# Math Module

Taichi provides a built-in `math`

module that supports frequently used mathematical functions and utility functions, including:

- Commonly-used mathematical functions that are analogous to those in Python's built-in
`math`

module. - Small vector and matrix types that are analogous to those in the OpenGL shading language (GLSL).
- Some GLSL-standard functions.
- Complex number operations in the form of 2D vectors.

## Mathematical functions

You must call the mathematical functions provided by Taichi's `math`

module from within the Taichi scope. For example:

`import taichi as ti`

import taichi.math as tm

ti.init()

@ti.kernel

def test():

a = 1.0

x = tm.sin(a)

y = tm.floor(a)

z = tm.degrees(a)

w = tm.log2(a)

...

These functions also take vectors and matrices as arguments and operate on them element-wise:

`@ti.kernel`

def test():

a = ti.Vector([1.0, 2.0, 3.0])

x = tm.sin(a) # [0.841471, 0.909297, 0.141120]

y = tm.floor(a) # [1.000000, 2.000000, 3.000000]

z = tm.degrees(a) # [57.295780, 114.591560, 171.887344]

b = ti.Vector([2.0, 3.0, 4.0])

w = tm.atan2(b, a) # [1.107149, 0.982794, 0.927295]

...

##### note

Taichi's math module overlaps to a large extent with Python's built-in math module. Ensure that you follow a few extra rules when using Taichi's math module:

- You must call the functions provided by Taichi's math module from within the Taichi scope.
- Functions in Taichi's math module also take vectors or matrices as arguments.
- The precision of a function in Taichi's math module depends on the settings of
`default_fp`

and`arch`

(backend) in`ti.init()`

.

## Small vector and matrix types

Taichi's math module provides a few small vector and matrix types:

`vec2/vec3/vec4`

: 2D/3D/4D floating-point vector types.`ivec2/ivec3/ivec4`

: 2D/3D/4D integer vector types.`uvec2/uvec3/uvec4`

: 2D/3D/4D unsigned integer vector types.`mat2/mat3/mat4`

: 2D/3D/4D floating-point square matrix types.

To create one of the vector/matrix types above, use template function `ti.types.vector()`

or `ti.types.matrix()`

. For example, `vec2`

is defined as follows:

`vec2 = ti.types.vector(2, float)`

The number of precision bits of such a type is determined by `default_fp`

or `default_ip`

in the `ti.init()`

method call. For example, if `ti.init(default_fp=ti.f64)`

is called, then `vec2/vec3/vec4`

and `mat2/mat3/mat4`

defined in the Taichi scope all have a 64-bit floating-point precision.

You can use these types to instantiate vectors/matrices or annotate data types for function arguments and struct members. See the Type System for more information. Here we emphasize that they have very flexible initialization routines:

`mat2 = ti.math.mat2`

vec3 = ti.math.mat3

vec4 = ti.math.vec4

m = mat2(1) # [[1., 1.], [1., 1.]]

m = mat2(1, 2, 3, 4) # [[1., 2.], [3, 4.]]

m = mat2([1, 2], [3, 4]) # [[1., 2.], [3, 4.]]

m = mat2([1, 2, 3, 4]) # [[1., 2.], [3, 4.]]

v = vec3(1, 2, 3)

m = mat2(v, 4) # [[1., 2.], [3, 4.]]

u = vec4([1, 2], [3, 4])

u = vec4(v, 4.0)

Another important feature of vector types created by `ti.types.vector()`

is that they support **vector swizzling** just as GLSL vectors do. This means you can use `xyzw`

, `rgba`

, `stpq`

to access their elements with indices ≤ four:

`v = ti.math.vec4(1, 2, 3, 4)`

u = v.xyz # vec3(1, 2, 3)

u = v.xxx # vec3(1, 1, 1)

u = v.wzyx # vec4(4, 3, 2, 1)

u = v.rraa # vec4(1, 1, 2, 2)

### Relations between `ti.Vector`

, `ti.types.vector`

and `ti.math.vec3`

`ti.Vector`

is a function that accepts a 1D array and returns a matrix instance that has only one column. For example,`ti.Vector([1, 2, 3, 4, 5])`

.`ti.types.vector`

is a function that accepts an integer and a primitive type and returns a vector type. For example:`vec5f = ti.types.vector(5, float)`

.`vec5f`

can then be used to instantiate 5D vectors or annotate data types of function arguments and struct members:`@ti.kernel`

def test(v: vec5f):

print(v.xyz)Unlike

`ti.Vector`

, whose input data must be a 1D array, vector types created by`ti.types.vector()`

have more flexible ways to initialize, as explained above.`ti.math.vec3`

is created by`vec3 = ti.types.vector(3, float)`

.

## GLSL-standard functions

Taichi's math module also supports a few GLSL standard functions. These functions follow the GLSL standard, except that they accept arbitrary vectors and matrices as arguments and operate on them element-wise. For example:

`import taichi as ti`

import taichi.math as tm

@ti.kernel

def example():

v = tm.vec3(0., 1., 2.)

w = tm.smoothstep(0.0, 1.0, v)

w = tm.clamp(w, 0.2, 0.8)

w = tm.reflect(v, tm.normalize(tm.vec3(1)))

##### note

Texture support in Taichi is implemented in the `ti.types.texture_types`

module.

## Complex number operations

Taichi's math module also supports basic complex arithmetic operations on 2D vectors.

You can use a 2D vector of type `ti.math.vec2`

to represent a complex number. In this way, additions and subtractions of complex numbers come in the form of 2D vector additions and subtractions. You can call `ti.math.cmul()`

and `ti.math.cdiv()`

to conduct multiplication and division of complex numbers:

`import taichi as ti`

import taichi.math as tm

ti.init()

@ti.kernel

def test():

x = tm.vec2(1, 1) # complex number 1+1j

y = tm.vec2(0, 1) # complex number 1j

z = tm.cmul(x, y) # vec2(-1, 1) = -1+1j

w = tm.cdiv(x, y) # vec2(2, 0) = 2+0j

You can also compute the power, logarithm, and exponential of a complex number:

@ti.kernel

def test():

x = tm.vec2(1, 1)

y = tm.cpow(x, 2)

z = tm.clog(x)

w = tm.cexp(x)