Functions for mathematical operations on vectors, matrices and quaternions.
vmath.vector3
and vmath.vector4
) supports addition and subtraction
with vectors of the same type. Vectors can be negated and multiplied (scaled) or divided by numbers.vmath.quat
) supports multiplication with other quaternions.vmath.matrix4
) can be multiplied with numbers, other matrices
and vmath.vector4
values.x
, y
and z
. Example: v.y
x
, y
, z
, and w
. Example: v.w
x
, y
, z
, and w
. Example: q.w
m00
to m33
where the first number is the row (starting from 0) and the second
number is the column. Columns can be accessed with c0
to c3
, returning a vector4
.
Example: m.m21
which is equal to m.c1.z
v[3]
Creates a vector of arbitrary size. The vector is initialized with numeric values from a table. The table values are converted to floating point values. If a value cannot be converted, a 0 is stored in that value position in the vector.
t - table of numbers
v - new vector
How to create a vector with custom data to be used for animation easing:
local values = { 0, 0.5, 0 }
local vec = vmath.vector(values)
print(vec) --> vmath.vector (size: 3)
print(vec[2]) --> 0.5
Creates a new zero vector with all components set to 0.
v - new zero vector
local vec = vmath.vector3()
pprint(vec) --> vmath.vector3(0, 0, 0)
print(vec.x) --> 0
Creates a new vector with all components set to the supplied scalar value.
n - scalar value to splat
v - new vector
local vec = vmath.vector3(1.0)
print(vec) --> vmath.vector3(1, 1, 1)
print(vec.x) --> 1
Creates a new vector with all components set to the corresponding values from the supplied vector. I.e. This function creates a copy of the given vector.
v1 - existing vector
v - new vector
local vec1 = vmath.vector3(1.0)
local vec2 = vmath.vector3(vec1)
if vec1 == vec2 then
-- yes, they are equal
print(vec2) --> vmath.vector3(1, 1, 1)
end
Creates a new vector with the components set to the supplied values.
x - x coordinate
y - y coordinate
z - z coordinate
v - new vector
local vec = vmath.vector3(1.0, 2.0, 3.0)
print(vec) --> vmath.vector3(1, 2, 3)
print(-vec) --> vmath.vector3(-1, -2, -3)
print(vec * 2) --> vmath.vector3(2, 4, 6)
print(vec + vmath.vector3(2.0)) --> vmath.vector3(3, 4, 5)
print(vec - vmath.vector3(2.0)) --> vmath.vector3(-1, 0, 1)
Creates a new zero vector with all components set to 0.
v - new zero vector
local vec = vmath.vector4()
print(vec) --> vmath.vector4(0, 0, 0, 0)
print(vec.w) --> 0
Creates a new vector with all components set to the supplied scalar value.
n - scalar value to splat
v - new vector
local vec = vmath.vector4(1.0)
print(vec) --> vmath.vector4(1, 1, 1, 1)
print(vec.w) --> 1
Creates a new vector with all components set to the corresponding values from the supplied vector. I.e. This function creates a copy of the given vector.
v1 - existing vector
v - new vector
local vect1 = vmath.vector4(1.0)
local vect2 = vmath.vector4(vec1)
if vec1 == vec2 then
-- yes, they are equal
print(vec2) --> vmath.vector4(1, 1, 1, 1)
end
Creates a new vector with the components set to the supplied values.
x - x coordinate
y - y coordinate
z - z coordinate
w - w coordinate
v - new vector
local vec = vmath.vector4(1.0, 2.0, 3.0, 4.0)
print(vec) --> vmath.vector4(1, 2, 3, 4)
print(-vec) --> vmath.vector4(-1, -2, -3, -4)
print(vec * 2) --> vmath.vector4(2, 4, 6, 8)
print(vec + vmath.vector4(2.0)) --> vmath.vector4(3, 4, 5, 6)
print(vec - vmath.vector4(2.0)) --> vmath.vector4(-1, 0, 1, 2)
Creates a new identity quaternion. The identity
quaternion is equal to:
vmath.quat(0, 0, 0, 1)
q - new identity quaternion
local quat = vmath.quat()
print(quat) --> vmath.quat(0, 0, 0, 1)
print(quat.w) --> 1
Creates a new quaternion with all components set to the corresponding values from the supplied quaternion. I.e. This function creates a copy of the given quaternion.
