We present six-degree-of-freedom (6DoF) haptic rendering algorithms using translational ($PD_t$) and generalized penetration depth ($PD_g$). Our rendering algorithm can handle any type of object/object haptic interaction using penalty-based response and makes no assumption about the underlying geometry and topology. Moreover, our rendering algorithm can effectively deal with multiple contacts. Our penetration depth algorithms for $PD_t$ and $PD_g$ are based on a contact-space projection technique combined with iterative, local optimization on the contact-space. We circumvent the local minima problem, imposed by the local optimization, using motion coherence present in the haptic simulation. Our experimental results show that our methods can produce high-fidelity force feedback for general polygonal models consisting of tens of thousands of triangles at near-haptic rates, and are successfully integrated into an off-the-shelf 6DoF haptic device. We also discuss the benefits of using different formulations of penetration depth in the context of 6DoF haptics.