Axon 01: Mathematical Foundations

Welcome to the Mathematical Foundations Axon. This track bridges high-school mathematics into applied robotics and modern machine learning through first-principles derivations, working code, and interactive sandboxes.

Every result is derived rather than asserted. Code is written in the language of whichever field the concept serves — Java and WPILib for the robotics concepts, Python and PyTorch for the machine learning ones, and both where a concept genuinely feeds each.

Modules in this Axon

1. Geometry for Robotics

• Concept 01: Coordinates, Poses & Pythagorean Distance
• Concept 02: Lines, Segments & Intersections
• Concept 03: Linear Interpolation, Lookup Tables & Blending
• Concept 04: Bounding Boxes, Overlap & Collision
• Concept 05: Polygons, Areas & Field Zones

2. Trigonometry & Angles

• Concept 01: The Unit Circle & Trigonometric Ratios
• Concept 02: Rotating a Vector
• Concept 03: Coordinate Frames (Field, Robot & Camera)
• Concept 04: Inverse Trig & 4-Quadrant Heading with atan2
• Concept 05: Angle Wrapping & Shortest Angular Distance
• Concept 06: Law of Sines, Law of Cosines & Two-Link Arms
• Concept 07: 3D Rotations, Gimbal Lock & Quaternions

3. Linear Algebra & Matrix Transformations

• Concept 01: Vectors, Displacements & Scalar Scaling
• Concept 02: Dot Products, Projections & Alignment
• Concept 03: Matrices as Coordinate Transformers
• Concept 04: Determinants, Inverses & Singularity

4. Calculus, Motion & Optimization

• Concept 01: Rates of Change & Derivatives
• Concept 02: Acceleration, Jerk & S-Curves
• Concept 03: Accumulation, Area & Numerical Integration
• Concept 04: Multivariable Gradients & Hill Climbing

5. Probability & Uncertainty

• Concept 01: Sensor Noise & Normal Distributions
• Concept 02: Bayes’ Rule & 1D Sensor Fusion
• Concept 03: Discrete Distributions & Softmax
• Concept 04: Expected Value & Decision Making