import hashes, math, random, strformat, strutils
export math
func between*(value, min, max: float32): bool =
## Returns true if value is between min and max or equal to them.
(value >= min) and (value <= max)
func sign*(v: float32): float32 =
## Returns the sign of a number, -1 or 1.
if v >= 0:
return 1.0
return -1.0
func quantize*(v: float32, n: float32): float32 =
## Makes v be multipe of n. Rounding to integer quantize by 1.0.
sign(v) * floor(abs(v) / n) * n
func lerp*(a: float32, b: float32, v: float32): float32 =
## Interpolates value between a and b.
## * 0 -> a
## * 1 -> b
## * 0.5 -> between a and b
a * (1.0 - v) + b * v
type Vec2* = object
## 2D vector
x*: float32
y*: float32
func vec2*(x, y: float32): Vec2 {.inline.} =
result.x = x
result.y = y
func vec2*(v: float32): Vec2 {.inline.} =
result.x = v
result.y = v
func vec2*(a: Vec2): Vec2 {.inline.} =
result.x = a.x
result.y = a.y
func `+`*(a: Vec2, b: Vec2): Vec2 {.inline.} =
result.x = a.x + b.x
result.y = a.y + b.y
func `-`*(a: Vec2, b: Vec2): Vec2 {.inline.} =
result.x = a.x - b.x
result.y = a.y - b.y
func `*`*(a: Vec2, b: float32): Vec2 {.inline.} =
result.x = a.x * b
result.y = a.y * b
func `*`*(a: float32, b: Vec2): Vec2 {.inline.} =
b * a
func `/`*(a: Vec2, b: float32): Vec2 {.inline.} =
result.x = a.x / b
result.y = a.y / b
func `+=`*(a: var Vec2, b: Vec2) {.inline.} =
a.x += b.x
a.y += b.y
func `-=`*(a: var Vec2, b: Vec2) {.inline.} =
a.x -= b.x
a.y -= b.y
func `*=`*(a: var Vec2, b: float32) {.inline.} =
a.x *= b
a.y *= b
func `/=`*(a: var Vec2, b: float32) {.inline.} =
a.x /= b
a.y /= b
func zero*(a: var Vec2) {.inline.} =
a.x = 0
a.y = 0
func `-`*(a: Vec2): Vec2 {.inline.} =
result.x = -a.x
result.y = -a.y
func hash*(a: Vec2): Hash =
hash((a.x, a.y))
func lengthSq*(a: Vec2): float32 {.inline.} =
a.x * a.x + a.y * a.y
func length*(a: Vec2): float32 {.inline.} =
math.sqrt(a.lengthSq)
func `length=`*(a: var Vec2, b: float32) {.inline.}=
a *= b / a.length
func normalize*(a: Vec2): Vec2 {.inline.} =
a / a.length
func dot*(a: Vec2, b: Vec2): float32 {.inline.} =
a.x*b.x + a.y*b.y
func dir*(at: Vec2, to: Vec2): Vec2 {.inline.} =
(at - to).normalize()
func dir*(th: float32): Vec2 {.inline.} =
vec2(cos(th), sin(th))
func dist*(at: Vec2, to: Vec2): float32 {.inline.} =
(at - to).length
func distSq*(at: Vec2, to: Vec2): float32 {.inline.} =
(at - to).lengthSq
func lerp*(a: Vec2, b: Vec2, v: float32): Vec2 {.inline.} =
a * (1.0 - v) + b * v
func quantize*(v: Vec2, n: float32): Vec2 {.inline.} =
result.x = sign(v.x) * floor(abs(v.x) / n) * n
result.y = sign(v.y) * floor(abs(v.y) / n) * n
func inRect*(v: Vec2, a: Vec2, b: Vec2): bool {.inline.} =
## Check to see if v is inside a rectange formed by a and b.
## It does not matter how a and b are arranged.
let
min = vec2(min(a.x, b.x), min(a.y, b.y))
max = vec2(max(a.x, b.x), max(a.y, b.y))
v.x > min.x and v.x < max.x and v.y > min.y and v.y < max.y
func `[]`*(a: Vec2, i: int): float32 =
assert(i == 0 or i == 1)
if i == 0:
return a.x
elif i == 1:
return a.y
func `[]=`*(a: var Vec2, i: int, b: float32) =
assert(i == 0 or i == 1)
if i == 0:
a.x = b
elif i == 1:
a.y = b
proc randVec2*(): Vec2 =
let a = rand(PI*2)
let v = rand(1.0)
vec2(cos(a)*v, sin(a)*v)
func `$`*(a: Vec2): string =
&"({a.x:.4f}, {a.y:.4f})"
func fixAngle*(angle: float32): float32 =
## Make angle be from -PI to PI radians.
var angle = angle
while angle > PI:
angle -= PI*2
while angle < -PI:
angle += PI*2
angle
func angle*(a: Vec2): float32 =
## Angle of a Vec2.
math.arctan2(a.y, a.x)
func angleBetween*(a: Vec2, b: Vec2): float32 =
## Angle between 2 Vec2.
fixAngle(math.arctan2(a.y - b.y, a.x - b.x))