q1 - existing quaternion
q - new quaternion
local quat1 = vmath.quat(1, 2, 3, 4)
local quat2 = vmath.quat(quat1)
if quat1 == quat2 then
-- yes, they are equal
print(quat2) --> vmath.quat(1, 2, 3, 4)
end
Creates a new quaternion with the components set according to the supplied parameter values.
x - x coordinate
y - y coordinate
z - z coordinate
w - w coordinate
q - new quaternion
local quat = vmath.quat(1, 2, 3, 4)
print(quat) --> vmath.quat(1, 2, 3, 4)
The resulting quaternion describes the rotation that, if applied to the first vector, would rotate the first vector to the second. The two vectors must be unit vectors (of length 1). The result is undefined if the two vectors point in opposite directions
v1 - first unit vector, before rotation
v2 - second unit vector, after rotation
q - quaternion representing the rotation from first to second vector
local v1 = vmath.vector3(1, 0, 0)
local v2 = vmath.vector3(0, 1, 0)
local rot = vmath.quat_from_to(v1, v2)
print(vmath.rotate(rot, v1)) --> vmath.vector3(0, 0.99999994039536, 0)
The resulting quaternion describes a rotation of angle
radians around the axis described by the unit vector v
.
v - axis
angle - angle
q - quaternion representing the axis-angle rotation
local axis = vmath.vector3(1, 0, 0)
local rot = vmath.quat_axis_angle(axis, 3.141592653)
local vec = vmath.vector3(1, 1, 0)
print(vmath.rotate(rot, vec)) --> vmath.vector3(1, -1, -8.7422776573476e-08)
The resulting quaternion describes the rotation from the identity quaternion (no rotation) to the coordinate system as described by the given x, y and z base unit vectors.
x - x base vector
y - y base vector
z - z base vector
q - quaternion representing the rotation of the specified base vectors
-- Axis rotated 90 degrees around z.
local rot_x = vmath.vector3(0, -1, 0)
local rot_y = vmath.vector3(1, 0, 0)
local z = vmath.vector3(0, 0, 1)
local rot1 = vmath.quat_basis(rot_x, rot_y, z)
local rot2 = vmath.quat_from_to(vmath.vector3(0, 1, 0), vmath.vector3(1, 0, 0))
if rot1 == rot2 then
-- These quaternions are equal!
print(rot2) --> vmath.quat(0, 0, -0.70710676908493, 0.70710676908493)
end
The resulting quaternion describes a rotation of angle
radians around the x-axis.
angle - angle in radians around x-axis
q - quaternion representing the rotation around the x-axis
local rot = vmath.quat_rotation_x(3.141592653)
local vec = vmath.vector3(1, 1, 0)
print(vmath.rotate(rot, vec)) --> vmath.vector3(1, -1, -8.7422776573476e-08)
The resulting quaternion describes a rotation of angle
radians around the y-axis.
angle - angle in radians around y-axis
q - quaternion representing the rotation around the y-axis
local rot = vmath.quat_rotation_y(3.141592653)
local vec = vmath.vector3(1, 1, 0)
print(vmath.rotate(rot, vec)) --> vmath.vector3(-1, 1, 8.7422776573476e-08)
The resulting quaternion describes a rotation of angle
radians around the z-axis.
angle - angle in radians around z-axis
q - quaternion representing the rotation around the z-axis
local rot = vmath.quat_rotation_z(3.141592653)
local vec = vmath.vector3(1, 1, 0)
print(vmath.rotate(rot, vec)) --> vmath.vector3(-0.99999988079071, -1, 0)
The resulting identity matrix describes a transform with no translation or rotation.
m - identity matrix
local mat = vmath.matrix4()
print(mat) --> vmath.matrix4(1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1)
-- get column 0:
print(mat.c0) --> vmath.vector4(1, 0, 0, 0)
-- get the value in row 3 and column 2:
print(mat.m32) --> 0
Creates a new matrix with all components set to the corresponding values from the supplied matrix. I.e. the function creates a copy of the given matrix.
m1 - existing matrix
m - matrix which is a copy of the specified matrix
local mat1 = vmath.matrix4_rotation_x(3.141592653)
local mat2 = vmath.matrix4(mat1)
if mat1 == mat2 then
-- yes, they are equal
print(mat2) --> vmath.matrix4(1, 0, 0, 0, 0, -1, 8.7422776573476e-08, 0, 0, -8.7422776573476e-08, -1, 0, 0, 0, 0, 1)
end
Constructs a frustum matrix from the given values. The left, right, top and bottom coordinates of the view cone are expressed as distances from the center of the near clipping plane. The near and far coordinates are expressed as distances from the tip of the view frustum cone.
