本文以六角格子為切割方式來推展到階級結構，將之應用在碰撞偵測上，再利用節點的概念，提出一種有效的線性編碼法-HCD，發展到三維空間，並導入晶格的觀念，利用八面體的構造發展出三維六角格子的階級結構，對於物體的模擬與近似，有非常大的幫助。以往物體的資料皆是以浮點方式儲存，本研究利用六角對稱座標系的概念，整個物體的位置以整數儲存，可有效達到資料壓縮的效果。我們更導入k-dops的概念，使得六角階級結構在低階層下，即有極佳的模擬效果。

In this paper, hexagonal grid is extended to hierarchical structure. This technique can be applied to collision detection. By using concept of node, we develop an effectually linear decode called “HCD”. To develop to three dimensions, the structure of octahedron is applied to develop hexagonal hierarchical structure in three dimensions. This is helpful to simulate objects and approximate objects. The object’s data is commonly deposited in float. In this paper, Symmetrical Hexagonal Frame makes whole object to be deposited in integer. So the data of object can be compressed to smaller size. Otherwise, by concept of k-dops, we can close to object’s real surfaces with hierarchical hexagonal structures at low level.

第一章 緒論．．．．．．．．．．．．．．．．．．．．．．．1
第二章 傳統矩形格子上的碰撞偵測．．．．．．．．．．．．．8
2-1 簡介四分樹資料結構．．．．．．．．．．．．．．．．8
2-2 四分樹資料結構的建構．．．．．．．．．．．．．．．10
2-3 以Morton序列建立資料結構加速碰撞偵測．．．．．．．11
2-4 以節點座標法建立資料結構加速碰撞偵測．．．．．．．16
2-5 節點參考座標法與摩頓序列的比較．．．．．．．．．．21
第三章 六角結構上的定位系統．．．．．．．．．．．．．．．24
3-1 對稱性六角座標系．．．．．．．．．．．．．．．．．24
3-2 廣義平衡三元化位址．．．．．．．．．．．．．．．．26
3-3 六角格子上的節點參考座標法．．．．．．．．．．．．28
3-4 應用於三維的HCD系統．．．．．．．．．．．．．．．30
3-4-1 八面體座標系統．．．．．．．．．．．．．．．．31
3-4-2 對稱性八面體座標系統的座標值判斷．．．．．．．33
3-4-3 應用八面體座標系於三維HCD系統．．．．．．．．．35
3-5 HCD與直角座標系之轉換．．．．．．．．．．．．．．37
3-6 六角結構上之HCD系統與GBT序列之比較．．．．．．．．40
第四章 以六角階級結構作碰撞偵測．．．．．．．．．．．．．42
4-1 六角格子上的階級結構．．．．．．．．．．．．．．．42
4-1-1 HCD上的階級結構．．．．．．．．．．．．．．．．．45
4-2 利用階級結構加速碰撞偵測．．．．．．．．．．．．．53
4-2-1 碰撞偵測的步驟．．．．．．．．．．．．．．．．．53
4-2-2 擴展到三維空間．．．．．．．．．．．．．．．．．56
4-2-3 實際模擬碰撞偵測．．．．．．．．．．．．．．．．58
4-2-4 中心位置與半徑的求法．．．．．．．．．．．．．．59
4-3 在矩形結構中與在六角結構中模擬物體的比較．．．．．60
第五章 以六角結構上的k-dops來近似物體．．．．．．．．．．65
5-1 邊界體積階級結構與k-dops的觀念．．．．．．．．．．65
5-1-1 邊界體積的應用．．．．．．．．．．．．．．．．．65
5-1-2 k-dops的原理．．．．．．．．．．．．．．．．．．66
5-2 以k-dops近似物體．．．．．．．．．．．．．．．．．68
5-3 三維物體的近似．．．．．．．．．．．．．．．．．．69
5-4 在k-dops下兩種結構的比較．．．．．．．．．．．．．74
第六章 結論與建議．．．．．．．．．．．．．．．．．．．．76
參考文獻．．．．．．．．．．．．．．．．．．．．．．．．．．78

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