func angleBetween*(a, b: float32): float32 =
## Angle between angle a and angle b.
(b - a).fixAngle
func turnAngle*(a, b, speed: float32): float32 =
## Move from angle a to angle b with step of v.
var
turn = fixAngle(b - a)
if abs(turn) < speed:
return b
elif turn > speed:
turn = speed
elif turn < -speed:
turn = -speed
a + turn
type Vec3* = object
## 3D vector
x*: float32
y*: float32
z*: float32
func vec3*(x, y, z: float32): Vec3 =
result.x = x
result.y = y
result.z = z
func vec3*(v: float32): Vec3 =
result.x = v
result.y = v
result.z = v
func vec3*(a: Vec3): Vec3 =
result.x = a.x
result.y = a.y
result.z = a.z
const X_DIR* = vec3(1.0, 0.0, 0.0)
const Y_DIR* = vec3(0.0, 1.0, 0.0)
const Z_DIR* = vec3(0.0, 0.0, 1.0)
func `+`*(a: Vec3, b: Vec3): Vec3 =
result.x = a.x + b.x
result.y = a.y + b.y
result.z = a.z + b.z
func `-`*(a: Vec3, b: Vec3): Vec3 =
result.x = a.x - b.x
result.y = a.y - b.y
result.z = a.z - b.z
func `-`*(a: Vec3): Vec3 =
result.x = -a.x
result.y = -a.y
result.z = -a.z
func `*`*(a: Vec3, b: float32): Vec3 =
result.x = a.x * b
result.y = a.y * b
result.z = a.z * b
func `*`*(a: float32, b: Vec3): Vec3 =
b * a
func `/`*(a: Vec3, b: float32): Vec3 =
result.x = a.x / b
result.y = a.y / b
result.z = a.z / b
func `/`*(a: float32, b: Vec3): Vec3 =
result.x = a / b.x
result.y = a / b.y
result.z = a / b.z
func `+=`*(a: var Vec3, b: Vec3) =
a.x += b.x
a.y += b.y
a.z += b.z
func `-=`*(a: var Vec3, b: Vec3) =
a.x -= b.x
a.y -= b.y
a.z -= b.z
func `*=`*(a: var Vec3, b: float32) =
a.x *= b
a.y *= b
a.z *= b
func `/=`*(a: var Vec3, b: float32) =
a.x /= b
a.y /= b
a.z /= b
func zero*(a: var Vec3) =
a.x = 0
a.y = 0
a.z = 0
func `-`*(a: var Vec3): Vec3 =
result.x = -a.x
result.y = -a.y
result.z = -a.z
func hash*(a: Vec3): Hash =
hash((a.x, a.y, a.z))
func lengthSq*(a: Vec3): float32 =
a.x * a.x + a.y * a.y + a.z * a.z
func length*(a: Vec3): float32 =
math.sqrt(a.lengthSq)
func `length=`*(a: var Vec3, b: float32) =
a *= b / a.length
func floor*(a: Vec3): Vec3 =
vec3(floor(a.x), floor(a.y), floor(a.z))
func round*(a: Vec3): Vec3 =
vec3(round(a.x), round(a.y), round(a.z))
func ceil*(a: Vec3): Vec3 =
vec3(ceil(a.x), ceil(a.y), ceil(a.z))
func normalize*(a: Vec3): Vec3 =
a / math.sqrt(a.x*a.x + a.y*a.y + a.z*a.z)
func cross*(a: Vec3, b: Vec3): Vec3 =
result.x = a.y*b.z - a.z*b.y
result.y = a.z*b.x - a.x*b.z
result.z = a.x*b.y - a.y*b.x
func computeNormal*(a, b, c: Vec3): Vec3 =
cross(c - b, b - a).normalize()
func dot*(a: Vec3, b: Vec3): float32 =
a.x*b.x + a.y*b.y + a.z*b.z
func dir*(at: Vec3, to: Vec3): Vec3 =
(at - to).normalize()
func dist*(at: Vec3, to: Vec3): float32 =
(at - to).length
func distSq*(at: Vec3, to: Vec3): float32 =
(at - to).lengthSq
func lerp*(a: Vec3, b: Vec3, v: float32): Vec3 =
a * (1.0 - v) + b * v
func quantize*(v: Vec3, n: float32): Vec3 =
result.x = sign(v.x) * floor(abs(v.x) / n) * n
result.y = sign(v.y) * floor(abs(v.y) / n) * n
result.z = sign(v.z) * floor(abs(v.z) / n) * n
func angleBetween*(a, b: Vec3): float32 =
var dot = dot(a, b)
dot = dot / (a.length * b.length)
arccos(dot)
func `[]`*(a: Vec3, i: int): float32 =
assert(i == 0 or i == 1 or i == 2)
if i == 0:
return a.x
elif i == 1:
return a.y
elif i == 2:
return a.z
func `[]=`*(a: var Vec3, i: int, b: float32) =
assert(i == 0 or i == 1 or i == 2)
if i == 0:
a.x = b
elif i == 1:
a.y = b
elif i == 2:
a.z = b
func xy*(a: Vec3): Vec2 =
vec2(a.x, a.y)
func xz*(a: Vec3): Vec2 =
vec2(a.x, a.z)
func yx*(a: Vec3): Vec2 =
vec2(a.y, a.x)
func yz*(a: Vec3): Vec2 =
vec2(a.y, a.z)
func zx*(a: Vec3): Vec2 =
vec2(a.y, a.x)
func zy*(a: Vec3): Vec2 =
vec2(a.z, a.y)
func almostEquals*(a, b: Vec3, precision = 1e-6): bool =
let c = a - b
abs(c.x) < precision and abs(c.y) < precision and abs(c.z) < precision
proc randVec3*(): Vec3 =
## Generates a random unit vector based on
## http://mathworld.wolfram.com/SpherePointPicking.html
let
u = rand(0.0 .. 1.0)
v = rand(0.0 .. 1.0)
th = 2 * PI * u
ph = arccos(2 * v - 1)
vec3(
cos(th) * sin(ph),
sin(th) * sin(ph),
cos(ph)
)
func `$`*(a: Vec3): string =
&"({a.x:.8f}, {a.y:.8f}, {a.z:.8f})"
