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Abstract--Rotation of three-dimensional objects by a two-dimensional mouse is a typical task in computer-aided design, operation simulations, and desktop virtual reality. The most commonly used rotation technique is a virtual trackball surrounding the object and operated by the mouse pointer. This article reviews and provides a mathematical foundation for virtual trackballs. The first, but still popular, virtual trackball was described by Chen et al. [CHECK END OF SENTENCE]. We show that the virtual trackball by Chen et al. does not rotate the object along the intended great circular arc on the virtual trackball and we give a correction. Another popular virtual trackball is Shoemake's quaternion implementation [CHECK END OF SENTENCE], which we show to be a special case of the virtual trackball by Chen et al.. Shoemake extends the scope of the virtual trackball to the full screen. Unfortunately, Shoemake's virtual trackball is inhomogeneous and discontinuous with consequences for usability. Finally, we review Bell's virtual trackball [CHECK END OF SENTENCE] and discuss studies of the usability of virtual trackballs.