Titlebook: Making Images with Mathematics; Alexei Sourin Textbook 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Sp

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978-3-030-69834-8The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerl

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Making Images with Mathematics978-3-030-69835-5Series ISSN 1863-7310 Series E-ISSN 2197-1781

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From Ancient Greeks to Pixels,The chapter explains how we see the world and how computer makes images. Beginning with Ancient Greek Geometry, it travels to modern geometry, introduces the subject of computer graphics and visualization, explains how the graphics pipeline works, and how a geometric point turns into a color spot on a computer screen.

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Geometric Shapes,This chapter presents the mathematical foundations of shape modeling. Curves, surfaces, and solid objects are considered as set of points, which are obtained by sampling various types of mathematical functions. Using the concept of sweeping, many varieties of shapes are defined based on only a few simple foundation principles.

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Transformations,This chapter considers how the same formulas, used for making shapes, can define their transformations. The rationale for using matrix transformations is explained and affine and projection matrix transformations are presented. Generalization of geometric sweeping implemented with matrices is further discussed.

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Adding Visual Appearance to Geometry,In this chapter, we consider how visual appearance including colors, shadows, material properties and textures can be added to geometry and how its photorealistic appearance can be achieved. The formulas, previously used for defining geometry, now will define variable colors as a new modality of immersion into the world of geometric definitions.

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Putting Everything Together,In this chapter, the ways of making interactive, real-time and immersive visualization environments are considered including technical and physiological design and implementation issues. Still the same transformations, and actually the same basic mathematical principles, will be used in the fast visualization methods.