matrices, etc

engine/box.go

@@ -36,6 +36,22 @@ func (b Box) Size() Int3 {
 	return b.Max.Sub(b.Min)
 }
 
+// Add offsets the box by vector p.
+func (b Box) Add(p Int3) Box {
+	return Box{
+		Min: b.Min.Add(p),
+		Max: b.Max.Add(p),
+	}
+}
+
+// Sub offsets the box by (-p).
+func (b Box) Sub(p Int3) Box {
+	return Box{
+		Min: b.Min.Sub(p),
+		Max: b.Max.Sub(p),
+	}
+}
+
 // Back returns an image.Rectangle representing the back of the box, using
 // the given projection π.
 func (b Box) Back(π IntProjection) image.Rectangle {

engine/matrix.go

@@ -1,6 +1,9 @@
 package engine
 
-import "image"
+import (
+	"errors"
+	"image"
+)
 
 // IntMatrix3 implements a 3x3 integer matrix.
 type IntMatrix3 [3][3]int
@@ -36,7 +39,7 @@ func (a IntMatrix3) Concat(b IntMatrix3) IntMatrix3 {
 }
 
 // IntMatrix3x4 implements a 3 row, 4 column integer matrix, capable of
-//describing any integer affine transformation.
+// describing any integer affine transformation.
 type IntMatrix3x4 [3][4]int
 
 // Apply applies the matrix to a vector to obtain a transformed vector.
@@ -48,6 +51,21 @@ func (a IntMatrix3x4) Apply(v Int3) Int3 {
 	}
 }
 
+// ToRatMatrix3 returns the 3x3 submatrix as a rational matrix equivalent.
+func (a IntMatrix3x4) ToRatMatrix3() RatMatrix3 {
+	return RatMatrix3{
+		0: [3]Rat{{a[0][0], 1}, {a[0][1], 1}, {a[0][2], 1}},
+		1: [3]Rat{{a[1][0], 1}, {a[1][1], 1}, {a[1][2], 1}},
+		2: [3]Rat{{a[2][0], 1}, {a[2][1], 1}, {a[2][2], 1}},
+	}
+}
+
+// Translation returns the translation component of the matrix (last column)
+// i.e. what you get if you Apply the matrix to the zero vector.
+func (a IntMatrix3x4) Translation() Int3 {
+	return Int3{X: a[0][3], Y: a[1][3], Z: a[2][3]}
+}
+
 // IntMatrix2x3 implements a 2 row, 3 column matrix (as two row vectors).
 type IntMatrix2x3 struct{ X, Y Int3 }
 
