# Vector math API documentation

Version: beta

Functions for mathematical operations on vectors, matrices and quaternions.

• The vector types (`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.
• The quaternion type (`vmath.quat`) supports multiplication with other quaternions.
• The matrix type (`vmath.matrix4`) can be multiplied with numbers, other matrices and `vmath.vector4` values.
• All types performs equality comparison by each component value.

The following components are available for the various types:

vector3
`x`, `y` and `z`. Example: `v.y`
vector4
`x`, `y`, `z`, and `w`. Example: `v.w`
quaternion
`x`, `y`, `z`, and `w`. Example: `q.w`
matrix4
`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`
vector
indexed by number 1 to the vector length. Example: `v`
FUNCTION
vmath.conj()

calculates the conjugate of a quaternion

vmath.cross()

calculates the cross-product of two vectors

vmath.dot()

calculates the dot-product of two vectors

vmath.inv()

calculates the inverse matrix.

vmath.length()

calculates the length of a vector or quaternion

vmath.length_sqr()

calculates the squared length of a vector or quaternion

vmath.lerp()

lerps between two vectors

vmath.lerp()

lerps between two quaternions

vmath.lerp()

lerps between two numbers

vmath.matrix4()

creates a new identity matrix

vmath.matrix4()

creates a new matrix from another existing matrix

vmath.matrix4_axis_angle()

creates a matrix from an axis and an angle

vmath.matrix4_from_quat()

creates a matrix from a quaternion

vmath.matrix4_frustum()

creates a frustum matrix

vmath.matrix4_look_at()

creates a look-at view matrix

vmath.matrix4_orthographic()

creates an orthographic projection matrix

vmath.matrix4_perspective()

creates a perspective projection matrix

vmath.matrix4_rotation_x()

creates a matrix from rotation around x-axis

vmath.matrix4_rotation_y()

creates a matrix from rotation around y-axis

vmath.matrix4_rotation_z()

creates a matrix from rotation around z-axis

vmath.matrix4_translation()

creates a translation matrix from a position vector

vmath.mul_per_elem()

performs an element wise multiplication of two vectors

vmath.normalize()

normalizes a vector

vmath.ortho_inv()

calculates the inverse of an ortho-normal matrix.

vmath.project()

projects a vector onto another vector

vmath.quat()

creates a new identity quaternion

vmath.quat()

creates a new quaternion from another existing quaternion

vmath.quat()

creates a new quaternion from its coordinates

vmath.quat_axis_angle()

creates a quaternion to rotate around a unit vector

vmath.quat_basis()

creates a quaternion from three base unit vectors

vmath.quat_from_to()

creates a quaternion to rotate between two unit vectors

vmath.quat_rotation_x()

creates a quaternion from rotation around x-axis

vmath.quat_rotation_y()

creates a quaternion from rotation around y-axis

vmath.quat_rotation_z()

creates a quaternion from rotation around z-axis

vmath.rotate()

rotates a vector by a quaternion

vmath.slerp()

slerps between two vectors

vmath.slerp()

slerps between two quaternions

vmath.vector()

create a new vector from a table of values

vmath.vector3()

creates a new zero vector

vmath.vector3()

creates a new vector from scalar value

vmath.vector3()

creates a new vector from another existing vector

vmath.vector3()

creates a new vector from its coordinates

vmath.vector4()

creates a new zero vector

vmath.vector4()

creates a new vector from scalar value

vmath.vector4()

creates a new vector from another existing vector

vmath.vector4()

creates a new vector from its coordinates

## Functions

#### vmath.conj()

`vmath.conj(q1)`

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* = [w, -v]`

PARAMETERS

 `q1` quatertion quaternion of which to calculate the conjugate

RETURNS

 `q` quatertion the conjugate

EXAMPLES

```local quat = vmath.quat(1, 2, 3, 4)
print(vmath.conj(quat)) --> vmath.quat(-1, -2, -3, 4)
```

#### vmath.cross()

`vmath.cross(v1,v2)`

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.

PARAMETERS

 `v1` vector3 first vector `v2` vector3 second vector

RETURNS

 `v` vector3 a new vector representing the cross product

EXAMPLES

```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)
```

#### vmath.dot()

`vmath.dot(v1,v2)`

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.

• If the dot product is positive then the angle between the vectors is below 90 degrees.
• If the dot product is zero the vectors are perpendicular (at right-angles to each other).
• If the dot product is negative then the angle between the vectors is more than 90 degrees.