type Vec4* = object
## 4D Vector.
x*: float32
y*: float32
z*: float32
w*: float32
func vec4*(x, y, z, w: float32): Vec4 =
result.x = x
result.y = y
result.z = z
result.w = w
func vec4*(v: float32): Vec4 =
result.x = v
result.y = v
result.z = v
result.w = v
func `+`*(a: Vec4, b: Vec4): Vec4 =
result.x = a.x + b.x
result.y = a.y + b.y
result.z = a.z + b.z
result.w = a.w + b.w
func `-`*(a: Vec4, b: Vec4): Vec4 =
result.x = a.x - b.x
result.y = a.y - b.y
result.z = a.z - b.z
result.w = a.w - b.w
func `-`*(a: Vec4): Vec4 =
result.x = -a.x
result.y = -a.y
result.z = -a.z
result.w = -a.w
func `*`*(a: Vec4, b: float32): Vec4 =
result.x = a.x * b
result.y = a.y * b
result.z = a.z * b
result.w = a.w * b
func `*`*(a: float32, b: Vec4): Vec4 =
b * a
func `/`*(a: Vec4, b: float32): Vec4 =
result.x = a.x / b
result.y = a.y / b
result.z = a.z / b
result.w = a.w / b
func `/`*(a: float32, b: Vec4): Vec4 =
result.x = a / b.x
result.y = a / b.y
result.z = a / b.z
result.w = a / b.w
func `+=`*(a: var Vec4, b: Vec4) =
a.x += b.x
a.y += b.y
a.z += b.z
a.w += b.w
func `-=`*(a: var Vec4, b: Vec4) =
a.x -= b.x
a.y -= b.y
a.z -= b.z
a.w -= b.w
func `*=`*(a: var Vec4, b: float32) =
a.x *= b
a.y *= b
a.z *= b
a.w *= b
func `/=`*(a: var Vec4, b: float32) =
a.x /= b
a.y /= b
a.z /= b
a.w /= b
func zero*(a: var Vec4) =
a.x = 0
a.y = 0
a.z = 0
a.w = 0
func hash*(a: Vec4): Hash =
hash((a.x, a.y, a.z, a.w))
func `[]`*(a: Vec4, i: int): float32 =
assert(i == 0 or i == 1 or i == 2)
if i == 0:
return a.x
elif i == 1:
return a.y
elif i == 2:
return a.z
elif i == 3:
return a.w
func `[]=`*(a: var Vec4, i: int, b: float32) =
assert(i == 0 or i == 1 or i == 2)
if i == 0:
a.x = b
elif i == 1:
a.y = b
elif i == 2:
a.z = b
elif i == 3:
a.w = b
func lerp*(a: Vec4, b: Vec4, v: float32): Vec4 =
a * (1.0 - v) + b * v
func xyz*(a: Vec4): Vec3 =
vec3(a.x, a.y, a.z)
func `$`*(a: Vec4): string =
&"({a.x:.8f}, {a.y:.8f}, {a.z:.8f}, {a.w:.8f})"
func vec3*(a: Vec2, z = 0.0): Vec3 =
vec3(a.x, a.y, z)
func vec4*(a: Vec3, w = 0.0): Vec4 =
vec4(a.x, a.y, a.z, w)
func vec4*(a: Vec2, z = 0.0, w = 0.0): Vec4 =
vec4(a.x, a.y, z, w)
type Mat3* = array[9, float32] ## 3x3 Matrix
template `[]`*(a: Mat3, i, j: int): float32 = a[i * 3 + j ]
template `[]=`*(a: Mat3, i, j: int, v: float32) = a[i * 3 + j] = v
func mat3*(a, b, c, d, e, f, g, h, i: float32): Mat3 =
result[0] = a
result[1] = b
result[2] = c
result[3] = d
result[4] = e
result[5] = f
result[6] = g
result[7] = h
result[8] = i
func mat3*(a: Mat3): Mat3 =
a
func identity*(a: var Mat3) =
a[0] = 1
a[1] = 0
a[2] = 0
a[3] = 0
a[4] = 1
a[5] = 0
a[6] = 0
a[7] = 0
a[8] = 1
func mat3*(): Mat3 =
result.identity()
func transpose*(a: Mat3): Mat3 =
result[0] = a[0]
result[1] = a[3]
result[2] = a[6]
result[3] = a[1]
result[4] = a[4]
result[5] = a[7]
result[6] = a[2]
result[7] = a[5]
result[8] = a[8]
func `$`*(a: Mat3): string =
&"""[{a[0]:.4f}, {a[1]:.4f}, {a[2]:.4f},
{a[3]:.4f}, {a[4]:.4f}, {a[5]:.4f},
{a[6]:.4f}, {a[7]:.4f}, {a[8]:.4f}]"""
func `*`*(a: Mat3, b: Mat3): Mat3 =
result[0, 0] += b[0, 0] * a[0, 0] + b[0, 1] * a[1, 0] + b[0, 2] * a[2, 0]
result[0, 1] += b[0, 0] * a[0, 1] + b[0, 1] * a[1, 1] + b[0, 2] * a[2, 1]
result[0, 2] += b[0, 0] * a[0, 2] + b[0, 1] * a[1, 2] + b[0, 2] * a[2, 2]
result[1, 0] += b[1, 0] * a[0, 0] + b[1, 1] * a[1, 0] + b[1, 2] * a[2, 0]
result[1, 1] += b[1, 0] * a[0, 1] + b[1, 1] * a[1, 1] + b[1, 2] * a[2, 1]
result[1, 2] += b[1, 0] * a[0, 2] + b[1, 1] * a[1, 2] + b[1, 2] * a[2, 2]
result[2, 0] += b[2, 0] * a[0, 0] + b[2, 1] * a[1, 0] + b[2, 2] * a[2, 0]
result[2, 1] += b[2, 0] * a[0, 1] + b[2, 1] * a[1, 1] + b[2, 2] * a[2, 1]
result[2, 2] += b[2, 0] * a[0, 2] + b[2, 1] * a[1, 2] + b[2, 2] * a[2, 2]
func scale*(a: Mat3, v: Vec2): Mat3 =
result[0] = v.x * a[0]
result[1] = v.x * a[1]
result[2] = v.x * a[2]
result[3] = v.y * a[3]
result[4] = v.y * a[4]
result[5] = v.y * a[5]
result[6] = a[6]
result[7] = a[7]
result[8] = a[8]
func scale*(a: Mat3, v: Vec3): Mat3 =
result[0] = v.x * a[0]
result[1] = v.x * a[1]
result[2] = v.x * a[2]
result[3] = v.y * a[3]
result[4] = v.y * a[4]
result[5] = v.y * a[5]
result[6] = v.z * a[6]
result[7] = v.z * a[7]
result[8] = v.z * a[8]
