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.The following components are available for the various types:
x, y and z. Example: v.yx, y, z, and w. Example: v.wx, y, z, and w. Example: q.wm00 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.zv[3]calculates the conjugate of a quaternion
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 -
quatertion quaternion of which to calculate the conjugate
q -
quatertion the conjugate
local quat = vmath.quat(1, 2, 3, 4) print(vmath.conj(quat)) --> vmath.quat(-1, -2, -3, 4)
calculates the cross-product of two vectors
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 -
vector3 first vector
v2 -
vector3 second vector
v -
vector3 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)
calculates the dot-product of two vectors
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 -
vector3 | vector4 first vector
v2 -
vector3 | vector4 second vector
n -
number dot product
if vmath.dot(vector1, vector2) == 0 then -- The two vectors are perpendicular (at right-angles to each other) ... end
calculates the inverse matrix.
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 -
matrix4 matrix to invert
m -
matrix4 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)
calculates the length of a vector or quaternion
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 -
vector3 | vector4 | quat value of which to calculate the length
n -
number 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
calculates the squared length of a vector or quaternion
Returns the squared length of the supplied vector or quaternion.
v -
vector3 | vector4 | quat value of which to calculate the squared length
n -
number squared length
if vmath.length_sqr(vector1) < vmath.length_sqr(vector2) then -- Vector 1 has less magnitude than vector 2 ... end
lerps between two vectors
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 -
number interpolation parameter, 0-1
v1 -
vector3 | vector4 vector to lerp from
v2 -
vector3 | vector4 vector to lerp to
v -
vector3 | vector4 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
lerps between two quaternions
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 -
number interpolation parameter, 0-1
q1 -
quaternion quaternion to lerp from
q2 -
quaternion quaternion to lerp to
q -
quaternion 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
lerps between two numbers
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 -
number interpolation parameter, 0-1
n1 -
number number to lerp from
n2 -
number number to lerp to
n -
number 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
creates a new identity matrix
The resulting identity matrix describes a transform with no translation or rotation.
m -
matrix4 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 from another existing matrix
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 -
matrix4 existing matrix
m -
matrix4 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
creates a matrix from an axis and an angle
The resulting matrix describes a rotation around the axis by the specified angle.
v -
vector3 axis
angle -
number angle in radians
m -
matrix4 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)
creates a matrix from a quaternion
The resulting matrix describes the same rotation as the quaternion, but does not have any translation (also like the quaternion).
q -
quaternion quaternion to create matrix from
m -
matrix4 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)
creates a frustum matrix
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 -
number coordinate for left clipping plane
right -
number coordinate for right clipping plane
bottom -
number coordinate for bottom clipping plane
top -
number coordinate for top clipping plane
near -
number coordinate for near clipping plane
far -
number coordinate for far clipping plane
m -
matrix4 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)
creates a look-at view matrix
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 -
vector3 eye position
look_at -
vector3 look-at position
up -
vector3 up vector
m -
matrix4 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
Creates an orthographic projection matrix. This is useful to construct a projection matrix for a camera or rendering in general.
left -
number coordinate for left clipping plane
right -
number coordinate for right clipping plane
bottom -
number coordinate for bottom clipping plane
top -
number coordinate for top clipping plane
near -
number coordinate for near clipping plane
far -
number coordinate for far clipping plane
m -
matrix4 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
Creates a perspective projection matrix. This is useful to construct a projection matrix for a camera or rendering in general.
fov -
number angle of the full vertical field of view in radians
aspect -
number aspect ratio
near -
number coordinate for near clipping plane
far -
number coordinate for far clipping plane
m -
matrix4 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)
creates a matrix from rotation around x-axis
The resulting matrix describes a rotation around the x-axis by the specified angle.
angle -
number angle in radians around x-axis
m -
matrix4 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)
creates a matrix from rotation around y-axis
The resulting matrix describes a rotation around the y-axis by the specified angle.
angle -
number angle in radians around y-axis
m -
matrix4 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)
creates a matrix from rotation around z-axis
The resulting matrix describes a rotation around the z-axis by the specified angle.
angle -
number angle in radians around z-axis
m -
matrix4 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)
creates a translation matrix from a position vector
The resulting matrix describes a translation of a point in euclidean space.
position -
vector3 | type:vector4 position vector to create matrix from
m -
matrix4 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)
performs an element wise multiplication of two vectors
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 -
vector3 | vector4 first vector
v2 -
vector3 | vector4 second vector
v -
vector3 | vector4 multiplied vector
local blend_color = vmath.mul_per_elem(color1, color2)
normalizes a vector
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 -
vector3 | vector4 | quat vector to normalize
v -
vector3 | vector4 | quat 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
calculates the inverse of an ortho-normal matrix.
