical Concepts) (The 3-D Global Spatial Data Model)

The principle of least squares states that the sum of the squares of the residuals— multiplied by their appropriate weights—will be a minimum for that set of answers (parameters) that has the greatest probability of being correct. The concept is simultaneously simple and complex because it applies with equal validity to computing a simple mean […]

Applications to the Global Spatial Data Model (GSDM) (Summary of Mathematical Concepts)

Given that this topic is devoted to describing the global spatial data model (GSDM), it will be shown in later topics how the various mathematical concepts in this topic are combined in a single comprehensive model, the GSDM. Very briefly: •    Observations are independent measurements of fundamental physical quantities. •    Observations are manipulated in conformance […]

Introduction (Geometrical Models for Spatial Data Computations) (The 3-D Global Spatial Data Model)

As previously defined, a mathematical model is a set of rules used to make a conceptual connection between abstract concepts and human experience. A model is judged “good” to the extent it is both simple and appropriate. When working with spatial data, the simplest model is a one-dimensional (1-D) distance. Models of increasing complexity include […]

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