# Vectors¶

A vector in Taichi can have two forms:

• as a temporary local variable. An n component vector consists of n scalar values.
• as an element of a global field. In this case, the field is an N-dimensional array of n component vectors.

In fact, Vector is simply an alias of Matrix, just with m = 1. See Matrices and Fields and matrices for more details.

## Declaration¶

### As global vector fields¶

ti.Vector.field(n, dtype, shape = None, offset = None)
Parameters: n – (scalar) the number of components in the vector dtype – (DataType) data type of the components shape – (optional, scalar or tuple) shape of the vector field, see Fields and matrices offset – (optional, scalar or tuple) see Coordinate offsets

For example, this creates a 3-D vector field of the shape of 5x4:

# Python-scope
a = ti.Vector.field(3, dtype=ti.f32, shape=(5, 4))

Note

In Python-scope, ti.field declares a scalar field (Scalar fields), while ti.Vector.field declares a vector field.

### As a temporary local variable¶

ti.Vector([x, y, ...])
Parameters: x – (scalar) the first component of the vector y – (scalar) the second component of the vector

For example, this creates a 3D vector with components (2, 3, 4):

# Taichi-scope
a = ti.Vector([2, 3, 4])

## Accessing components¶

### As global vector fields¶

a[p, q, ...][i]
Parameters: a – (ti.Vector.field) the vector p – (scalar) index of the first field dimension q – (scalar) index of the second field dimension i – (scalar) index of the vector component

This extracts the first component of vector a[6, 3]:

x = a[6, 3][0]

# or
vec = a[6, 3]
x = vec[0]

Note

Always use two pairs of square brackets to access scalar elements from vector fields.

• The indices in the first pair of brackets locate the vector inside the vector fields;
• The indices in the second pair of brackets locate the scalar element inside the vector.

For 0-D vector fields, indices in the first pair of brackets should be [None].

### As a temporary local variable¶

a[i]
Parameters: a – (Vector) the vector i – (scalar) index of the component

For example, this extracts the first component of vector a:

x = a[0]

This sets the second component of a to 4:

a[1] = 4

### XYZW vector component accessors¶

We also provide four handy accessors for the first four vector components:

a.x

Same as a[0].

a.y

Same as a[1].

a.z

Same as a[2].

a.w

Same as a[3].

Note

XYZW accessors can be used for both reading and writing:

v = ti.Vector([2, 3, 4])
print(v.x)  # 2
print(v.y)  # 3
print(v.z)  # 4
v.y = 8
print(v.y)  # 8

XYZW accessors can be used in both Taichi-scope and Python-scope.

XYZW accessors don’t work for ti.Matrix.

For GLSL-alike shuffling accessors, consider using taichi_glsl:

import taichi_glsl as tl

v = tl.vec(2, 3, 4)
print(v.xy)  # [2 3]
print(v._xYzX_z)  # [0 2 -3 4 -2 0 4]

## Methods¶

a.norm(eps = 0)
Parameters: a – (ti.Vector) eps – (optional, scalar) a safe-guard value for sqrt, usually 0. See the note below. (scalar) the magnitude / length / norm of vector

For example,

a = ti.Vector([3, 4])
a.norm() # sqrt(3*3 + 4*4 + 0) = 5

a.norm(eps) is equivalent to ti.sqrt(a.dot(a) + eps)

Note

To safeguard the operator’s gradient on zero vectors during differentiable programming, set eps to a small, positive value such as 1e-5.

a.norm_sqr()
Parameters: a – (ti.Vector) (scalar) the square of the magnitude / length / norm of vector

For example,

a = ti.Vector([3, 4])
a.norm_sqr() # 3*3 + 4*4 = 25

a.norm_sqr() is equivalent to a.dot(a)

a.normalized()
Parameters: a – (ti.Vector) (ti.Vector) the normalized / unit vector of a

For example,

a = ti.Vector([3, 4])
a.normalized() # [3 / 5, 4 / 5]

a.normalized() is equivalent to a / a.norm().

a.dot(b)
Parameters: a – (ti.Vector) b – (ti.Vector) (scalar) the dot (inner) product of a and b

E.g.,

a = ti.Vector([1, 3])
b = ti.Vector([2, 4])
a.dot(b) # 1*2 + 3*4 = 14
a.cross(b)
Parameters: a – (ti.Vector, 2 or 3 components) b – (ti.Vector of the same size as a) (scalar (for 2D inputs), or 3D Vector (for 3D inputs)) the cross product of a and b

We use a right-handed coordinate system. E.g.,

a = ti.Vector([1, 2, 3])
b = ti.Vector([4, 5, 6])
c = ti.cross(a, b)
# c = [2*6 - 5*3, 4*3 - 1*6, 1*5 - 4*2] = [-3, 6, -3]

p = ti.Vector([1, 2])
q = ti.Vector([4, 5])
r = ti.cross(a, b)
# r = 1*5 - 4*2 = -3
a.outer_product(b)
Parameters: a – (ti.Vector) b – (ti.Vector) (ti.Matrix) the outer product of a and b

E.g.,

a = ti.Vector([1, 2])
b = ti.Vector([4, 5, 6])
c = ti.outer_product(a, b) # NOTE: c[i, j] = a[i] * b[j]
# c = [[1*4, 1*5, 1*6], [2*4, 2*5, 2*6]]

Note

The outer product should not be confused with the cross product (ti.cross). For example, a and b do not have to be 2- or 3-component vectors for this function.

a.cast(dt)
Parameters: a – (ti.Vector) dt – (DataType) (ti.Vector) vector with all components of a casted into type dt

E.g.,

# Taichi-scope
a = ti.Vector([1.6, 2.3])
a.cast(ti.i32) # [2, 3]

See Type system for more details.

Note

Vectors are special matrices with only 1 column. In fact, ti.Vector is just an alias of ti.Matrix.

a.n
Parameters: a – (ti.Vector or ti.Vector.field) (scalar) return the dimensionality of vector a

E.g.,

# Taichi-scope
a = ti.Vector([1, 2, 3])
a.n  # 3
# Python-scope
a = ti.Vector.field(3, dtype=ti.f32, shape=(4, 5))
a.n  # 3

See :ref:`meta` for more details.

Note

When used as a global vector field, it will additionally contain all the metadata that a scalar field would have, E.g.:

# Python-scope
a = ti.Vector.field(3, dtype=ti.f32, shape=(4, 5))
a.shape  # (4, 5)
a.dtype  # ti.f32

## Element-wise operations (WIP)¶

TODO: add element wise operations docs