This field of geometry will introduce the use of 'coordinates', which makes it easier to apply our skills in algebra to understand problems in geometry. This book will be looking at how a coordinate system helps our understanding of Euclidean spaces, though this 'Cartesian philosophy' is not just limited to a specific type of space.

Many call this field 'analytic geometry', but 'analytic' usually relates to techniques from a field called 'mathematical analysis' which is completely unrelated to this book.


1. Simple coordinate systems
- coordinates
- cartesian coordinate system
- polar coordinate system

- shoelace formula

2. Cartesian tigonomety
- unit circle





\part{Advanced}
3. Advanced coordinate systems (assuming familiarity with euclidean geometry and linear algebra)

The first chapter of this book aims to provide begginers with an idea of a coordinate system and it can be used to connect geometry to other areas of mathematics. This chapter assumes familiarity with linear algebra and Euclidean geometry to gain an enriched understanding of the depth of coordinate systems. We will have to back track a bit and define everything rigorously (as true mathematicians do).

- coordinates ( covered intuitively previously, cover rigorously for the first time)
f be a surjective function so that $f(x(p))=p$ where $x(p)$ is the coordinate of p where p is a point in space


- orthogonal coordinate system
- curvilinear coordinate system

- \mathbb{R}^2 cartesian coordinate system
- polar coordinate system
- \mathbb{R}^3 cartesian coordinate system
- cylindrical coordinate system
- spherical coordinate system

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- orthonormal basis ( should be covered in linear algebra)
- orthonormal frame
- coordinate transform
- change of basis
- orthonormal frame of coordinate system
- orthonormal frame of trajectory
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