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Visualized Deformation of Joinery to Understand Jointing Process by Homotopy Theory and Attaching Maps

Visualization is effective to understand complicated and sophisticated phenomena. Joiner, which joints pieces of woods and has complex process of jointing, is analyzed and modeled for visualization using homotopy theory and attaching maps. Joining the two pieces of woods changes their structures differently, which depends on jointing techniques. The paper describes difference between the tenon-mortise and lap joints in the fundamental group of homotopy. As the properties of the fundamental group are invariants, they are preserved through modeling from the abstract level to the specific one. When describing the process of jointing, two piece of woods are indirectly connected through a virtual attaching space using an attaching map, which avoids unnecessary consideration of piece relations. Using these concepts of homotopy, the process of jointing two pieces of woods are modeled from the abstract level in homotopy to the middle level of cellular structured spaces, to the specific one of physical presentation.