We have already touched on some of the core geometric objects and the facts associated with thim in our Algebra book. Here we delve into it further and then generalize.
Plane geometry is where we begin. There are many interesting geometric puzzles we could pursue, but we will try to limit ourselves to some standard ideas. Our approach is to give a sense as to why these are true, some of which will be proper proofs, some of which will not be. In our proofs section, we will highlight how it can all be made rigorous, but in the main text we want to help develp a geometric intuition. Symmetry and transformations are often key ideas here.
Constructions allow us to explore the objects without numbers. While algebra and coordinates are key to keeping our sanity, there is something interesting to see what can and cannot be done when constrained to non-measurement methods.
Much practical work with geometry involves trigonometry. We already came across trigonometric functions before. Geometry is one major application of these functions and we explore those functions. Triangles will become a treasured tool in your geometric explorations. We capstone our chapter with computations of the circumference of the Earth and computing the distance to of astronomical objects (parallax).
After lines, circles and their cousins are the next simplest obejcts to explore. These are grouped under the title of conic sections, named so as these can be represented as planar slices of cones.
How do we deal with more generic curves in the plane? We can parametrize them and travel them as a function of time. That is, we can look at them as the path of some moving object through our space. How do we write that down and what can we say about it? Calculus is required for some of these techniques.
Geometric ideas extend into higher dimensions. We will take a look of some the generalizations. In some ways, this is a precursor to our study of Linear Algebra.
We often talk about Euclidean Geometry to refer to the plane geometry we talked about. There are many other types of geometries. We start to explore some of them.