C++ – 3D – Rotation Matrix from direction vector (Forward, Up, Right)

3dcdirectxmatrixvector

I need to get rotation matrix from direction vector (vForward) I also have vRight and vUp vectors. All those vectors are unit vectors.

I just need to get rotation matrix.

To get rotation matrix for rotation in only one plane (xy) parallel to ground, I do this:
XMMATRIX xmResult;

Vec3f vFwd = pPlayer->VForward;
vFwd.z = 0;
vFwd.Normalize();

xmResult = XMMatrixSet( vFwd.y,     -vFwd.x,    0,      0,
    vFwd.x,      vFwd.y,    0,      0,
    0,          0,          1,      0,
    0,          0,          0,      1);

Above code only get rotation matrix to rotate around Z axis:

I would like to get the code to rotate around all axis.
This is coordinate system I'm forced to use. I know it is strange:

Coordinate System I'm using

This is how I'm using my matrix later in code:

XMStoreFloat3((XMFLOAT3*)&vStart, XMVector3Transform(XMLoadFloat3((XMFLOAT3*)&vStart), xmTransformation));
XMStoreFloat3((XMFLOAT3*)&vEnd, XMVector3Transform(XMLoadFloat3((XMFLOAT3*)&vEnd), xmTransformation));

Best Answer

Depending on how you use your matrices, Right, Up and Forward should correspond to the rows or columns of your matrix.

xmResult = XMMatrixSet( vRight.x, vRight.y, vRight.z, 0, vFwd.x, vFwd.y, vFwd.z, 0, vUp.x, vUp.y, vUp.z, 0, 0, 0, 0, 1);

Related Solutions

Calculating ‘up vector’ from transformation matrix in 3D

Translation actually does affect it. Let's say in the example the transformation matrix didn't do any scaling or rotation, but did translate it 2 units in the Z direction. Then when you transform (0,1,0) you get (0,1,2), and then normalizing it gives (0,1/sqrt(5), 2/sqrt(5)).

What you want to do is take the difference between the transformation of (0,1,0) and the transformation of (0,0,0), and then normalize the resulting vector. In the above example you would take (0,1,2) minus (0,0,2) (0,0,2 being the transformation of the zero vector) to get (0,1,0) as desired.

3D Z-Y-X rotation of vector

The ZYX Euler angle rotation matrix is defined as

R_ZYX(dz, dy, dx) = R(Z, dz) * R(Y, dy) * R(X, dx)

There are two different ways of reading the order of the rotations: either from left to right or from right to left. When read from left to right the rotations are about the local axes of the coordinate frame, as you are correctly doing. It's only when read from right to left that the rotations are about a fixed coordinate frame.

Now to answer the question, let's compute the angle dy you should be rotating with if you want all coordinates of the rotated vector to have the same absolute value.

Let r be the length of v and let a be the absolute coordinate value. By Pythogoras, a^2 + a^2 + a^2 = r^2, hence a = r / sqrt(3). The angle of the rotated vector relative to the XY plane is dy = asin(a / r) = asin(1 / sqrt(3)), which is about 35.3 degrees. This angle is different from the 45 degrees (or asin(1 / sqrt(2)) in radians) that you are currently using.

A test (using Python and the gameobjects library):

from gameobjects import *
from math import *
import random

V = vector3.Vector3
T = matrix44.Matrix44

def R_x(dx): return T.x_rotation(dx)
def R_y(dy): return T.y_rotation(dy)
def R_z(dz): return T.z_rotation(dz)

def fmt(v): return "(%.3f, %.3f, %.3f)" % (v[0], v[1], v[2])

dx = 0
dy = asin(1 / sqrt(3.0))
dz = pi / 4

v = V(10, 0, 0)

print "ZYX Euler angle transformations:"
print fmt((R_z(dz) * R_y(dy) * R_x(dx)).transform(v))
dy = pi / 4
print fmt((R_z(dz) * R_y(dy) * R_x(dx)).transform(v))

The output:

ZYX Euler angle transformations:
(5.774, 5.774, -5.774)
(5.000, 5.000, -7.071)

The last line for dz = dy = pi / 4 shows that the program agrees with your Euler angle implementation.

Best Answer

Related Solutions

Calculating ‘up vector’ from transformation matrix in 3D

3D Z-Y-X rotation of vector

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