CFrames are represented as a 3x4 matrix (3x3 rotation matrix and a vector combined):

```
| r00 r01 r02 x |
| r10 r11 r12 y |
| r20 r21 r22 z |
```

Whenever a 4x4 matrix is needed, the last row is presumed to be 0,0,0,1

The 3x3 rotation matrix should be orthonormal (columns have a magnitude of 1, and each column is orthogonal to the others).

The x,y,z part (fourth column) of the CFrame represents the position, and the other components represent the rotation.

The first column [r00, r10, r20] represents the direction right relative to the orientation of the CFrame (.XVector). The second column [r01, r11, r21] represents the direction up relative to the orientation of the CFrame (.YVector). The third column [r02, r12, r22] represents the direction forward relative to the orientation of the CFrame (.ZVector).

For each column, roblox has a property to describe it (.XVector, .YVector, .ZVector, .Position (these are all mentioned later)).

When a CFrame is visualized, it makes more sense. In this picture the red arrow is for the right direction, the blue arrow is for the forward direction, the green arrow is for up direction, and the grey cube is for position.

The next sections will be about CFrame methods and properties, as well as methods to create a CFrame.

There are many methods that create a CFrame.

Creates a CFrame which when you add the 4th row, becomes the identity matrix.

```
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
```

Position is 0,0,0 Rotation is the identity matrix

Creates a CFrame with position x,y,z, rotation stays as the identity matrix.

```
| 1 0 0 x |
| 0 1 0 y |
| 0 0 1 z |
```

Creates a CFrame with the position being the components of vector.

```
| 1 0 0 vector.x |
| 0 1 0 vector.y |
| 0 0 1 vector.z |
```

The default for up is Vector3.new(0,1,0).

Creates a CFrame with position being the position argument, and the rotation being how it should be oriented to look at the lookAt argument. The up argument is used for what is "up".

This won't be accurate if the positions are the same, so in that case you will want to use the identity matrix.

Creates a CFrame with all provided arguments.

```
| r00 r01 r02 x |
| r10 r11 r12 y |
| r20 r21 r22 z |
```

Creates a CFrame from position x,y,z and quaternion qX,qY,qZ,qW.

The default for vZ is vX:Cross(vY).Unit.

Creates a new CFrame with Position pos, XVector vX,YVector vY, and ZVector vZ.

```
| vX.X vY.X vZ.X pos.X |
| vX.Y vY.Y vZ.Y pos.Y |
| vX.Z vY.Z vZ.Z pos.Z |
```

Rotation is in radians.

Creates a new CFrame with position 0,0,0 and angles x,y,z.

Angles are applied in Z,Y,X order.

Rotation is in radians.

Creates a new CFrame with position 0,0,0 and angles x,y,z.

Angles are applied in Z,X,Y order.

Creates a CFrame with position 0,0,0 and angles defined by axis angle v,r.

The position of the CFrame as a vector.

The elements of the vector are the x,y,z elements of the CFrame.

Represents the X part of the position vector.

Represents the Y part of the position vector.

Represents the Z part of the position vector.

Represents the 'rightward' direction of the CFrame.

Elements - r00,r10,r20

Represents the 'upward' direction of the CFrame.

Elements - r10,r11,r12

Represents the 'forward' direction of the CFrame.

Element - r20,r21,r22

Represents the 'rightward' direction of the CFrame.

Elements - r00,r10,r20

Represents the 'upward' direction of the CFrame.

Elements - r01,r11,r21

Represents the negative 'forward' direction of the CFrame.

Elements - -r02,-r12,-r22

Helpful link:

```
developer.roblox.com/en-us/articles/CFrame-Math-Operations
```

Adds the elements of the vector to the position part of the CFrame.

```
| r00 r01 r02 x+vector.X |
| r10 r11 r12 y+vector.Y |
| r20 r21 r22 z+vector.Z |
```

Subtracts the elements of the vector from the position part of the CFrame.

```
| r00 r01 r02 x-vector.X |
| r10 r11 r12 y-vector.Y |
| r20 r21 r22 z-vector.Z |
```

Moves the CFrame according to the rotation of the CFrame and the Vector3, which can be though of as an offset.

```
local function cftimesv3(cf,v3)
return cf.Position+
cf.XVector*v3.X+
cf.YVector*v3.Y+
cf.ZVector*v3.Z
end
```

The linked article at the beginning of this section does a good job of explaining what is being done.