func translate*(v: Vec2): Mat3 =
result[0, 0] = 1
result[1, 1] = 1
result[2, 0] = v.x
result[2, 1] = v.y
result[2, 2] = 1
func scale*(v: Vec2): Mat3 =
result[0, 0] = v.x
result[1, 1] = v.y
result[2, 2] = 1
func rotationMat3*(angle: float32): Mat3 =
# Create a matrix from an angle.
let
sin = sin(angle)
cos = cos(angle)
result[0, 0] = cos
result[0, 1] = -sin
result[0, 2] = 0
result[1, 0] = sin
result[1, 1] = cos
result[1, 2] = 0
result[2, 0] = 0
result[2, 1] = 0
result[2, 2] = 1
func rotate*(a: Mat3, angle: float32): Mat3 =
# Rotates a matrix by an angle.
a * rotationMat3(angle)
func `*`*(a: Mat3, b: Vec2): Vec2 =
result.x = a[0, 0]*b.x + a[1, 0]*b.y + a[2, 0]
result.y = a[0, 1]*b.x + a[1, 1]*b.y + a[2, 1]
func `*`*(a: Mat3, b: Vec3): Vec3 =
result.x = a[0, 0]*b.x + a[1, 0]*b.y + a[2, 0]*b.z
result.y = a[0, 1]*b.x + a[1, 1]*b.y + a[2, 1]*b.z
result.z = a[0, 2]*b.x + a[1, 2]*b.y + a[2, 2]*b.z
func inverse*(a: Mat3): Mat3 =
let determinant = (
a[0, 0] * (a[1, 1] * a[2, 2] - a[2, 1] * a[1, 2]) -
a[0, 1] * (a[1, 0] * a[2, 2] - a[1, 2] * a[2, 0]) +
a[0, 2] * (a[1, 0] * a[2, 1] - a[1, 1] * a[2, 0])
)
let invDet = 1 / determinant
result[0, 0] = (a[1, 1] * a[2, 2] - a[2, 1] * a[1, 2]) * invDet
result[0, 1] = -(a[0, 1] * a[2, 2] - a[0, 2] * a[2, 1]) * invDet
result[0, 2] = (a[0, 1] * a[1, 2] - a[0, 2] * a[1, 1]) * invDet
result[1, 0] = -(a[1, 0] * a[2, 2] - a[1, 2] * a[2, 0]) * invDet
result[1, 1] = (a[0, 0] * a[2, 2] - a[0, 2] * a[2, 0]) * invDet
result[1, 2] = -(a[0, 0] * a[1, 2] - a[1, 0] * a[0, 2]) * invDet
result[2, 0] = (a[1, 0] * a[2, 1] - a[2, 0] * a[1, 1]) * invDet
result[2, 1] = -(a[0, 0] * a[2, 1] - a[2, 0] * a[0, 1]) * invDet
result[2, 2] = (a[0, 0] * a[1, 1] - a[1, 0] * a[0, 1]) * invDet
type Mat4* = array[16, float32] ## 4x4 Matrix - OpenGL row order
template `[]`*(a: Mat4, i, j: int): float32 = a[i * 4 + j ]
template `[]=`*(a: Mat4, i, j: int, v: float32) = a[i * 4 + j] = v
func mat4*(v0, v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11, v12, v13,
v14, v15: float32): Mat4 =
result[0] = v0
result[1] = v1
result[2] = v2
result[3] = v3
result[4] = v4
result[5] = v5
result[6] = v6
result[7] = v7
result[8] = v8
result[9] = v9
result[10] = v10
result[11] = v11
result[12] = v12
result[13] = v13
result[14] = v14
result[15] = v15
func mat4*(a: Mat4): Mat4 =
a
func identity*(): Mat4 =
result[0] = 1
result[1] = 0
result[2] = 0
result[3] = 0
result[4] = 0
result[5] = 1
result[6] = 0
result[7] = 0
result[8] = 0
result[9] = 0
result[10] = 1
result[11] = 0
result[12] = 0
result[13] = 0
result[14] = 0
result[15] = 1
func mat4*(): Mat4 =
identity()
func transpose*(a: Mat4): Mat4 =
result[0] = a[0]
result[1] = a[4]
result[2] = a[8]
result[3] = a[12]
result[4] = a[1]
result[5] = a[5]
result[6] = a[9]
result[7] = a[13]
result[8] = a[2]
result[9] = a[6]
result[10] = a[10]
result[11] = a[14]
result[12] = a[3]
result[13] = a[7]
result[14] = a[11]
result[15] = a[15]
func determinant*(a: Mat4): float32 =
var
a00 = a[0]
a01 = a[1]
a02 = a[2]
a03 = a[3]
a10 = a[4]
a11 = a[5]
a12 = a[6]
a13 = a[7]
a20 = a[8]
a21 = a[9]
a22 = a[10]
a23 = a[11]
a30 = a[12]
a31 = a[13]
a32 = a[14]
a33 = a[15]
(
a30*a21*a12*a03 - a20*a31*a12*a03 - a30*a11*a22*a03 + a10*a31*a22*a03 +
a20*a11*a32*a03 - a10*a21*a32*a03 - a30*a21*a02*a13 + a20*a31*a02*a13 +
a30*a01*a22*a13 - a00*a31*a22*a13 - a20*a01*a32*a13 + a00*a21*a32*a13 +
a30*a11*a02*a23 - a10*a31*a02*a23 - a30*a01*a12*a23 + a00*a31*a12*a23 +
a10*a01*a32*a23 - a00*a11*a32*a23 - a20*a11*a02*a33 + a10*a21*a02*a33 +
a20*a01*a12*a33 - a00*a21*a12*a33 - a10*a01*a22*a33 + a00*a11*a22*a33
)
func inverse*(a: Mat4): Mat4 =
var
a00 = a[0]
a01 = a[1]
a02 = a[2]
a03 = a[3]
a10 = a[4]
a11 = a[5]
a12 = a[6]
a13 = a[7]
a20 = a[8]
a21 = a[9]
a22 = a[10]
a23 = a[11]
a30 = a[12]
a31 = a[13]
a32 = a[14]
a33 = a[15]
