The term safety in robotics is often understood as a synonym for avoidance. Although this perspective has led to progress in path planning and reactive control, a generalization of this perspective is necessary to include task semantics relevant to contact-rich manipulation tasks, especially during teleoperation and to ensure the safety of learned policies. We introduce the semantics-aware distance function and a corresponding computational method based on the Kelvin Transformation. This allows us to compute smooth distance approximations in an unbounded domain by instead solving a Laplace equation in a bounded domain. The semantics-aware distance generalizes signed distance functions by allowing the zero level set to lie inside of the object in regions where contact is allowed, effectively incorporating task semantics, such as object affordances, in an adaptive implicit representation of safe sets. In numerical experiments we show the computational viability of our method for real applications and visualize the computed function on a wrench with various semantic regions.