Lesson 20 · Rotating a shape into the world

Rotating a shape into the world

A polygon is stored around its own origin, but bodies sit and spin somewhere in the world. Today you give a body an angle and transform a local vertex into world space by rotating then translating.

The goal

Add an orientation to a body and map a shape's local vertex into world space.

Start here - the target

TO DO

Scenario: Transforming a local point to world space

Givena body at position {5, 5} with angle pi/2 (a quarter turn counterclockwise)

Whenthe local vertex {1, 0} is transformed into world space

Thenthe world point is {5, 6} (within 1e-9)

Andthe local vertex {0, 1} maps to {4, 5}, and with angle 0 the vertex {1, 0} maps to {6, 5}

Background

So far bodies only had a position; a rigid body also has an orientation, an angle it is turned to. A shape’s vertices are stored in its own local space, centered on the origin and unrotated, which keeps the shape definition simple and reusable. To find where a vertex actually is in the world, you apply the body’s transform: rotate the local point by the body’s angle, then translate it by the body’s position. Order matters - rotate first, around the origin, then move.

Rotation uses the standard 2D rotation matrix, which for a quarter turn (pi/2) sends {1, 0} to {0, 1}. Because trigonometry brings in irrational values, compare rotated results with a small tolerance rather than for exact equality - a habit for every rotation test from here on. This ToWorld transform is what lets the collision code in the next chapter reason about polygons wherever they sit and however they are turned.

Make it work

// add an Angle float64 field to Body
func Rotate(v Vec2, angle float64) Vec2 {
  c, s := math.Cos(angle), math.Sin(angle)
  // standard 2D rotation matrix applied to v
  return Vec2{v.X*c - v.Y*s, v.X*s + v.Y*c}
}
// world = rotate the local point by the body's angle, then translate by position
func (b *Body) ToWorld(local Vec2) Vec2 { return Add(Rotate(local, b.Angle), b.Position) }

CheckpointDONE

A body has an orientation and can place its shape's vertices in the world. Commit and stop here.