Conventionally, geometric computing problems are treated as algebraic computing problems by representing a geometric object in a global reference coordinate system. This approach has two problems. The first problem is that the intuitive view of the interaction of objects is lost, and the second problem is that algebraic computing is prone to errors for degenerated cases related to the interaction of (e.g., notable case of "divided by zero"). In this paper, we propose a new approach to geometric computing especially to analyze the interaction of two objects (e.g., two triangles). The main idea behind this new approach is a specially-tailored local coordinate system for two interacting objects is defined, which makes the projection of the objects on this local coordinate system represent the true geometry of the objects. This idea significantly departs from the conventional approach which is primarily based on the concept of the global coordinate system. Three examples are provided to illustrate the effectiveness of the proposed approach. Among them, one example is related to the robustness of methods for analyzing the relations of two interacting objects, which is still an open issue in the field of geometric computing and suggests that the proposed approach could have some benefit to robustness in analysis.