This can be thought of as 4x4 rotation matrix multiplication. CFrames don't store the 4th row, but it assumed to be 0,0,0,1.

It is important to note that matrix multiplication **isn't** commutative. So a times b most likely won't equal b times a

The first cframe will be represented as

```
| r00 r01 r02 rX |
| r10 r11 r12 rY |
| r20 r21 r22 rZ |
```

and the second cframe as

```
| m00 m01 m02 mX |
| m10 m11 m12 mY |
| m20 m21 m22 mZ |
```

result:

```
| r00*m00+r01*m10+r02*m20 r00*m01+r01*m11+r02*m21 r00*m02+r01*m12+r02*m22 r00*mX+r01*mY+r02*mZ+rX |
| r10*m00+r11*m10+r12*m20 r10*m01+r11*m11+r12*m21 r10*m02+r11*m12+r12*m22 r10*mX+r11*mY+r12*mZ+rY |
| r20*m00+r21*m10+r22*m20 r20*m01+r21*m11+r22*m21 r20*m02+r21*m12+r22*m22 r20*mX+r21*mY+r22*mZ+rZ |
```

'Vector3s' refers to 1 or more Vector3s

'CFrames' refers to 1 or more CFrames

Returns all 12 components of the CFrame in this order.

x,y,z,r00,r01,r02,r10,r11,r12,r20,r21,r22

The first three are the position. The fourth, seventh, and tenth numbers are the XVector. The fifth, eighth, and eleventh numbers are the YVector. The sixth, ninth, and twelfth numbers are the ZVector.

Returns the inverse of the CFrame, or returns the CFrame that when multiplied with the original CFrame will result in the identity matrix. The following expressions are equivalent:

```
A:Inverse()*A
A*A:Inverse()
CFrame.new()
```

Equivalent to

```
CFrame*cf
```

For each argument

Equivalent to:

```
CFrame:Inverse()*cf
```

For each argument

This will often be useful when you need to solve for B in something like this:

```
A*B = C
```

where A is the first CFrame and C is the second CFrame.

B will be A:Inverse()*C because

```
A*B = C
-- Multiply each side by A inverse
A^-1*A*B = A^-1*C
-- A^-1*A = I
-- I*B = B
B = A^-1*C
-- or with :Inverse()
B = A:Inverse()*C
```

Equivalent to:

```
CFrame*v3
```

For each argument

Equivalent to:

```
CFrame:Inverse()*v3
```

For each argument

Equivalent to:

```
(CFrame-CFrame.Position)*v3
```

For each argument

Equivalent to:

```
(CFrame:Inverse()-CFrame:Inverse().Position)*v3
```

For each argument

Converts the rotation matrix to euler angles.

Angles returned are in radians.

Applying the angles returned in Z,Y,X order will create approximately the same the rotation.

The first result is the X component of the rotation. The second result is the Y component of the rotation. The third result is the Z component of the rotation.

Converts the rotation matrix to euler angles.

Angles returned are in radians.

Applying the angles returned in Z,X,Y order will create approximately the same the rotation.

The first result is the X component of the rotation. The second result is the Y component of the rotation. The third result is the Z component of the rotation.

Converts the rotation matrix to axis angle.

Linearly interpolates between CFrame and goal by amount alpha.

Converts the CFrame to a CFrame which will be an orthonormal CFrame.

How to think about CFrames (by AxisAngle)

```
devforum.roblox.com/t/how-to-think-about-cframes/11743
```

I mentioned quaternions briefly, but here is another article about quaternions and rotation (by suremark)

```
devforum.roblox.com/t/rotations-with-quaternions/13209
```