left - coordinate for left clipping plane
right - coordinate for right clipping plane
bottom - coordinate for bottom clipping plane
top - coordinate for top clipping plane
near - coordinate for near clipping plane
far - coordinate for far clipping plane
m - matrix representing the frustum
-- Construct a projection frustum with a vertical and horizontal
-- FOV of 45 degrees. Useful for rendering a square view.
local proj = vmath.matrix4_frustum(-1, 1, -1, 1, 1, 1000)
render.set_projection(proj)
The resulting matrix is created from the supplied look-at parameters. This is useful for constructing a view matrix for a camera or rendering in general.
eye - eye position
look_at - look-at position
up - up vector
m - look-at matrix
-- Set up a perspective camera at z 100 with 45 degrees (pi/2) FOV
-- Aspect ratio 4:3
local eye = vmath.vector3(0, 0, 100)
local look_at = vmath.vector3(0, 0, 0)
local up = vmath.vector3(0, 1, 0)
local view = vmath.matrix4_look_at(eye, look_at, up)
render.set_view(view)
local proj = vmath.matrix4_perspective(3.141592/2, 4/3, 1, 1000)
render.set_projection(proj)
Creates an orthographic projection matrix. This is useful to construct a projection matrix for a camera or rendering in general.
left - coordinate for left clipping plane
right - coordinate for right clipping plane
bottom - coordinate for bottom clipping plane
top - coordinate for top clipping plane
near - coordinate for near clipping plane
far - coordinate for far clipping plane
m - orthographic projection matrix
-- Set up an orthographic projection based on the width and height
-- of the game window.
local w = render.get_width()
local h = render.get_height()
local proj = vmath.matrix4_orthographic(- w / 2, w / 2, -h / 2, h / 2, -1000, 1000)
render.set_projection(proj)
Creates a perspective projection matrix. This is useful to construct a projection matrix for a camera or rendering in general.
fov - angle of the full vertical field of view in radians
aspect - aspect ratio
near - coordinate for near clipping plane
far - coordinate for far clipping plane
m - perspective projection matrix
-- Set up a perspective camera at z 100 with 45 degrees (pi/2) FOV
-- Aspect ratio 4:3
local eye = vmath.vector3(0, 0, 100)
local look_at = vmath.vector3(0, 0, 0)
local up = vmath.vector3(0, 1, 0)
local view = vmath.matrix4_look_at(eye, look_at, up)
render.set_view(view)
local proj = vmath.matrix4_perspective(3.141592/2, 4/3, 1, 1000)
render.set_projection(proj)
The resulting matrix describes the same rotation as the quaternion, but does not have any translation (also like the quaternion).
q - quaternion to create matrix from
m - matrix represented by quaternion
local vec = vmath.vector4(1, 1, 0, 0)
local quat = vmath.quat_rotation_z(3.141592653)
local mat = vmath.matrix4_from_quat(quat)
print(mat * vec) --> vmath.matrix4_frustum(-1, 1, -1, 1, 1, 1000)
The resulting matrix describes a rotation around the axis by the specified angle.
v - axis
angle - angle in radians
m - matrix represented by axis and angle
local vec = vmath.vector4(1, 1, 0, 0)
local axis = vmath.vector3(0, 0, 1) -- z-axis
local mat = vmath.matrix4_axis_angle(axis, 3.141592653)
print(mat * vec) --> vmath.vector4(-0.99999994039536, -1.0000001192093, 0, 0)
The resulting matrix describes a rotation around the x-axis by the specified angle.
angle - angle in radians around x-axis
m - matrix from rotation around x-axis
local vec = vmath.vector4(1, 1, 0, 0)
local mat = vmath.matrix4_rotation_x(3.141592653)
print(mat * vec) --> vmath.vector4(1, -1, -8.7422776573476e-08, 0)
The resulting matrix describes a rotation around the y-axis by the specified angle.
angle - angle in radians around y-axis
m - matrix from rotation around y-axis
local vec = vmath.vector4(1, 1, 0, 0)
local mat = vmath.matrix4_rotation_y(3.141592653)
print(mat * vec) --> vmath.vector4(-1, 1, 8.7422776573476e-08, 0)
The resulting matrix describes a rotation around the z-axis by the specified angle.
