Abstract: Two methods for rendering haptic surfaces through spatial modulation of ultrasound are described: Singular Value Decomposition (SVD) with Tikhonov Regularization and Mini-Batch, Stochastic Gradient Descent with Momentum (MSGDM). Further, adjustment in the placement of transducers, arrays can be generated which allow for variable focus points using a limited set of driving signals. This is accomplished by generating a set of mutually-exclusive transducer placement layouts which when driven together produce a set of sufficiently orthogonal fields at various points of interest. Further, using basic transformations on sampling locations a field can be translated or rotated in 3-dimensional space. This enables the creation of spatiotemporal modulated haptics with reduced computation. Further, mid-air haptics using ultrasound can be generated via amplitude modulation or via spatial modulation of control points defined at spatial locations in an acoustic field above a phased array device.