Cartography of Space
Coordinate Geometry & Analytic Proof
The Human Problem
Geometry without coordinates describes shapes. Coordinate geometry lets us describe those shapes algebraically. By giving every point a mathematical address (x, y), a line becomes an equation, and a geometric relationship becomes a calculation. Space itself acquires a mathematical address.
POINT →
COORDINATE →
DISTANCE →
SLOPE →
LINE →
EQUATION →
GEOMETRIC RELATIONSHIP
The 50 Cartography Cells
From Grid to Analytic Proof
I. The Cartesian Grid
CELL 01-05
The Cartesian Plane
- (x, y) Ordered Pair
- X and Y Axes, Origin
- The Four Quadrants
CELL 06-08
Distance & Midpoint
- d = √((x₂-x₁)² + (y₂-y₁)²)
- Midpoint: Average of coords
- Pythagoras translated
CELL 09-13
Section Formula & Area
- Internal/External Division
- Centroid Coordinates
- Area of ∆ & Collinearity
II. Slope & Straight Lines
CELL 14-19
The Idea of Slope
- m = (y₂-y₁)/(x₂-x₁)
- Rise over Run
- m = tan θ
- Horizontal & Vertical lines
CELL 20-27
Equation of a Line
y = mx + c
y - y₁ = m(x - x₁)
x/a + y/b = 1
Ax + By + C = 0
- Slope-Intercept Form
- Point-Slope & Two-Point
- General & Normal Forms
CELL 28-32
Line Relationships
- Parallel: m₁ = m₂
- Perpendicular: m₁m₂ = -1
- Distance from Point to Line
III. Analytic Geometry
CELL 33-36
Locus & Graphing
- Locus: Condition → Equation
- Equation ↔ Geometric Object
- Graphing from two points
CELL 37-40
Geometric Tests
- Classifying triangles/quads
- Using distance & slope
- Section as weighted average
CELL 41-44
3D Preview
- (x, y, z) Coordinates
- 3D Distance Formula
- Preview of Transformations
IV. The Cartographer's Toolkit
CELL 45-50
Systems & Analytic Proof
- Intersection of Lines
- Coordinate Modelling
- Coordinate Proofs
- The Complete Toolkit
Exit Portal
The decisive transformation is complete: GEOMETRY → ALGEBRA. We have mapped space onto a grid. Now, what happens when we look for patterns in how numbers grow, repeat, and progress infinitely? It is time to discover the engine of sequences.