Rotate 3D object to mouse with Three.js

Question

I want to rotate an object in 3D space, so that the front side always looks to the mouse.

function onMouseMove(event){
             mouse3D = projector.unprojectVector(
                 new THREE.Vector3( event.clientX, event.clientY, 0.5 ), camera );
}

var angle = ??;
box.rotation.y = angle;

First is the unprojection correct ? And secondly how to calculate the angle ? Is it just tan(mouseX/mouseY) ? I'm trying to get more into the 3D mathematics, so a little bit explanation would be nice.

Thanks in advance.

Answer 1

// Direction we are already facing (without rotation)
var forward = new Vector3(0,0,-1);

// Direction we want to be facing (towards mouse pointer)
var target = new Vector3().sub(mouse3D, box.position).normalize();

// Axis and angle of rotation
var axis = new Vector3().cross(forward, target);
var sinAngle = axis.length(); // |u x v| = |u|*|v|*sin(a)
var cosAngle = forward.dot(target); // u . v = |u|*|v|*cos(a)
var angle = Math.atan2(sinAngle, cosAngle); // atan2(sin(a),cos(a)) = a
axis.normalize();

// Overwrite rotation
box.rotation.makeRotationAxis(axis, angle);

Alternatively, you could use quaternions:

// Overwrite rotation
box.useQuaternion = true;
box.quaternion.setFromAxisAngle(axis, angle);