angle - angle in radians around z-axis
m - matrix from rotation around z-axis
local vec = vmath.vector4(1, 1, 0, 0)
local mat = vmath.matrix4_rotation_z(3.141592653)
print(mat * vec) --> vmath.vector4(-0.99999994039536, -1.0000001192093, 0, 0)
The resulting matrix describes a translation of a point in euclidean space.
position - position vector to create matrix from
m - matrix from the supplied position vector
-- Set camera view from custom view and translation matrices
local mat_trans = vmath.matrix4_translation(vmath.vector3(0, 10, 100))
local mat_view = vmath.matrix4_rotation_y(-3.141592/4)
render.set_view(mat_view * mat_trans)
The resulting matrix is the inverse of the supplied matrix.
For ortho-normal matrices, e.g. regular object transformation,
use vmath.ortho_inv()
instead.
The specialized inverse for ortho-normalized matrices is much faster
than the general inverse.
m1 - matrix to invert
m - inverse of the supplied matrix
local mat1 = vmath.matrix4_rotation_z(3.141592653)
local mat2 = vmath.inv(mat1)
-- M * inv(M) = identity matrix
print(mat1 * mat2) --> vmath.matrix4(1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1)
The resulting matrix is the inverse of the supplied matrix.
The supplied matrix has to be an ortho-normal matrix, e.g.
describe a regular object transformation.
For matrices that are not ortho-normal
use the general inverse vmath.inv()
instead.
m1 - ortho-normalized matrix to invert
m - inverse of the supplied matrix
local mat1 = vmath.matrix4_rotation_z(3.141592653)
local mat2 = vmath.ortho_inv(mat1)
-- M * inv(M) = identity matrix
print(mat1 * mat2) --> vmath.matrix4(1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1)
The returned value is a scalar defined as:
P ⋅ Q = |P| |Q| cos θ
where θ is the angle between the vectors P and Q.
v1 - first vector
v2 - second vector
n - dot product
if vmath.dot(vector1, vector2) == 0 then
-- The two vectors are perpendicular (at right-angles to each other)
...
end
Returns the squared length of the supplied vector or quaternion.
v - value of which to calculate the squared length
n - squared length
if vmath.length_sqr(vector1) < vmath.length_sqr(vector2) then
-- Vector 1 has less magnitude than vector 2
...
end
Returns the length of the supplied vector or quaternion. If you are comparing the lengths of vectors or quaternions, you should compare the length squared instead as it is slightly more efficient to calculate (it eliminates a square root calculation).
v - value of which to calculate the length
n - length
if vmath.length(self.velocity) < max_velocity then
-- The speed (velocity vector) is below max.
-- TODO: max_velocity can be expressed as squared
-- so we can compare with length_sqr() instead.
...
end
Normalizes a vector, i.e. returns a new vector with the same direction as the input vector, but with length 1. The length of the vector must be above 0, otherwise a division-by-zero will occur.
v1 - vector to normalize
v - new normalized vector
local vec = vmath.vector3(1, 2, 3)
local norm_vec = vmath.normalize(vec)
print(norm_vec) --> vmath.vector3(0.26726123690605, 0.5345224738121, 0.80178368091583)
print(vmath.length(norm_vec)) --> 0.99999994039536
Given two linearly independent vectors P and Q, the cross product, P × Q, is a vector that is perpendicular to both P and Q and therefore normal to the plane containing them. If the two vectors have the same direction (or have the exact opposite direction from one another, i.e. are not linearly independent) or if either one has zero length, then their cross product is zero.
v1 - first vector
v2 - second vector
v - a new vector representing the cross product
local vec1 = vmath.vector3(1, 0, 0)
local vec2 = vmath.vector3(0, 1, 0)
print(vmath.cross(vec1, vec2)) --> vmath.vector3(0, 0, 1)
local vec3 = vmath.vector3(-1, 0, 0)
print(vmath.cross(vec1, vec3)) --> vmath.vector3(0, -0, 0)
Linearly interpolate between two vectors. The function treats the vectors as positions and interpolates between the positions in a straight line. Lerp is useful to describe transitions from one place to another over time. The function does not clamp t between 0 and 1.
t - interpolation parameter, 0-1
v1 - vector to lerp from
v2 - vector to lerp to
v - the lerped vector
function init(self)
self.t = 0
end
function update(self, dt)
self.t = self.t + dt
if self.t <= 1 then
local startpos = vmath.vector3(0, 600, 0)
local endpos = vmath.vector3(600, 0, 0)
local pos = vmath.lerp(self.t, startpos, endpos)
go.set_position(pos, "go")
end
end
Linearly interpolate between two quaternions. Linear interpolation of rotations are only useful for small rotations. For interpolations of arbitrary rotations, vmath.slerp yields much better results. The function does not clamp t between 0 and 1.
