Vector Domain

The Vector domain handles 2D and 3D vectors (Vec2, Vec3), the fundamental atoms of spatial math.

A Vector is just a number with a direction. It's the difference between saying "10 miles" (Scalar) and "10 miles North" (Vector).

• Position: "Where am I?" (Point)
• Velocity: "Where am I going?" (Movement)
• Force: "What's pushing me?" (Physics)

Graphics and physics use vectors for clean math (v1 + v2) and benefit from hardware acceleration (SIMD).

Overview

use aivi.vector (Vec2, Vec3)

// Define using the `v2` tag
v1 = (1.0, 2.0)v2
v2 = (3.0, 4.0)v2

// Add components parallelly
v3 = v1 + v2 // (4.0, 6.0)

Features

Vec2 = { x: Float, y: Float }
Vec3 = { x: Float, y: Float, z: Float }
Vec4 = { x: Float, y: Float, z: Float, w: Float }

Scalar = Float

Domain Definition

domain Vector over Vec2 = {
  (+) : Vec2 -> Vec2 -> Vec2
  (+) v1 v2 = { x: v1.x + v2.x, y: v1.y + v2.y }
  
  (-) : Vec2 -> Vec2 -> Vec2
  (-) v1 v2 = { x: v1.x - v2.x, y: v1.y - v2.y }
  
  (*) : Vec2 -> Scalar -> Vec2
  (*) v s = { x: v.x * s, y: v.y * s }
  
  (/) : Vec2 -> Scalar -> Vec2
  (/) v s = { x: v.x / s, y: v.y / s }
}

domain Vector over Vec3 = {
  (+) : Vec3 -> Vec3 -> Vec3
  (+) v1 v2 = { x: v1.x + v2.x, y: v1.y + v2.y, z: v1.z + v2.z }
  
  (-) : Vec3 -> Vec3 -> Vec3
  (-) v1 v2 = { x: v1.x - v2.x, y: v1.y - v2.y, z: v1.z - v2.z }
  
  (*) : Vec3 -> Scalar -> Vec3
  (*) v s = { x: v.x * s, y: v.y * s, z: v.z * s }
  
  (/) : Vec3 -> Scalar -> Vec3
  (/) v s = { x: v.x / s, y: v.y / s, z: v.z / s }
}

Helper Functions

Function	Explanation
magnitude v Vec2 -> Float	Returns the Euclidean length of v.
normalize v Vec2 -> Vec2	Returns a unit vector in the direction of v.
dot a b Vec2 -> Vec2 -> Float	Returns the dot product of a and b.
cross a b Vec3 -> Vec3 -> Vec3	Returns the 3D cross product orthogonal to a and b.

Usage Examples

use aivi.vector

position = { x: 10.0, y: 20.0 }
velocity = { x: 1.0, y: 0.5 }

newPos = position + velocity * 0.016  // 60fps frame
direction = normalize velocity