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 -
matrix4 ortho-normalized matrix to invert
m -
matrix4 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)
projects a vector onto another vector
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 -
vector3 vector to be projected on the second
v2 -
vector3 vector onto which the first will be projected, must not have zero length
n -
number 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
creates a new identity quaternion
Creates a new identity quaternion. The identity quaternion is equal to:
vmath.quat(0, 0, 0, 1)
q -
quaternion new identity quaternion
local quat = vmath.quat() print(quat) --> vmath.quat(0, 0, 0, 1) print(quat.w) --> 1
creates a new quaternion from another existing quaternion
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 -
quaternion existing quaternion
q -
quaternion 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 from its coordinates
Creates a new quaternion with the components set according to the supplied parameter values.
x -
number x coordinate
y -
number y coordinate
z -
number z coordinate
w -
number w coordinate
q -
quaternion new quaternion
local quat = vmath.quat(1, 2, 3, 4) print(quat) --> vmath.quat(1, 2, 3, 4)
creates a quaternion to rotate around a unit vector
The resulting quaternion describes a rotation of angle
radians around the axis described by the unit vector v.
v -
vector3 axis
angle -
number angle
q -
quaternion 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)
creates a quaternion from three base unit vectors
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 -
vector3 x base vector
y -
vector3 y base vector
z -
vector3 z base vector
q -
quaternion 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
creates a quaternion to rotate between two unit vectors
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 -
vector3 first unit vector, before rotation
v2 -
vector3 second unit vector, after rotation
q -
quaternion 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)
creates a quaternion from rotation around x-axis
The resulting quaternion describes a rotation of angle
radians around the x-axis.
angle -
number angle in radians around x-axis
q -
quaternion 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)
creates a quaternion from rotation around y-axis
The resulting quaternion describes a rotation of angle
radians around the y-axis.
angle -
number angle in radians around y-axis
q -
quaternion 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)
creates a quaternion from rotation around z-axis
The resulting quaternion describes a rotation of angle
radians around the z-axis.
angle -
number angle in radians around z-axis
q -
quaternion 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)
rotates a vector by a quaternion
Returns a new vector from the supplied vector that is rotated by the rotation described by the supplied quaternion.
q -
quatertion quaternion
v1 -
vector3 vector to rotate
v -
vector3 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)
slerps between two vectors
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 -
number interpolation parameter, 0-1
v1 -
vector3 | vector4 vector to slerp from
v2 -
vector3 | vector4 vector to slerp to
v -
vector3 | vector4 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
slerps between two quaternions
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 -
number interpolation parameter, 0-1
q1 -
quaternion quaternion to slerp from
q2 -
quaternion quaternion to slerp to
q -
quaternion 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
create a new vector from a table of values
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 table of numbers
v -
vector 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
Creates a new zero vector with all components set to 0.
v -
vector3 new zero vector
local vec = vmath.vector3() pprint(vec) --> vmath.vector3(0, 0, 0) print(vec.x) --> 0
creates a new vector from scalar value
Creates a new vector with all components set to the supplied scalar value.
n -
number scalar value to splat
v -
vector3 new vector
local vec = vmath.vector3(1.0) print(vec) --> vmath.vector3(1, 1, 1) print(vec.x) --> 1
creates a new vector from another existing vector
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 -
vector3 existing vector
v -
vector3 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 from its coordinates
Creates a new vector with the components set to the supplied values.
x -
number x coordinate
y -
number y coordinate
z -
number z coordinate
v -
vector3 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
Creates a new zero vector with all components set to 0.
v -
vector4 new zero vector
local vec = vmath.vector4() print(vec) --> vmath.vector4(0, 0, 0, 0) print(vec.w) --> 0
creates a new vector from scalar value
Creates a new vector with all components set to the supplied scalar value.
n -
number scalar value to splat
v -
vector4 new vector
local vec = vmath.vector4(1.0) print(vec) --> vmath.vector4(1, 1, 1, 1) print(vec.w) --> 1
creates a new vector from another existing vector
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 -
vector4 existing vector
v -
vector4 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 from its coordinates
Creates a new vector with the components set to the supplied values.
x -
number x coordinate
y -
number y coordinate
z -
number z coordinate
w -
number w coordinate
v -
vector4 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)