PARAMETERS

 `v1` vector3 | vector4 first vector `v2` vector3 | vector4 second vector

RETURNS

 `n` number dot product

EXAMPLES

```if vmath.dot(vector1, vector2) == 0 then
-- The two vectors are perpendicular (at right-angles to each other)
...
end
```

#### vmath.inv()

`vmath.inv(m1)`

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.

PARAMETERS

 `m1` matrix4 matrix to invert

RETURNS

 `m` matrix4 inverse of the supplied matrix

EXAMPLES

```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)
```

#### vmath.length()

`vmath.length(v)`

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).

PARAMETERS

 `v` vector3 | vector4 | quat value of which to calculate the length

RETURNS

 `n` number length

EXAMPLES

```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
```

#### vmath.length_sqr()

`vmath.length_sqr(v)`

Returns the squared length of the supplied vector or quaternion.

PARAMETERS

 `v` vector3 | vector4 | quat value of which to calculate the squared length

RETURNS

 `n` number squared length

EXAMPLES

```if vmath.length_sqr(vector1) < vmath.length_sqr(vector2) then
-- Vector 1 has less magnitude than vector 2
...
end
```

#### vmath.lerp()

`vmath.lerp(t,v1,v2)`

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.

PARAMETERS

 `t` number interpolation parameter, 0-1 `v1` vector3 | vector4 vector to lerp from `v2` vector3 | vector4 vector to lerp to

RETURNS

 `v` vector3 | vector4 the lerped vector

EXAMPLES

```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
```

#### vmath.lerp()

`vmath.lerp(t,q1,q2)`

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.

PARAMETERS

 `t` number interpolation parameter, 0-1 `q1` quaternion quaternion to lerp from `q2` quaternion quaternion to lerp to

RETURNS

 `q` quaternion the lerped quaternion

EXAMPLES

```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
```

#### vmath.lerp()

`vmath.lerp(t,n1,n2)`

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.

PARAMETERS

 `t` number interpolation parameter, 0-1 `n1` number number to lerp from `n2` number number to lerp to

RETURNS

 `n` number the lerped number

EXAMPLES

```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
```

#### vmath.matrix4()

`vmath.matrix4()`

The resulting identity matrix describes a transform with no translation or rotation.

PARAMETERS

RETURNS

 `m` matrix4 identity matrix

EXAMPLES

```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
```

#### vmath.matrix4()

`vmath.matrix4(m1)`

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.

PARAMETERS

 `m1` matrix4 existing matrix

RETURNS

 `m` matrix4 matrix which is a copy of the specified matrix

EXAMPLES

```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
```

#### vmath.matrix4_axis_angle()

`vmath.matrix4_axis_angle(v,angle)`

The resulting matrix describes a rotation around the axis by the specified angle.

PARAMETERS

 `v` vector3 axis `angle` number angle in radians

RETURNS

 `m` matrix4 matrix represented by axis and angle

EXAMPLES

```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)
```

#### vmath.matrix4_from_quat()

`vmath.matrix4_from_quat(q)`

The resulting matrix describes the same rotation as the quaternion, but does not have any translation (also like the quaternion).

PARAMETERS

 `q` quaternion quaternion to create matrix from

RETURNS

 `m` matrix4 matrix represented by quaternion

EXAMPLES

```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)
```

#### vmath.matrix4_frustum()

`vmath.matrix4_frustum(left,right,bottom,top,near,far)`

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.

PARAMETERS

 `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

RETURNS

 `m` matrix4 matrix representing the frustum

EXAMPLES

```-- 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)
```

#### vmath.matrix4_look_at()

`vmath.matrix4_look_at(eye,look_at,up)`

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.

PARAMETERS

 `eye` vector3 eye position `look_at` vector3 look-at position `up` vector3 up vector

RETURNS

 `m` matrix4 look-at matrix

EXAMPLES

```-- 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)
```

#### vmath.matrix4_orthographic()

`vmath.matrix4_orthographic(left,right,bottom,top,near,far)`

Creates an orthographic projection matrix. This is useful to construct a projection matrix for a camera or rendering in general.

PARAMETERS

 `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

RETURNS

 `m` matrix4 orthographic projection matrix

EXAMPLES

```-- 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)
```

#### vmath.matrix4_perspective()

`vmath.matrix4_perspective(fov,aspect,near,far)`

Creates a perspective projection matrix. This is useful to construct a projection matrix for a camera or rendering in general.

PARAMETERS

 `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

RETURNS

 `m` matrix4 perspective projection matrix

EXAMPLES

```-- 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)
```

#### vmath.matrix4_rotation_x()

`vmath.matrix4_rotation_x(angle)`

The resulting matrix describes a rotation around the x-axis by the specified angle.