@@ -58,3 +76,53 @@ func (a IntMatrix2x3) Apply(v Int3) image.Point {
 		Y: v.Dot(a.Y),
 	}
 }
+
+// RatMatrix3 implements a 3x3 matrix with rational number entries.
+type RatMatrix3 [3][3]Rat
+
+// IdentityRatMatrix3x4 is the identity matrix for RatMatrix3x4.
+var IdentityRatMatrix3 = RatMatrix3{
+	0: [3]Rat{0: {1, 1}},
+	1: [3]Rat{1: {1, 1}},
+	2: [3]Rat{2: {1, 1}},
+}
+
+// IntApply applies the matrix to the integer vector v. Any remainder is lost.
+func (a RatMatrix3) IntApply(v Int3) Int3 {
+	x, y, z := IntRat(v.X), IntRat(v.Y), IntRat(v.Z)
+	return Int3{
+		X: x.Mul(a[0][0]).Add(y.Mul(a[0][1])).Add(z.Mul(a[0][2])).Int(),
+		Y: x.Mul(a[1][0]).Add(y.Mul(a[1][1])).Add(z.Mul(a[1][2])).Int(),
+		Z: x.Mul(a[2][0]).Add(y.Mul(a[2][1])).Add(z.Mul(a[2][2])).Int(),
+	}
+}
+
+// Inverse returns the inverse of the matrix.
+func (a RatMatrix3) Inverse() (RatMatrix3, error) {
+	adj := RatMatrix3{
+		0: [3]Rat{
+			0: a[1][1].Mul(a[2][2]).Sub(a[1][2].Mul(a[2][1])),
+		},
+		1: [3]Rat{
+			0: a[1][0].Mul(a[2][2]).Sub(a[1][2].Mul(a[2][0])).Neg(),
+		},
+		2: [3]Rat{
+			0: a[1][0].Mul(a[2][1]).Sub(a[1][1].Mul(a[2][0])),
+		},
+		// other columns after determinant...
+	}
+	det := a[0][0].Mul(adj[0][0]).Add(a[0][1].Mul(adj[1][0])).Add(a[0][2].Mul(adj[2][0]))
+	if det.N == 0 {
+		return RatMatrix3{}, errors.New("matrix is singular")
+	}
+	adj[0][0] = adj[0][0].Div(det)
+	adj[1][0] = adj[1][0].Div(det)
+	adj[2][0] = adj[2][0].Div(det)
+	adj[0][1] = a[0][1].Mul(a[2][2]).Sub(a[0][2].Mul(a[2][1])).Neg().Div(det)
+	adj[0][2] = a[0][1].Mul(a[1][2]).Sub(a[0][2].Mul(a[1][1])).Div(det)
+	adj[1][1] = a[0][0].Mul(a[2][2]).Sub(a[0][2].Mul(a[2][0])).Div(det)
+	adj[1][2] = a[0][0].Mul(a[1][2]).Sub(a[0][2].Mul(a[1][0])).Neg().Div(det)
+	adj[2][1] = a[0][0].Mul(a[2][1]).Sub(a[0][1].Mul(a[2][0])).Neg().Div(det)
+	adj[2][2] = a[0][0].Mul(a[1][1]).Sub(a[0][1].Mul(a[1][0])).Div(det)
+	return adj, nil
+}

engine/prisms.go

@@ -2,7 +2,9 @@ package engine
 
 import (
 	"encoding/gob"
+	"fmt"
 	"image"
+	"log"
 
 	"github.com/hajimehoshi/ebiten/v2"
 )
@@ -28,31 +30,57 @@ func init() {
 	gob.Register(&Prism{})
 }
 
-// PrismMap
+// PrismMap is a generalised tilemap/wallmap/etc.
 type PrismMap struct {
 	ID
 	Disabled
 	Hidden
-
+	Ersatz        bool
 	Map           map[Int3]*Prism // pos -> prism
 	DrawOrderBias image.Point     // dot with pos.XY() = bias value
 	DrawOffset    image.Point     // offset applies to whole map
-	PosToWorld    IntMatrix3x4    // p.pos -> voxelspace
-	PrismSize     Int3            // in voxelspace
+	PosToWorld    IntMatrix3x4    // p.pos -> world voxelspace
+	PrismSize     Int3            // in world voxelspace units
 	Sheet         Sheet
 
-	game *Game
+	game      *Game
+	pwinverse RatMatrix3
 }
 
 func (m *PrismMap) CollidesWith(b Box) bool {
-	// Back corner of a prism p is:
-	// m.PrismPos.Apply(p.pos)
+	if m.Ersatz {
+		return false
+	}
+	// To find the prisms need to test, we need to invert PosToWorld.
 
+	// Step 1: subtract whatever the translation component of PosToWorld is,
+	// reducing the rest of the problem to the 3x3 submatrix.
+	b = b.Sub(m.PosToWorld.Translation())
+	// Step 2: invert the rest of the fucking matrix.
+	// (Spoilers: I did this already in Prepare)
+	b.Min = m.pwinverse.IntApply(b.Min)
+	b.Max = m.pwinverse.IntApply(b.Max.Sub(Int3{1, 1, 1}))
+	for k := b.Min.Z; k < b.Max.Z; k++ {
+		for j := b.Min.Y; j < b.Max.Y; j++ {
+			for i := b.Min.X; i < b.Max.X; i++ {
+				// TODO: take into account the prism shape...
+				if _, found := m.Map[Int3{i, j, k}]; found {
+					return true
+				}
+			}
+		}
+	}
 	return false
 }
 
