A three-dimensional transformation of a three-dimensional object is calculated by a solver using an algorithm that reduces deviation between projected two-dimensional locations of three-dimensional contact points on a surface of the object after object transformation and two dimensional locations of touch points placed on a multi-touch display device. The algorithm is biased to generate a three-dimensional transformation that preferentially rotates the object in one direction over an opposite direction when all three-dimensional contact points are located at substantially the same depth along a Z-axis away from an image plane of the multi-touch display device and when transformation is then triggered by touch points moving such that a Euclidean distance between at least two of the touch points decreases.