Three Dimensional Geometry and Projection Operators - Notes

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, i.e., smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra.
- The simplest examples of such smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space.
- It is the branch of mathematics that studies the geometry of curves, surfaces, and manifolds (the higher-dimensional analogs of surfaces).
- The discipline owes its name to its use of ideas and techniques from differential calculus, though the modern subject often uses algebraic and purely geometric techniques instead.
Differential geometry finds applications throughout mathematics and the natural sciences.

What is the use of Euclidean geometry?
- Majorly used in the field of architecture to build a variety of structures and buildings; designing is a huge application.
- In surveying, it is used to do the levelling of the ground.
Euclidean geometry is the study of geometrical shapes and figures based on different axioms and theorems.
- It is basically introduced for flat surfaces and is better explained especially for the shapes of geometrical figures and planes.
- This part of geometry was employed by the Greek mathematician Euclid, who has described it in his book Elements; therefore this geometry is also called Euclid geometry.

This chapter title indicates a focus on foundational objects in 3D Euclidean space: points, line segments, vectors, rays, lines, and planes.
Subsection 1.1: Introduction
- The main objects studied in three dimensional geometry are:
- Point
- Line Segment
- Vector
- Ray
- Line
- Plane

Point
- A point is an object that has position only; it has no length, width, or thickness; it is represented by a dot (even dot has size).
Line Segment
- A straight line segment is the part of a straight line between two of its points including the two points themselves.
Vector
- A vector (or arrow) is a line segment with a specified direction; specifically, it is a directed line segment with a specified initial point and terminal point.
Ray
- The ray is the part of a straight line at a given point and extending limitless in one direction.
Line
- The line is an object that has indefinite length but no width or thickness; it is represented by the path of a piece of chalk on the blackboard or by a stretched rubber band.
- A straight line is unlimited in extent; it extends indefinitely in either direction.
- Figure 10.1.1 illustrates: straight line for a point moving in the same direction, curved line for a point moving continuously changing direction, and combinations of straight and curved lines.
- We assign the letter L to denote a line.
Plane
- A plane has infinite length and width but no thickness.
- It may be represented by a blackboard, side of a box, flat piece of paper, or by a white page in a book.
- These are representations of a plane but not a plane because they have a finite area.
- We assign the letter π to denote a plane.

In three dimensional Euclidean space, one of the most interesting types of coordinate systems is the three dimensional rectangular coordinate system; this system helps study the geometric model of the physical universe in which we live.
The three dimensions are commonly called length, width, and depth (or height), although any three directions can be chosen, and they do not lie in the same plane.
This three dimensional space will be denoted by oxed{\,\,\mathbb{R}^3\,\,}.
The positive direction of the x-axis is pointing towards the observer and the three axis forms a right-handed system; this is why the system is known as a right-handed coordinate system.
- They are usually labelled x, y, z.
- The point at which they cross is called the origin O(0,0,0).
In mathematics, the three dimensional rectangular or Cartesian coordinate system is described by means of three mutually perpendicular coordinate axes, namely the x-axis, y-axis, and z-axis.
- It is extended indefinitely in two opposite directions, one direction for positive numbers and the other for negative numbers.
- Now, we are going to represent the standard position for the three dimensional coordinate system.

Page references mentioned: Figure 12.1.2 and Figure 13.1.2 on page 19 (as visuals for the line/curve concepts).
Page 20 statement: Equivalently; through a fixed point called the origin O(0,0,0) there are mutually perpendicular straight lines called the x-axis, the y-axis, and the z-axis.

3D Euclidean space: oxed{\,\,\mathbb{R}^3\,\,}
Origin: O(0,0,0)
Axes: x, y, z with the right-handed orientation
Planes: denoted by oxed{\pi} (plane symbol)
Points, lines, line segments, rays, vectors, and planes are foundational objects in 3D geometry used to build more complex geometric models and projections.

Differential geometry uses calculus and linear algebra to study smooth manifolds; its language extends beyond simple Euclidean objects to curves, surfaces, and higher-dimensional spaces.
Euclidean geometry provides the fixed, intuitive ground for defining basic objects (points, lines, planes) used as building blocks in higher-dimensional geometry and in applications like architecture and surveying.

Understanding the rigidity and limitations of Euclidean space underpins real-world design, construction, and spatial reasoning.
The right-handed coordinate system is a convention that ensures consistency in vector operations like cross products and orientation in three-dimensional space.