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Mathematical Foundations of Scientific Visualization: Computer Graphics and Beyond

Ike Bell

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Published in Mathematical Foundations Of Scientific Visualization Computer Graphics And Massive Data Exploration (Mathematics And Visualization)

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Abstract: Scientific visualization is a rapidly growing field that uses computer graphics techniques to create visual representations of scientific data. This data can come from a variety of sources, such as simulations, experiments, and observations. Scientific visualization can be used to explore data, identify patterns, and communicate results.

The mathematical foundations of scientific visualization are based on a number of different areas of mathematics, including linear algebra, calculus, and differential equations. Linear algebra is used to represent the geometric objects that are displayed in scientific visualizations. Calculus is used to define the mathematical operations that are performed on the data. Differential equations are used to model the physical processes that are being simulated.

Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration (Mathematics and Visualization)

by Tigran Bagdasaryan

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Language	:	English
File size	:	12672 KB
Screen Reader	:	Supported
Print length	:	360 pages

In addition to these core mathematical areas, scientific visualization also draws on a number of other disciplines, such as computer science, engineering, and psychology. Computer science provides the algorithms and data structures that are used to process and display the data. Engineering provides the hardware and software that is used to create the visualizations. Psychology provides insights into how people perceive and interact with visualizations.

The mathematical foundations of scientific visualization are essential for understanding how to create visualizations that are both accurate and effective. By understanding the mathematical principles behind scientific visualization, researchers can develop new techniques for visualizing data and communicating results.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear transformations. Vectors are used to represent points in space, while matrices are used to represent linear transformations. Linear transformations are used to rotate, scale, and translate objects.

Linear algebra is used extensively in scientific visualization to represent the geometric objects that are displayed. For example, a vector can be used to represent the position of a point in space, while a matrix can be used to represent the transformation that is applied to the point.

Calculus

Calculus is a branch of mathematics that deals with limits, derivatives, and integrals. Limits are used to find the value of a function as it approaches a certain point. Derivatives are used to find the rate of change of a function. Integrals are used to find the area under a curve.

Calculus is used extensively in scientific visualization to define the mathematical operations that are performed on the data. For example, a derivative can be used to find the velocity of a particle, while an integral can be used to find the volume of a region.

Differential Equations

Differential equations are equations that relate the rate of change of a function to the value of the function itself. Differential equations are used to model a wide variety of physical processes, such as the motion of a particle, the flow of a fluid, and the growth of a population.

Differential equations are used extensively in scientific visualization to model the physical processes that are being simulated. For example, a differential equation can be used to model the motion of a particle in a fluid.

Computer Science

Computer science provides the algorithms and data structures that are used to process and display the data. Algorithms are used to perform specific tasks, such as finding the minimum or maximum value of a dataset. Data structures are used to organize and store the data.

Computer science is essential for scientific visualization because it provides the tools that are needed to process and display the data. Without computer science, it would be impossible to create the visualizations that are used to explore data, identify patterns, and communicate results.

Engineering

Engineering provides the hardware and software that is used to create the visualizations. Hardware includes the computers, graphics cards, and displays that are used to create the visualizations. Software includes the operating systems, programming languages, and visualization libraries that are used to create the visualizations.

Engineering is essential for scientific visualization because it provides the tools that are needed to create the visualizations. Without engineering, it would be impossible to create the visualizations that are used to explore data, identify patterns, and communicate results.

Psychology

Psychology provides insights into how people perceive and interact with visualizations. This information is used to design visualizations that are effective at communicating information.

Psychology is essential for scientific visualization because it provides the insights that are needed to create visualizations that are effective at communicating information. Without psychology, it would be impossible to create the visualizations that are used to explore data, identify patterns, and communicate results.

References

A. K. Peters, to Scientific Visualization, Springer, 1997.
C. D. Hansen and C. R. Johnson, Visualizing Data, Academic Press, 2005.
M. H. Gross and R. S. Fishman, Computational Geometry for Scientific Visualization, Springer, 2006.