var
b00 = a00*a11 - a01*a10
b01 = a00*a12 - a02*a10
b02 = a00*a13 - a03*a10
b03 = a01*a12 - a02*a11
b04 = a01*a13 - a03*a11
b05 = a02*a13 - a03*a12
b06 = a20*a31 - a21*a30
b07 = a20*a32 - a22*a30
b08 = a20*a33 - a23*a30
b09 = a21*a32 - a22*a31
b10 = a21*a33 - a23*a31
b11 = a22*a33 - a23*a32
# Calculate the inverse determinant.
var invDet = 1.0/(b00*b11 - b01*b10 + b02*b09 + b03*b08 - b04*b07 + b05*b06)
result[00] = (+a11*b11 - a12*b10 + a13*b09)*invDet
result[01] = (-a01*b11 + a02*b10 - a03*b09)*invDet
result[02] = (+a31*b05 - a32*b04 + a33*b03)*invDet
result[03] = (-a21*b05 + a22*b04 - a23*b03)*invDet
result[04] = (-a10*b11 + a12*b08 - a13*b07)*invDet
result[05] = (+a00*b11 - a02*b08 + a03*b07)*invDet
result[06] = (-a30*b05 + a32*b02 - a33*b01)*invDet
result[07] = (+a20*b05 - a22*b02 + a23*b01)*invDet
result[08] = (+a10*b10 - a11*b08 + a13*b06)*invDet
result[09] = (-a00*b10 + a01*b08 - a03*b06)*invDet
result[10] = (+a30*b04 - a31*b02 + a33*b00)*invDet
result[11] = (-a20*b04 + a21*b02 - a23*b00)*invDet
result[12] = (-a10*b09 + a11*b07 - a12*b06)*invDet
result[13] = (+a00*b09 - a01*b07 + a02*b06)*invDet
result[14] = (-a30*b03 + a31*b01 - a32*b00)*invDet
result[15] = (+a20*b03 - a21*b01 + a22*b00)*invDet
func `*`*(a, b: Mat4): Mat4 =
var
a00 = a[0]
a01 = a[1]
a02 = a[2]
a03 = a[3]
a10 = a[4]
a11 = a[5]
a12 = a[6]
a13 = a[7]
a20 = a[8]
a21 = a[9]
a22 = a[10]
a23 = a[11]
a30 = a[12]
a31 = a[13]
a32 = a[14]
a33 = a[15]
var
b00 = b[0]
b01 = b[1]
b02 = b[2]
b03 = b[3]
b10 = b[4]
b11 = b[5]
b12 = b[6]
b13 = b[7]
b20 = b[8]
b21 = b[9]
b22 = b[10]
b23 = b[11]
b30 = b[12]
b31 = b[13]
b32 = b[14]
b33 = b[15]
result[00] = b00*a00 + b01*a10 + b02*a20 + b03*a30
result[01] = b00*a01 + b01*a11 + b02*a21 + b03*a31
result[02] = b00*a02 + b01*a12 + b02*a22 + b03*a32
result[03] = b00*a03 + b01*a13 + b02*a23 + b03*a33
result[04] = b10*a00 + b11*a10 + b12*a20 + b13*a30
result[05] = b10*a01 + b11*a11 + b12*a21 + b13*a31
result[06] = b10*a02 + b11*a12 + b12*a22 + b13*a32
result[07] = b10*a03 + b11*a13 + b12*a23 + b13*a33
result[08] = b20*a00 + b21*a10 + b22*a20 + b23*a30
result[09] = b20*a01 + b21*a11 + b22*a21 + b23*a31
result[10] = b20*a02 + b21*a12 + b22*a22 + b23*a32
result[11] = b20*a03 + b21*a13 + b22*a23 + b23*a33
result[12] = b30*a00 + b31*a10 + b32*a20 + b33*a30
result[13] = b30*a01 + b31*a11 + b32*a21 + b33*a31
result[14] = b30*a02 + b31*a12 + b32*a22 + b33*a32
result[15] = b30*a03 + b31*a13 + b32*a23 + b33*a33
func `*`*(a: Mat4, b: Vec3): Vec3 =
result.x = a[0]*b.x + a[4]*b.y + a[8]*b.z + a[12]
result.y = a[1]*b.x + a[5]*b.y + a[9]*b.z + a[13]
result.z = a[2]*b.x + a[6]*b.y + a[10]*b.z + a[14]
func `*`*(a: Mat4, b: Vec4): Vec4 =
result.x = a[0]*b.x + a[4]*b.y + a[8]*b.z + a[12]*b.w
result.y = a[1]*b.x + a[5]*b.y + a[9]*b.z + a[13]*b.w
result.z = a[2]*b.x + a[6]*b.y + a[10]*b.z + a[14]*b.w
result.w = a[3]*b.x + a[7]*b.y + a[11]*b.z + a[15]*b.w
func right*(a: Mat4): Vec3 =
result.x = a[0]
result.y = a[1]
result.z = a[2]
func `right=`*(a: var Mat4, b: Vec3) =
a[0] = b.x
a[1] = b.y
a[2] = b.z
func up*(a: Mat4): Vec3 =
result.x = a[4]
result.y = a[5]
result.z = a[6]
func `up=`*(a: var Mat4, b: Vec3) =
a[4] = b.x
a[5] = b.y
a[6] = b.z
func forward*(a: Mat4): Vec3 =
result.x = a[8]
result.y = a[9]
result.z = a[10]
func `forward=`*(a: var Mat4, b: Vec3) =
a[8] = b.x
a[9] = b.y
a[10] = b.z
func pos*(a: Mat4): Vec3 =
result.x = a[12]
result.y = a[13]
result.z = a[14]
func `pos=`*(a: var Mat4, b: Vec3) =
a[12] = b.x
a[13] = b.y
a[14] = b.z
func rotationOnly*(a: Mat4): Mat4 =
result = a
result.pos = vec3(0, 0, 0)
func dist*(a, b: Mat4): float32 =
var
x = a[12] - b[12]
y = a[13] - b[13]
z = a[14] - b[14]
sqrt(x*x + y*y + z*z)
#[
func translate*(a: Mat4, v: Vec3): Mat4 =
var
a00 = a[0]
a01 = a[1]
a02 = a[2]
a03 = a[3]
a10 = a[4]
a11 = a[5]
a12 = a[6]
a13 = a[7]
a20 = a[8]
a21 = a[9]
a22 = a[10]
a23 = a[11]
result[0] = a00
result[1] = a01
result[2] = a02
result[3] = a03
result[4] = a10
result[5] = a11
result[6] = a12
result[7] = a13
result[8] = a20
result[9] = a21
result[10] = a22
result[11] = a23
result[12] = a00*v.x + a10*v.y + a20*v.z + a[12]
