Look-at quaternion using up vector

Question

I have a camera (in a custom 3D engine) that accepts a quaternion for the rotation transform. I have two 3D points representing a camera and an object to look at. I want to

Answer 1

Assume you initially have three ortonormal vectors: worldUp, worldFront and worldSide, and lets use your equations for lookAt, sideAxis and rotatedUp. The worldSide vector will not be necessary to achieve the result.

Break the operation in two. First, rotate around worldUp. Then rotate around sideAxis, which will now actually be parallel to the rotated worldSide.

Axis1 = worldUp
Angle1 = (see below)

Axis2 = cross(lookAt, worldUp) = sideAxis
Angle2 = (see below)

Each of these rotations correspond to a quaternion using:

Q = cos(Angle/2) + i * Axis_x * sin(Angle/2) + j * Axis_y * sin(Angle/2) + k * Axis_z * sin(Angle/2)

Multiply both Q1 and Q2 and you get the desired quaternion.

Details for the angles:

Let P(worldUp) be the projection matrix on the worldUp direction, i.e., P(worldUp).v = cos(worldUp,v).worldUp or using kets and bras, P(worldUp) = |worldUp >< worldUp|. Let I be the identity matrix.

1. Project lookAt in the plane perpendicular to worldUp and normalize it.

   tmp1 = (I - P(worldUp)).lookAt
   n1 = normalize(tmp1)

2. Angle1 = arccos(dot(worldFront,n1))

3. Angle2 = arccos(dot(lookAt,n1))

EDIT1:

Notice that there is no need to compute transcendental functions. Since the dot product of a pair of normalized vectors is the cosine of an angle and assuming that cos(t) = x, we have the trigonometric identities:

• cos(t/2) = sqrt((1 + x)/2)
• sin(t/2) = sqrt((1 - x)/2)