How to draw a line segment at run time using three.js

three.js is mainly for drawing in 3D. If you want to copy a 2D drawing application like paint then using the 2D canvas will probably be easier: canvas.getContext("2d");.

I will assume that you do want to draw in three.js. In which case I have put together this example. Follow these steps:

1. Click anywhere on the page and drag the mouse around to draw a line. The line is drawn on the z plane.
2. Click the Shapes button and notice how one shape is closer in and the other further, that is because one is above the z plane and the other behind.
3. Click the Rotate button, this will cause the camera to zoom out and rotate around the axes. Notice that when you pass through the z plane all your drawing is on that plane.

Have a look at the code, the main parts are:

You need to project the mouse click coordinates onto the plane. This is done with this function:

function get3dPointZAxis(event)
{
    var vector = new THREE.Vector3(
                ( event.clientX / window.innerWidth ) * 2 - 1,
                - ( event.clientY / window.innerHeight ) * 2 + 1,
                0.5 );
    projector.unprojectVector( vector, camera );
    var dir = vector.sub( camera.position ).normalize();
    var distance = - camera.position.z / dir.z;
    var pos = camera.position.clone().add( dir.multiplyScalar( distance ) );    
    return pos;
}

Then draw the line from previous to this point:

geometry.vertices.push(lastPoint);
geometry.vertices.push(pos);
var line = new THREE.Line(geometry, material);
scene.add(line);

Note that when you get close to passing through the z plane while rotating, the projection to Z is very far off and you go out of bounds, to prevent this the following check is done:

if( Math.abs(lastPoint.x - pos.x) < 500 && Math.abs(lastPoint.y - pos.y) < 500 && Math.abs(lastPoint.z - pos.z) < 500 )

For reference, I found information for projecting the mouse coordinate here (SO answer) and here (three.js sample).

Update

To draw a line from the mousedown position to the mouseup position see this demo. The code changes are to instead just do the draw between points on mouse up.

function stopDraw(event)
{
     if( lastPoint )
    {
        var pos = get3dPointZAxis(event);
        var material = new THREE.LineBasicMaterial({
            color: 0x0000ff
        });
        var geometry = new THREE.Geometry();
        if( Math.abs(lastPoint.x - pos.x) < 2000 && Math.abs(lastPoint.y - pos.y) < 2000 && Math.abs(lastPoint.z - pos.z) < 2000 )
        {
            geometry.vertices.push(lastPoint);
            geometry.vertices.push(pos);

            var line = new THREE.Line(geometry, material);
            scene.add(line);
            lastPoint = pos;        
        }
        else
        {
            console.debug(lastPoint.x.toString() + ':' + lastPoint.y.toString() + ':' + lastPoint.z.toString()  + ':' + 
                        pos.x.toString() + ':' + pos.y.toString()  + ':' + pos.z.toString());
        }
    }
}