This book explores three computational formalisms for solving geometric problems. Part I introduces a trigonometric-based formalism, enabling calculations of distances, angles, and areas using basic trigonometry. Part II focuses on complex numbers, representing points in the plane to manipulate geometric properties like collinearity and concurrency, making it particularly useful for planar problems and rotations. Part III covers vector formalism, applying linear algebra to both plane and solid geometry. Vectors are effective for solving problems related to perpendicularity, collinearity, and the calculation of distances, areas, and volumes.
Each formalism has its strengths and limitations, with complex numbers excelling in the plane and vectors being more versatile in three-dimensional space. This book equips readers to choose the best approach for various geometric challenges.
This book, designed for math majors, especially future educators, is also valuable for gifted high school students and educators seeking diverse proofs and teaching inspiration.
Contents:
- Trigonometric Approach:
- Foundations of Geometry
- Triangle Geometry
- Quadrilateral Geometry
- Complex Numbers Approach:
- Complex Numbers in Geometry
- Vectorial Approach:
- Vectors and Their Basic Properties
- Elementary Properties of Vectors
- The Inner Product and Its Applications
- The Cross Product and Its Applications
- The Mixed Scalar Product
- Applications to Plane Geometry
- Baricentric Coordinates
- Momenta and Applications
- Applications to Solid Geometry
Readership: This is a textbook aimed for an undergraduate College Geometry class. The targeted students are primarily mathematics majors. Other students can be computer science or physics majors needing a minor in mathematics. The book can also be useful for math educators who are interested in diverse proofs of geometric problems and need inspiration for their classwork or Math clubs.
Ovidiu Calin is a professor at Eastern Michigan University and a former visiting professor at Princeton University and University of Notre Dame, as well as a Fulbright fellow. He has delivered lectures in Canada, Japan, Hong Kong, Taiwan, Kuwait, UK and Romania. He is the author/coauthor of several monographs and books such as Stochastic Geometric Analysis with Applications (World Scientific, 2023), Stochastic Calculus with Applications (World Scientific, 2015, 2021), Deep Learning Architectures (Springer 2020), Deterministic and Stochastic Topics in Computational Finance (World Scientific, 2016), Geometric Modeling in Probability and Statistics (Springer, 2014), Heat Kernels for Elliptic and Sub-elliptic Operators (Birkhauser, 2010), Sub-Riemannian Geometry (Cambridge Press, 2009), Geometric Analysis on the Heisenberg Group and Its Generalizations (AMS/IP, 2007), Geometric Mechanics on Riemannian Manifolds (Birkhauser, 2004) as well as over 60 research papers.
