**Related Topics:** OpenGL Camera, Angles to Axes, Rotation About Axis

**NOTE:** *This page is constructing a rotation matrix for object's lookat, not for camera. If you look for camera's lookat, please visit Camera LookAt page.*

The rotation matrix can be constructed by a lookat vector with 2 or 3 points in 3D space. If an object at P_{1} is facing to P_{2}, then the lookat vector is P_{2} - P_{1}.

First, *Forward* axis vector is simply calculated by normalizing the lookat vector.

Second, *left* axis vector is computed by a cross product of a specified up direction vector and *forward* axis. This up direction vector is given to determine the roll angle of the object. And, it is not necessary perpendicular to the *forward* axis. If we don't consider roll rotation of the object, then we can use (0, 1, 0) instead. It means the object is always stood straight up.

The actual *up* axis vector, which is orthogonal to both *forward* and *left* axis, is computed by another cross product of *forward* and *left*. Both *left* and *up* axis should be normalized after cross product in order to have a unit length.

Here is a C++ example to calculate *left*, *up* and *forward* axis from a lookat vector. The first code block is a minimal implementation of Vector3 struct variable. The second code block is to compute 3 axes from 2 points (position and target vectors), and the last code block is to compute 3 axes from 3 points (position, target and up direction vectors).

```
// minimal implementation of Vector3 struct
struct Vector3
{
float x;
float y;
float z;
Vector3() : x(0), y(0), z(0) {}; // constructors
Vector3(float x, float y, float z) : x(x), y(y), z(z) {};
// functions
Vector3& normalize(); //
Vector3 operator-(const Vector3& rhs) const; // subtract rhs
Vector3 cross(const Vector3& rhs) const; // cross product
};
Vector3& Vector3::normalize() {
float invLength = 1 / sqrtf(x*x + y*y + z*z);
x *= invLength;
y *= invLength;
z *= invLength;
return *this;
}
Vector3 Vector3::operator-(const Vector3& rhs) const {
return Vector3(x-rhs.x, y-rhs.y, z-rhs.z);
}
Vector3 Vector3::cross(const Vector3& rhs) const {
return Vector3(y*rhs.z - z*rhs.y, z*rhs.x - x*rhs.z, x*rhs.y - y*rhs.x);
}
```

```
///////////////////////////////////////////////////////////////////////////////
// compute transform axis from object position and target point
///////////////////////////////////////////////////////////////////////////////
void lookAtToAxes(const Vector3& position, const Vector3& target,
Vector3& left, Vector3& up, Vector3& forward)
{
// compute the forward vector
forward = target - position;
forward.normalize();
// compute temporal up vector based on the forward vector
// watch out when look up/down at 90 degree
// for example, forward vector is on the Y axis
if(fabs(forward.x) < EPSILON && fabs(forward.z) < EPSILON)
{
// forward vector is pointing +Y axis
if(forward.y > 0)
up = Vector3(0, 0, -1);
// forward vector is pointing -Y axis
else
up = Vector3(0, 0, 1);
}
// in general, up vector is straight up
else
{
up = Vector3(0, 1, 0);
}
// compute the left vector
left = up.cross(forward); // cross product
left.normalize();
// re-calculate the orthonormal up vector
up = forward.cross(left); // cross product
}
```

```
///////////////////////////////////////////////////////////////////////////////
// compute transform axis from object position, target and up direction
///////////////////////////////////////////////////////////////////////////////
void lookAtToAxes(const Vector3& pos, const Vector3& target, const Vector3& upDir,
Vector3& left, Vector3& up, Vector3& forward)
{
// compute the forward vector
forward = target - pos;
forward.normalize();
// compute the left vector
left = upDir.cross(forward); // cross product
left.normalize();
// compute the orthonormal up vector
up = forward.cross(left); // cross product
}
```

Once all 3 axes are computed, you can construct a rotation matrix by replacing 3 columns of 4x4 matrix.