PARAMETERS

 `angle` number angle in radians around x-axis

RETURNS

 `m` matrix4 matrix from rotation around x-axis

EXAMPLES

```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)
```

#### vmath.matrix4_rotation_y()

`vmath.matrix4_rotation_y(angle)`

The resulting matrix describes a rotation around the y-axis by the specified angle.

PARAMETERS

 `angle` number angle in radians around y-axis

RETURNS

 `m` matrix4 matrix from rotation around y-axis

EXAMPLES

```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)
```

#### vmath.matrix4_rotation_z()

`vmath.matrix4_rotation_z(angle)`

The resulting matrix describes a rotation around the z-axis by the specified angle.

PARAMETERS

 `angle` number angle in radians around z-axis

RETURNS

 `m` matrix4 matrix from rotation around z-axis

EXAMPLES

```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)
```

#### vmath.matrix4_translation()

`vmath.matrix4_translation(position)`

The resulting matrix describes a translation of a point in euclidean space.

PARAMETERS

 `position` vector3 | type:vector4 position vector to create matrix from

RETURNS

 `m` matrix4 matrix from the supplied position vector

EXAMPLES

```-- 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)
```

#### vmath.mul_per_elem()

`vmath.mul_per_elem(v1,v2)`

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)`

PARAMETERS

 `v1` vector3 | vector4 first vector `v2` vector3 | vector4 second vector

RETURNS

 `v` vector3 | vector4 multiplied vector

EXAMPLES

```local blend_color = vmath.mul_per_elem(color1, color2)
```

#### vmath.normalize()

`vmath.normalize(v1)`

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.

PARAMETERS

 `v1` vector3 | vector4 | quat vector to normalize

RETURNS

 `v` vector3 | vector4 | quat new normalized vector

EXAMPLES

```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
```

#### vmath.ortho_inv()

`vmath.ortho_inv(m1)`

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.

PARAMETERS

 `m1` matrix4 ortho-normalized matrix to invert

RETURNS

 `m` matrix4 inverse of the supplied matrix

EXAMPLES

```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)
```

#### vmath.project()

`vmath.project(v1,v2)`

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.

PARAMETERS

 `v1` vector3 vector to be projected on the second `v2` vector3 vector onto which the first will be projected, must not have zero length

RETURNS

 `n` number the projected extent of the first vector onto the second

EXAMPLES

```local v1 = vmath.vector3(1, 1, 0)
local v2 = vmath.vector3(2, 0, 0)
print(vmath.project(v1, v2)) --> 0.5
```

#### vmath.quat()

`vmath.quat()`

Creates a new identity quaternion. The identity quaternion is equal to:

`vmath.quat(0, 0, 0, 1)`

PARAMETERS

RETURNS

 `q` quaternion new identity quaternion

EXAMPLES

```local quat = vmath.quat()
print(quat) --> vmath.quat(0, 0, 0, 1)
print(quat.w) --> 1
```

#### vmath.quat()

`vmath.quat(q1)`

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.

PARAMETERS

 `q1` quaternion existing quaternion

RETURNS

 `q` quaternion new quaternion

EXAMPLES

```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
```

#### vmath.quat()

`vmath.quat(x,y,z,w)`

Creates a new quaternion with the components set according to the supplied parameter values.

PARAMETERS

 `x` number x coordinate `y` number y coordinate `z` number z coordinate `w` number w coordinate

RETURNS

 `q` quaternion new quaternion

EXAMPLES

```local quat = vmath.quat(1, 2, 3, 4)
print(quat) --> vmath.quat(1, 2, 3, 4)
```

#### vmath.quat_axis_angle()

`vmath.quat_axis_angle(v,angle)`

The resulting quaternion describes a rotation of `angle` radians around the axis described by the unit vector `v`.

PARAMETERS

 `v` vector3 axis `angle` number angle

RETURNS

 `q` quaternion quaternion representing the axis-angle rotation

EXAMPLES

```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)
```

#### vmath.quat_basis()

`vmath.quat_basis(x,y,z)`

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.

PARAMETERS

 `x` vector3 x base vector `y` vector3 y base vector `z` vector3 z base vector

RETURNS

 `q` quaternion quaternion representing the rotation of the specified base vectors

EXAMPLES

```-- 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
```

#### vmath.quat_from_to()

`vmath.quat_from_to(v1,v2)`

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

PARAMETERS

 `v1` vector3 first unit vector, before rotation `v2` vector3 second unit vector, after rotation

RETURNS

 `q` quaternion quaternion representing the rotation from first to second vector

EXAMPLES

```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)
```

#### vmath.quat_rotation_x()

`vmath.quat_rotation_x(angle)`

The resulting quaternion describes a rotation of `angle` radians around the x-axis.