 func (m *PrismMap) Prepare(g *Game) error {
 	m.game = g
+	pwi, err := m.PosToWorld.ToRatMatrix3().Inverse()
+	if err != nil {
+		return fmt.Errorf("inverting PosToWorld: %w", err)
+	}
+	m.pwinverse = pwi
+	log.Printf("inverted PosToWorld: %v", pwi)
 	for v, p := range m.Map {
 		p.pos = v
 		p.m = m

engine/rational.go

@@ -0,0 +1,92 @@
+package engine
+
+// Rat is a (small) rational number implementation. Overflow can happen.
+type Rat struct{ N, D int }
+
+// IntRat returns the rational representation of n.
+func IntRat(n int) Rat { return Rat{N: n, D: 1} }
+
+// Int returns r.N / r.D.
+func (r Rat) Int() int { return r.N / r.D }
+
+// Rem returns r.N % r.D.
+func (r Rat) Rem() int { return r.N % r.D }
+
+// Canon puts the rational number into reduced form.
+func (r Rat) Canon() Rat {
+	if r.D == 0 {
+		panic("division by zero")
+	}
+	if r.N == 0 {
+		r.D = 1
+		return r
+	}
+	if r.D < 0 {
+		r.N, r.D = -r.N, -r.D
+	}
+	if d := gcd(abs(r.N), r.D); d > 1 {
+		r.N /= d
+		r.D /= d
+	}
+	return r
+}
+
+// Neg returns -r.
+func (r Rat) Neg() Rat {
+	r.N = -r.N
+	return r
+}
+
+// Add returns r + q.
+func (r Rat) Add(q Rat) Rat {
+	return Rat{
+		N: r.N*q.D + q.N*r.D,
+		D: r.D * q.D,
+	}.Canon()
+}
+
+// Sub returns r - q.
+func (r Rat) Sub(q Rat) Rat {
+	return Rat{
+		N: r.N*q.D - q.N*r.D,
+		D: r.D * q.D,
+	}.Canon()
+}
+
+// Mul returns r * q.
+func (r Rat) Mul(q Rat) Rat {
+	return Rat{
+		N: r.N * q.N,
+		D: r.D * q.D,
+	}.Canon()
+}
+
+// Invert returns 1/r if it exists, otherwise panics.
+func (r Rat) Invert() Rat {
+	return Rat{N: r.D, D: r.N}.Canon()
+}
+
+// Div returns r/q.
+func (r Rat) Div(q Rat) Rat {
+	return Rat{
+		N: r.N * q.D,
+		D: r.D * q.N,
+	}.Canon()
+}
+
+func abs(n int) int {
+	if n < 0 {
+		return -n
+	}
+	return n
+}
+
+func gcd(a, b int) int {
+	if a < b {
+		return gcd(b, a)
+	}
+	for b != 0 {
+		a, b = b, a%b
+	}
+	return a
+}

main.go

@@ -140,7 +140,6 @@ func level1() *engine.Scene {
 			&engine.PrismMap{
 				ID:            "hexagons",
 				DrawOrderBias: image.Pt(0, -1), // draw higher Y after lower Y
-				DrawOffset:    image.Pt(-8, 0),
 				PosToWorld: engine.IntMatrix3x4{
 					0: [4]int{24, 0, 0, 0},
 					1: [4]int{0, 16, 0, 0},
@@ -159,9 +158,9 @@ func level1() *engine.Scene {
 					engine.Pt3(4, -1, 0): {},
 					engine.Pt3(4, 0, 0):  {},
 
-					engine.Pt3(6, 0, -4): {},
-					engine.Pt3(6, 0, -3): {},
 					engine.Pt3(6, 0, -2): {},
+					engine.Pt3(6, 0, -1): {},
+					engine.Pt3(6, 0, 0):  {},
 				},
 			},
 			&engine.Tilemap{