t - interpolation parameter, 0-1
q1 - quaternion to lerp from
q2 - quaternion to lerp to
q - the lerped quaternion
function init(self)
self.t = 0
end
function update(self, dt)
self.t = self.t + dt
if self.t <= 1 then
local startrot = vmath.quat_rotation_z(0)
local endrot = vmath.quat_rotation_z(3.141592653)
local rot = vmath.lerp(self.t, startrot, endrot)
go.set_rotation(rot, "go")
end
end
Linearly interpolate between two values. Lerp is useful to describe transitions from one value to another over time. The function does not clamp t between 0 and 1.
t - interpolation parameter, 0-1
n1 - number to lerp from
n2 - number to lerp to
n - the lerped number
function init(self)
self.t = 0
end
function update(self, dt)
self.t = self.t + dt
if self.t <= 1 then
local startx = 0
local endx = 600
local x = vmath.lerp(self.t, startx, endx)
go.set_position(vmath.vector3(x, 100, 0), "go")
end
end
Spherically interpolates between two vectors. The difference to lerp is that slerp treats the vectors as directions instead of positions in space. The direction of the returned vector is interpolated by the angle and the magnitude is interpolated between the magnitudes of the from and to vectors. Slerp is computationally more expensive than lerp. The function does not clamp t between 0 and 1.
t - interpolation parameter, 0-1
v1 - vector to slerp from
v2 - vector to slerp to
v - the slerped vector
function init(self)
self.t = 0
end
function update(self, dt)
self.t = self.t + dt
if self.t <= 1 then
local startpos = vmath.vector3(0, 600, 0)
local endpos = vmath.vector3(600, 0, 0)
local pos = vmath.slerp(self.t, startpos, endpos)
go.set_position(pos, "go")
end
end
Slerp travels the torque-minimal path maintaining constant velocity, which means it travels along the straightest path along the rounded surface of a sphere. Slerp is useful for interpolation of rotations. Slerp travels the torque-minimal path, which means it travels along the straightest path the rounded surface of a sphere. The function does not clamp t between 0 and 1.
t - interpolation parameter, 0-1
q1 - quaternion to slerp from
q2 - quaternion to slerp to
q - the slerped quaternion
function init(self)
self.t = 0
end
function update(self, dt)
self.t = self.t + dt
if self.t <= 1 then
local startrot = vmath.quat_rotation_z(0)
local endrot = vmath.quat_rotation_z(3.141592653)
local rot = vmath.slerp(self.t, startrot, endrot)
go.set_rotation(rot, "go")
end
end
Calculates the conjugate of a quaternion. The result is a
quaternion with the same magnitudes but with the sign of
the imaginary (vector) parts changed:
q
q1 - quaternion of which to calculate the conjugate
q - the conjugate
local quat = vmath.quat(1, 2, 3, 4)
print(vmath.conj(quat)) --> vmath.quat(-1, -2, -3, 4)
Returns a new vector from the supplied vector that is rotated by the rotation described by the supplied quaternion.
q - quaternion
v1 - vector to rotate
v - the rotated vector
local vec = vmath.vector3(1, 1, 0)
local rot = vmath.quat_rotation_z(3.141592563)
print(vmath.rotate(rot, vec)) --> vmath.vector3(-1.0000002384186, -0.99999988079071, 0)
Calculates the extent the projection of the first vector onto the second.
The returned value is a scalar p defined as:
p = |P| cos θ / |Q|
where θ is the angle between the vectors P and Q.
v1 - vector to be projected on the second
v2 - vector onto which the first will be projected, must not have zero length
n - the projected extent of the first vector onto the second
local v1 = vmath.vector3(1, 1, 0)
local v2 = vmath.vector3(2, 0, 0)
print(vmath.project(v1, v2)) --> 0.5
Performs an element wise multiplication between two vectors of the same type
The returned value is a vector defined as (e.g. for a vector3):
v = vmath.mul_per_elem(a, b) = vmath.vector3(a.x * b.x, a.y * b.y, a.z * b.z)
v1 - first vector
v2 - second vector
v - multiplied vector
local blend_color = vmath.mul_per_elem(color1, color2)