PARAMETERS

 `angle` number angle in radians around x-axis

RETURNS

 `q` quaternion quaternion representing the rotation around the x-axis

EXAMPLES

```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)
```

#### vmath.quat_rotation_y()

`vmath.quat_rotation_y(angle)`

The resulting quaternion describes a rotation of `angle` radians around the y-axis.

PARAMETERS

 `angle` number angle in radians around y-axis

RETURNS

 `q` quaternion quaternion representing the rotation around the y-axis

EXAMPLES

```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)
```

#### vmath.quat_rotation_z()

`vmath.quat_rotation_z(angle)`

The resulting quaternion describes a rotation of `angle` radians around the z-axis.

PARAMETERS

 `angle` number angle in radians around z-axis

RETURNS

 `q` quaternion quaternion representing the rotation around the z-axis

EXAMPLES

```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)
```

#### vmath.rotate()

`vmath.rotate(q,v1)`

Returns a new vector from the supplied vector that is rotated by the rotation described by the supplied quaternion.

PARAMETERS

 `q` quatertion quaternion `v1` vector3 vector to rotate

RETURNS

 `v` vector3 the rotated vector

EXAMPLES

```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)
```

#### vmath.slerp()

`vmath.slerp(t,v1,v2)`

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.

PARAMETERS

 `t` number interpolation parameter, 0-1 `v1` vector3 | vector4 vector to slerp from `v2` vector3 | vector4 vector to slerp to

RETURNS

 `v` vector3 | vector4 the slerped vector

EXAMPLES

```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
```

#### vmath.slerp()

`vmath.slerp(t,q1,q2)`

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.

PARAMETERS

 `t` number interpolation parameter, 0-1 `q1` quaternion quaternion to slerp from `q2` quaternion quaternion to slerp to

RETURNS

 `q` quaternion the slerped quaternion

EXAMPLES

```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
```

#### vmath.vector()

`vmath.vector(t)`

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.

PARAMETERS

 `t` table table of numbers

RETURNS

 `v` vector new vector

EXAMPLES

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) --> 0.5
```

#### vmath.vector3()

`vmath.vector3()`

Creates a new zero vector with all components set to 0.

PARAMETERS

RETURNS

 `v` vector3 new zero vector

EXAMPLES

```local vec = vmath.vector3()
pprint(vec) --> vmath.vector3(0, 0, 0)
print(vec.x) --> 0
```

#### vmath.vector3()

`vmath.vector3(n)`

Creates a new vector with all components set to the supplied scalar value.

PARAMETERS

 `n` number scalar value to splat

RETURNS

 `v` vector3 new vector

EXAMPLES

```local vec = vmath.vector3(1.0)
print(vec) --> vmath.vector3(1, 1, 1)
print(vec.x) --> 1
```

#### vmath.vector3()

`vmath.vector3(v1)`

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.

PARAMETERS

 `v1` vector3 existing vector

RETURNS

 `v` vector3 new vector

EXAMPLES

```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
```

#### vmath.vector3()

`vmath.vector3(x,y,z)`

Creates a new vector with the components set to the supplied values.

PARAMETERS

 `x` number x coordinate `y` number y coordinate `z` number z coordinate

RETURNS

 `v` vector3 new vector

EXAMPLES

```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)
```

#### vmath.vector4()

`vmath.vector4()`

Creates a new zero vector with all components set to 0.

PARAMETERS

RETURNS

 `v` vector4 new zero vector

EXAMPLES

```local vec = vmath.vector4()
print(vec) --> vmath.vector4(0, 0, 0, 0)
print(vec.w) --> 0
```

#### vmath.vector4()

`vmath.vector4(n)`

Creates a new vector with all components set to the supplied scalar value.

PARAMETERS

 `n` number scalar value to splat

RETURNS

 `v` vector4 new vector

EXAMPLES

```local vec = vmath.vector4(1.0)
print(vec) --> vmath.vector4(1, 1, 1, 1)
print(vec.w) --> 1
```

#### vmath.vector4()

`vmath.vector4(v1)`

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.

PARAMETERS

 `v1` vector4 existing vector

RETURNS

 `v` vector4 new vector

EXAMPLES

```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
```

#### vmath.vector4()

`vmath.vector4(x,y,z,w)`

Creates a new vector with the components set to the supplied values.

PARAMETERS

 `x` number x coordinate `y` number y coordinate `z` number z coordinate `w` number w coordinate

RETURNS

 `v` vector4 new vector

EXAMPLES

```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)
```