result[13] = a01*v.x + a11*v.y + a21*v.z + a[13]
result[14] = a02*v.x + a12*v.y + a22*v.z + a[14]
result[15] = a03*v.x + a13*v.y + a23*v.z + a[15]
]#
func translate*(v: Vec3): Mat4 =
result[0] = 1
result[5] = 1
result[10] = 1
result[15] = 1
result[12] = v.x
result[13] = v.y
result[14] = v.z
func scale*(v: Vec3): Mat4 =
result[0] = v.x
result[5] = v.y
result[10] = v.z
result[15] = 1
func close*(a: Mat4, b: Mat4): bool =
for i in 0..15:
if abs(a[i] - b[i]) > 0.001:
return false
true
func hrp*(m: Mat4): Vec3 =
var heading, pitch, roll: float32
if m[1] > 0.998: # singularity at north pole
heading = arctan2(m[2], m[10])
pitch = PI / 2
roll = 0
elif m[1] < -0.998: # singularity at south pole
heading = arctan2(m[2], m[10])
pitch = -PI / 2
roll = 0
else:
heading = arctan2(-m[8], m[0])
pitch = arctan2(-m[6], m[5])
roll = arcsin(m[4])
result.x = heading
result.y = pitch
result.z = roll
func frustum*(left, right, bottom, top, near, far: float32): Mat4 =
var
rl = (right - left)
tb = (top - bottom)
fn = (far - near)
result[0] = (near*2) / rl
result[1] = 0
result[2] = 0
result[3] = 0
result[4] = 0
result[5] = (near*2) / tb
result[6] = 0
result[7] = 0
result[8] = (right + left) / rl
result[9] = (top + bottom) / tb
result[10] = -(far + near) / fn
result[11] = -1
result[12] = 0
result[13] = 0
result[14] = -(far*near*2) / fn
result[15] = 0
func perspective*(fovy, aspect, near, far: float32): Mat4 =
var
top = near * tan(fovy*PI / 360.0)
right = top * aspect
frustum(-right, right, -top, top, near, far)
func ortho*(left, right, bottom, top, near, far: float32): Mat4 =
var
rl = (right - left)
tb = (top - bottom)
fn = (far - near)
result[0] = 2 / rl
result[1] = 0
result[2] = 0
result[3] = 0
result[4] = 0
result[5] = 2 / tb
result[6] = 0
result[7] = 0
result[8] = 0
result[9] = 0
result[10] = -2 / fn
result[11] = 0
result[12] = -(left + right) / rl
result[13] = -(top + bottom) / tb
result[14] = -(far + near) / fn
result[15] = 1
func lookAt*(eye, center, up: Vec3): Mat4 =
var
eyex = eye[0]
eyey = eye[1]
eyez = eye[2]
upx = up[0]
upy = up[1]
upz = up[2]
centerx = center[0]
centery = center[1]
centerz = center[2]
if eyex == centerx and eyey == centery and eyez == centerz:
return identity()
var
# vec3.direction(eye, center, z)
z0 = eyex - center[0]
z1 = eyey - center[1]
z2 = eyez - center[2]
# normalize (no check needed for 0 because of early return)
var len = 1/sqrt(z0*z0 + z1*z1 + z2*z2)
z0 *= len
z1 *= len
z2 *= len
var
# vec3.normalize(vec3.cross(up, z, x))
x0 = upy*z2 - upz*z1
x1 = upz*z0 - upx*z2
x2 = upx*z1 - upy*z0
len = sqrt(x0*x0 + x1*x1 + x2*x2)
if len == 0:
x0 = 0
x1 = 0
x2 = 0
else:
len = 1/len
x0 *= len
x1 *= len
x2 *= len
var
# vec3.normalize(vec3.cross(z, x, y))
y0 = z1*x2 - z2*x1
y1 = z2*x0 - z0*x2
y2 = z0*x1 - z1*x0
len = sqrt(y0*y0 + y1*y1 + y2*y2)
if len == 0:
y0 = 0
y1 = 0
y2 = 0
else:
len = 1/len
y0 *= len
y1 *= len
y2 *= len
result[0] = x0
result[1] = y0
result[2] = z0
result[3] = 0
result[4] = x1
result[5] = y1
result[6] = z1
result[7] = 0
result[8] = x2
result[9] = y2
result[10] = z2
result[11] = 0
result[12] = -(x0*eyex + x1*eyey + x2*eyez)
result[13] = -(y0*eyex + y1*eyey + y2*eyez)
result[14] = -(z0*eyex + z1*eyey + z2*eyez)
result[15] = 1
func mat3*(m: Mat4): Mat3 =
## Gets rotation and translation, ignoring z coordinates.
result[0, 0] = m[0, 0]
result[0, 1] = m[0, 1]
result[0, 2] = 0
result[1, 0] = m[1, 0]
result[1, 1] = m[1, 1]
result[1, 2] = 0
result[2, 0] = m[3, 0]
result[2, 1] = m[3, 1]
result[2, 2] = 0
func mat3Rotation*(m: Mat4): Mat3 =
## Gets the rotational part of the 4x4 matrix.
result[0, 0] = m[0, 0]
result[0, 1] = m[0, 1]
result[0, 2] = m[0, 2]
result[1, 0] = m[1, 0]
result[1, 1] = m[1, 1]
result[1, 2] = m[1, 2]
result[2, 0] = m[2, 0]
result[2, 1] = m[2, 1]
result[2, 2] = m[2, 2]
func mat4*(m: Mat3): Mat4 =
## Takes a 2d Mat3 with position and converts to a 3d matrix.
result[0, 0] = m[0, 0]
result[0, 1] = m[0, 1]
result[0, 2] = 0
result[0, 3] = 0
result[1, 0] = m[1, 0]
result[1, 1] = m[1, 1]
result[1, 2] = 0
result[1, 3] = 0
result[2, 0] = 0
result[2, 1] = 0
result[2, 2] = 1
result[2, 3] = 0
result[3, 0] = m[2, 0]
result[3, 1] = m[2, 1]
