平面和空間的理論，皆可找到對應的關係，就像是平面的極心可以對應到空間中的螺旋理論。將探討機構的特性，雙滑塊機構的卡丹(Cardan)圓是否在空間中找到對應。以及合成機構問題中，平面極心三角形所衍伸出的三相關線交於一點、三相關點在一直線上，在空間中是否具有相同的性質，兩者皆為本論文要探討的問題。
就空間中的機構而言，機構的運動也可以用螺旋來表示，根據Bottema和Roth(1990)，螺旋可以用來形容機構在空間中的旋轉和平移，機構在瞬時運動過程中，所產生的螺旋就稱為瞬時螺旋，將瞬時螺旋集合起來，就會形成瞬時螺旋曲面，若從耦桿角度下觀察就稱為運動瞬時螺旋曲面；在有限的運動下，就稱為旋轉螺旋，將旋轉螺旋集合起來，就會形成旋轉螺旋曲面，若從耦桿角度下觀察就稱為運動旋轉螺旋曲面。設定三個位置的合成是由兩垂直且相交的螺旋所構成，建立其幾何關係方程式，自訂不同數值案例參數進行求解。
從RCCC連桿機構所產生的瞬時螺旋曲面及旋轉曲面，可歸納出桿長和運動瞬時曲面大小成正比關係，桿長也和旋轉螺旋曲面成正比關係；扭角會影響瞬時螺旋曲面及運動旋轉螺旋曲面的形狀。合成三個位置所產生的固定軸及移動軸曲面會依參數設定分成三種情形，集合其固定軸及移動軸的交線稱之為把手，也會構成三種情況。
觀察RCCC連桿機構所產生的瞬時螺旋曲面及運動螺旋曲面，和卡丹圓相似處為其曲面彼此相切，且運動瞬時螺旋曲面會沿著瞬時螺旋曲面滾動。三個位置合成的結果中，固定軸曲面與移動軸曲面具有類似雙曲面的趨勢，且將分為兩種情況，可對應到平面極心三角形所延伸的理論。

This thesis takes advantage of theories in planar kinematics and extends them to their counterparts in spatial kinematics. It is known that the pole in planar synthesis theory is the degeneration of the screw in spatial kinematics. This thesis seeks to find the spatial generalizations of curves arising in the motion of a planar double-slider linkage, such as the circles of moving/fixed centrodes, the circles of rotation curves, and the circles for three-position synthesis of sliders and inverted sliders.
For Spatial mechanisms, we use screw to describe the motion of rigid bodies. A screw used in instantaneous situation is called instantaneous screw. The locus of all instantaneous screws of a single degree-of-freedom motion is an instantaneous screw surface. A screw used in finite displacement is called rotation screw. The locus of all rotation screws is a rotation screw surface. For three-position synthesis of the spatial double-slider linkage, we obtain a double infinity of screw axes.
We investigate the motion of a special RCCC linkage. For the RC and CC dyad, joint axes of each dyad intersect perpendicularly. Various screw surfaces have been reported in this thesis. We observe that the screw surfaces associated with the special RCCC linkage are analogous to the curves associated with the planar double-slider linkage. In addition, the loci of the handle of the three-position synthesis problem are categorized into three types, and two of them resemble the shape of a hyperboloid.

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