The shape-based routing needs a routing architecture with a geometrical computation framework on it. This paper introduces a novel routing architecture, Oct-Touched Tile (OTT), with a geometrical computation method along the horizontal- and vertical-constraints. The architecture is represented by the tiles spreading over the 2-D plane. Each tile is flexible to satisfy the constraints imposed for non-overlapping and sizing request. In this framework, path finding and shape-based sizing are executed on the same architecture. In experiments, our system demonstrates the performance comparable to a commercial tool. In addition, we show potential of OTT by introducing several ideas of extensions to analog layout constraints.