result[3, 2] = 0
result[3, 3] = 1
func mat4Rotation*(m: Mat3): Mat4 =
## Gets the rotational part of the 3x3 matrix into a 4x4 matrix.
result[0, 0] = m[0, 0]
result[0, 1] = m[0, 1]
result[0, 2] = m[0, 2]
result[1, 0] = m[1, 0]
result[1, 1] = m[1, 1]
result[1, 2] = m[1, 2]
result[2, 0] = m[2, 0]
result[2, 1] = m[2, 1]
result[2, 2] = m[2, 2]
func `$`*(a: Mat4): string =
&"""[{a[0]:.5f}, {a[1]:.5f}, {a[2]:.5f}, {a[3]:.5f},
{a[4]:.5f}, {a[5]:.5f}, {a[6]:.5f}, {a[7]:.5f},
{a[8]:.5f}, {a[9]:.5f}, {a[10]:.5f}, {a[11]:.5f},
{a[12]:.5f}, {a[13]:.5f}, {a[14]:.5f}, {a[15]:.5f}]"""
type Quat* = object
x*: float32
y*: float32
z*: float32
w*: float32
func quat*(x, y, z, w: float32): Quat =
result.x = x
result.y = y
result.z = z
result.w = w
func conjugate*(q: Quat): Quat =
result.w = +q.w
result.x = -q.x
result.y = -q.y
result.z = -q.z
func length*(q: Quat): float32 =
sqrt(
q.w * q.w +
q.x * q.x +
q.y * q.y +
q.z * q.z)
func normalize*(q: Quat): Quat =
var m = q.length
result.x = q.x / m
result.y = q.y / m
result.z = q.z / m
result.w = q.w / m
func xyz*(q: Quat): Vec3 =
result.x = q.x
result.y = q.y
result.z = q.z
func `xyz=`*(q: var Quat, v: Vec3) =
q.x = v.x
q.y = v.y
q.z = v.z
func `-`*(a: var Quat): Quat =
result.x = -a.x
result.y = -a.y
result.z = -a.z
result.w = -a.w
func `+`*(a: Quat, b: Quat): Quat =
result.x = a.x + b.x
result.y = a.y + b.y
result.z = a.z + b.z
result.w = a.w + b.w
func `*`*(a, b: Quat): Quat =
## Multiply the quaternion by a quaternion.
#[
var q = quat(0,0,0,0)
q.w = dot(a.xyz, b.xyz)
var va = cross(a.xyz, b.xyz)
var vb = a.xyz * b.w
var vc = b.xyz * a.w
va = va + vb
q.xyz = va + vc
return q.normalize()
]#
result.x = a.x * b.w + a.w * b.x + a.y * b.z - a.z * b.y
result.y = a.y * b.w + a.w * b.y + a.z * b.x - a.x * b.z
result.z = a.z * b.w + a.w * b.z + a.x * b.y - a.y * b.x
result.w = a.w * b.w - a.x * b.x - a.y * b.y - a.z * b.z
func `*`*(q: Quat, v: float32): Quat =
## Multiply the quaternion by a float32.
result.x = q.x * v
result.y = q.y * v
result.z = q.z * v
result.w = q.w * v
func `*`*(q: Quat, v: Vec3): Vec3 =
## Multiply the quaternion by a vector.
var
x = v.x
y = v.y
z = v.z
qx = q.x
qy = q.y
qz = q.z
qw = q.w
ix = +qw * x + qy * z - qz * y
iy = +qw * y + qz * x - qx * z
iz = +qw * z + qx * y - qy * x
iw = -qx * x - qy * y - qz * z
result.x = ix * qw + iw * -qx + iy * -qz - iz * -qy
result.y = iy * qw + iw * -qy + iz * -qx - ix * -qz
result.z = iz * qw + iw * -qz + ix * -qy - iy * -qx
func `[]=`*(a: var Quat, i: int, b: float32) =
assert(i == 0 or i == 1 or i == 2 or i == 3)
if i == 0:
a.x = b
elif i == 1:
a.y = b
elif i == 2:
a.z = b
elif i == 3:
a.w = b
func mat3*(q: Quat): Mat3 =
var xx = q.x * q.x
var xy = q.x * q.y
var xz = q.x * q.z
var xw = q.x * q.w
var yy = q.y * q.y
var yz = q.y * q.z
var yw = q.y * q.w
var zz = q.z * q.z
var zw = q.z * q.w
result[0] = 1 - 2 * (yy + zz)
result[1] = 0 + 2 * (xy - zw)
result[2] = 0 + 2 * (xz + yw)
result[3] = 0 + 2 * (xy + zw)
result[4] = 1 - 2 * (xx + zz)
result[5] = 0 + 2 * (yz - xw)
result[6] = 0 + 2 * (xz - yw)
result[7] = 0 + 2 * (yz + xw)
result[8] = 1 - 2 * (xx + yy)
func mat4*(q: Quat): Mat4 =
var xx = q.x * q.x
var xy = q.x * q.y
var xz = q.x * q.z
var xw = q.x * q.w
var yy = q.y * q.y
var yz = q.y * q.z
var yw = q.y * q.w
var zz = q.z * q.z
var zw = q.z * q.w
result[00] = 1 - 2 * (yy + zz)
result[01] = 0 + 2 * (xy - zw)
result[02] = 0 + 2 * (xz + yw)
result[04] = 0 + 2 * (xy + zw)
result[05] = 1 - 2 * (xx + zz)
result[06] = 0 + 2 * (yz - xw)
result[08] = 0 + 2 * (xz - yw)
result[09] = 0 + 2 * (yz + xw)
result[10] = 1 - 2 * (xx + yy)
result[3] = 0
result[7] = 0
result[11] = 0
result[12] = 0
result[13] = 0
result[14] = 0
result[15] = 1.0
func recifuncalSqrt*(x: float32): float32 =
1.0/sqrt(x)
proc quat*(m: Mat4): Quat =
var
m00 = m[0]
m01 = m[4]
m02 = m[8]
m10 = m[1]
m11 = m[5]
m12 = m[9]
m20 = m[2]
m21 = m[6]
m22 = m[10]
var q: Quat
var t: float32
if m22 < 0:
if m00 > m11:
t = 1 + m00 - m11 - m22
q = quat(t, m01 + m10, m20 + m02, m12 - m21)
else:
t = 1 - m00 + m11 - m22
q = quat(m01 + m10, t, m12 + m21, m20 - m02)
else:
if m00 < - m11:
t = 1 - m00 - m11 + m22
