The Conic Kingdom
Analytic Geometry, Curves & Celestial Shapes
The Human Problem
Why do these shapes keep appearing? Circles in astronomy, parabolas in projectiles, ellipses in orbits, hyperbolas in navigation. The answer is astonishing: They can all emerge by slicing a cone. We discover that equations themselves can become worlds of shape. The curve is not merely a picture; it is the visible shadow of an invisible mathematical rule.
CONE →
PLANE →
INTERSECTION →
CURVE →
EQUATION →
GEOMETRIC PROPERTIES →
REAL-WORLD MODEL
The 50 Conic Cells
From the Double Cone to Orbital Mechanics
I. Conic Foundations
CELL 01-04
Conic Foundations
- Plane-Cone Intersection
- The Double Cone
- Four Main Curves
- Degenerate Conics
II. Circles
CELL 05-07
Circle Geometry
x² + y² = r²
- Circle as a Conic
- Standard Equation
- Reading Centre & Radius
CELL 08-10
Circle Equations
- General Equation Form
- Diameter Form
- Circle as a Locus
III. Parabolas
CELL 11-14
Parabola Geometry
- Focus-Directrix Definition
- Standard Forms (y² = 4ax)
- Four Orientations
CELL 15-17
Parabola Structure
- Vertex, Axis, Latus Rectum
- Parabola as a Function
- Physical Models (Projectiles)
IV. Ellipses
CELL 18-20
Ellipse Geometry
- Ellipse as Locus (Sum=Const)
- Major & Minor Axes
- Horizontal Standard Form
CELL 21-23
Ellipse Structure
- Vertical Standard Form
- Eccentricity (e = c/a)
- Elliptical Orbits (Kepler)
V. Hyperbolas
CELL 24-26
Hyperbola Geometry
- Locus (Difference=Const)
- Horizontal Standard Form
- Vertical Standard Form
CELL 27-29
Hyperbola Structure
- Eccentricity (e > 1)
- Asymptotes
- Focal Geometry
VI. Unification & Curve Interaction
CELL 30-32
Conic Unification
e=00<e<1e=1e>1
- The Conic Family
- Focus-Directrix General Form
- Eccentricity Classification
CELL 33-35
Translation & Locus
- Translation of Conics (h, k)
- Axes & Symmetry
- Locus → Equation Mastery
CELL 36-38
Lines & Curves
- Tangents (Preview)
- Normals (Preview)
- Chords
CELL 39-41
Conic Intersections
- Secants
- Intersection of Two Conics
- Line-Circle Systems
VII. Structure & Applications
CELL 42-45
Geometric Structure
- Reflective Prop (Parabola)
- Reflective Prop (Ellipse)
- Asymptotes & Completing Square
CELL 46-49
Conic Applications
🚢
- Physics & Orbits
- Engineering & Antennas
- Navigation (Hyperbolic)
CELL 50
The Conic Kingdom
- Synthesis of the 4 curves
- Algebraic Pattern Recognition
- The Unified Geometric Vision
Exit Portal
We have discovered that equations can trace out the curves of planetary orbits and the flight of projectiles. Now, we must ask: What happens when numbers form infinite, predictable trails? It is time to discover the engine of sequences.