Space

Point

• An infinitely small object
• Represented as an ordered triple of reals $(x,y,z)$

Space

• A collection of points
• Notated as $\{A,B,C\}$

Distance

• Function: $A,B\to \mathbb{R}|AB|$
  • Assigns a unique non negative number for any pair of points
• Axioms
  • $|AB|\ge 0$, $|AB|⟺A=B$
  • $|AB|=|BA|$
  • $|AB|\le |AC|+|CB|$ (triangle inequality)

Line Segments

• $AB=\{C\mid |AC|+|CB|=|AB|\}$
• A collection of all points where the triangle inequality is saturated (equal instead of less)
• $|AB|=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

Coordinate Axis

• $\mathcal{L}\sim \mathbb{R}$
• Every point on the line is uniquely associated with a distance from 0
• Can put three together orthogonally to get ${\mathbb{R}}^3$, a three dimensional coordinate system
  • Every point can be represented as a unique triple or reals $(a,b,c)$

Shapes

• In the form $S=\{p\mid \text{(in)equality(s)}\}$
  • The shape is at any input point that satisfies the conditions
• Common shapes
  • $Sphere=\{(x,y,z)\mid (x-a{)}^2+(y-b{)}^2+(z-c{)}^2=r^2\}$
    • An expansion of the geometric definition of a sphere (a collection of points equidistant from a center)
    • Sometimes need to complete the square for each variable to get it in that form
  • $Box=\{(x,y,z)\mid |x|=a,|y|=b,|z|=c\}$
  • Equations that only have two variables is that graph but extended infinitely in the third dimension
• Inequalities lead to solid shapes

Rigid Transformations

• Transforms of a coordinate system that preserve distance
• Includes
  • Translation: $(x,y,z)\to (x-x_0,y-y_0,z-z_0)$
  • Rotation
  • Reflection: $(x,y,z)\to ((-)x,(-)y,(-)z)$