q = quat(m20 + m02, m12 + m21, t, m01 - m10)
else:
t = 1 + m00 + m11 + m22
q = quat(m12 - m21, m20 - m02, m01 - m10, t)
q = q * (0.5 / sqrt(t))
assert abs(q.length - 1.0) < 0.001
q
func fromAxisAngle*(axis: Vec3, angle: float32): Quat =
var a = axis.normalize()
var s = sin(angle / 2)
result.x = a.x * s
result.y = a.y * s
result.z = a.z * s
result.w = cos(angle / 2)
func toAxisAngle*(q: Quat, axis: var Vec3, angle: var float32) =
var cosAngle = q.w
angle = arccos(cosAngle) * 2.0
var sinAngle = sqrt(1.0 - cosAngle * cosAngle)
if abs(sinAngle) < 0.0005:
sinAngle = 1.0
axis.x = q.x / sinAngle
axis.y = q.y / sinAngle
axis.z = q.z / sinAngle
func quat*(heading, pitch, roll: float32): Quat =
var t0 = cos(heading * 0.5)
var t1 = sin(heading * 0.5)
var t2 = cos(roll * 0.5)
var t3 = sin(roll * 0.5)
var t4 = cos(pitch * 0.5)
var t5 = sin(pitch * 0.5)
result.w = t0 * t2 * t4 + t1 * t3 * t5
result.x = t0 * t3 * t4 - t1 * t2 * t5
result.y = t0 * t2 * t5 + t1 * t3 * t4
result.z = t1 * t2 * t4 - t0 * t3 * t5
func quat*(hpr: Vec3): Quat =
quat(hpr.x, hpr.y, hpr.z)
func hrp*(q: Quat): Vec3 =
var ysqr = q.y * q.y
# roll
var t0 = +2.0 * (q.w * q.x + q.y * q.z)
var t1 = +1.0 - 2.0 * (q.x * q.x + ysqr)
result.z = arctan2(t0, t1)
# pitch
var t2 = +2.0 * (q.w * q.y - q.z * q.x)
if t2 > 1.0:
t2 = 1.0
if t2 < -1.0:
t2 = -1.0
result.y = arcsin(t2)
# heading
var t3 = +2.0 * (q.w * q.z + q.x * q.y)
var t4 = +1.0 - 2.0 * (ysqr + q.z * q.z)
result.x = arctan2(t3, t4)
func dot*(a: Quat, b: Quat): float32 =
a.x*b.x + a.y*b.y + a.z*b.z + a.w*b.w
func nlerp*(a: Quat, b: Quat, v: float32): Quat =
if dot(a, b) < 0:
var c = a
(-c * (1.0 - v) + b * v).normalize()
else:
(a * (1.0 - v) + b * v).normalize()
func `$`*(a: Quat): string =
&"q({a.x:.8f}, {a.y:.8f}, {a.z:.8f}, {a.w:.8f})"
func rotate*(angle: float32, axis: Vec3): Mat4 =
fromAxisAngle(axis, angle).mat4()
func rotateX*(angle: float32): Mat4 =
rotate(angle, vec3(1, 0, 0))
func rotateY*(angle: float32): Mat4 =
rotate(angle, vec3(0, 1, 0))
func rotateZ*(angle: float32): Mat4 =
rotate(angle, vec3(0, 0, 1))
func scaleMat*(scale: Vec3): Mat4 =
result[0] = scale.x
result[5] = scale.y
result[10] = scale.z
result[15] = 1.0
func scaleMat*(scale: float32): Mat4 =
scaleMat(vec3(scale, scale, scale))
type Rect* = object
x*: float32
y*: float32
w*: float32
h*: float32
func rect*(x, y, w, h: float32): Rect =
result.x = x
result.y = y
result.w = w
result.h = h
func rect*(pos, size: Vec2): Rect =
result.x = pos.x
result.y = pos.y
result.w = size.x
result.h = size.y
func xy*(rect: Rect): Vec2 =
## Gets the xy as a Vec2.
vec2(rect.x, rect.y)
func `xy=`*(rect: var Rect, v: Vec2) =
## Sets the xy from Vec2.
rect.x = v.x
rect.y = v.y
func wh*(rect: Rect): Vec2 =
## Gets the wh as a Vec2.
vec2(rect.w, rect.h)
func `wh=`*(rect: var Rect, v: Vec2) =
## Sets the wh from Vec2.
rect.w = v.x
rect.h = v.y
func `*`*(r: Rect, v: float): Rect =
## * all elements of a Rect.
rect(r.x * v, r.y * v, r.w * v, r.h * v)
func `/`*(r: Rect, v: float): Rect =
## / all elements of a Rect.
rect(r.x / v, r.y / v, r.w / v, r.h / v)
func `+`*(a, b: Rect): Rect =
## Add two boxes together.
result.x = a.x + b.x
result.y = a.y + b.y
result.w = a.w
result.h = a.h
func `$`*(a: Rect): string =
&"({a.x}, {a.y}: {a.w} x {a.h})"
func inside*(pos: Vec2, rect: Rect): bool =
## Checks if pos is inside rect.
(rect.x <= pos.x and pos.x <= rect.x + rect.w) and (
rect.y <= pos.y and pos.y <= rect.y + rect.h)
func overlap*(a, b: Rect): bool =
## Returns true if box a overlaps box b.
let
xOverlap = between(a.x, b.x, b.x + b.w) or between(b.x, a.x, a.x + a.w)
yOverlap = between(a.y, b.y, b.y + b.h) or between(b.y, a.y, a.y + a.h)
xOverlap and yOverlap
proc `or`*(a, b: Rect): Rect =
## Union of two rectangles.
result.x = min(a.x, b.x)
result.y = min(a.y, b.y)
result.w = max(a.x + a.w, b.x + b.w) - result.x
result.h = max(a.y + a.h, b.y + b.h) - result.y
proc `and`*(a, b: Rect): Rect =
## Intersection of two rectangles.
result.x = max(a.x, b.x)
result.y = max(a.y, b.y)
result.w = min(a.x + a.w, b.x + b.w) - result.x
result.h = min(a.y + a.h, b.y + b.h) - result.y