目前不完全指定位移與線幾何理論的發展，已經成功地找出指定一點、兩點和一直線位移螺旋系統之線性線集合。本論文將延續不完全指定位移與線幾何理論之研究，將線幾何學應用到目前所有已知的剛體元素不完全指定位移螺旋系統，並且討論線幾何學是否也可以應用在剛體元素位移為無限小也就是瞬時運動的情況。
線幾何學為研究線集合的工具，我們可以利用螺旋基底的線性線集合交集得到其螺旋系統的線性線集合，並且可從足夠的線集合資訊中合成出剛體扭轉螺旋。由於螺旋系統可將螺旋基底線性組合之後做為新的螺旋基底，我們利用螺旋系統此特性將螺旋基底變為純平移或純角位移螺旋，只要螺旋系統是由純平移或純角位移螺旋基底所構成，我們即可很明確的找出其螺旋系統之線性線集合。
本論文將線幾何學應用到平面、有方向性平面與平面上一直線，成功地找到其螺旋系統之線性線集合，並且從足夠的線集合資訊中合成出剛體扭轉螺旋。另外直接從瞬時運動學的角度找出了可造成剛體元素瞬時運動之螺旋系統與螺旋系統之線性線集合，並且也從足夠的線集合資訊中合成出剛體瞬時螺旋，同時討論了線性線集合與剛體元素瞬時運動之螺旋系統之間的物理意義。
從本論文的結果可以得知，目前所有已知的指定剛體元素位移，無論其位移為有限值或無限小都具有螺旋系統的性質，並且都可以將線幾何學應用於其螺旋系統，此結果完善的顯示出線幾何學與指定剛體元素位移螺旋系統之間的關聯。

Developments in line geometry have successfully found the linear line sets pertaining to the incompletely specified displacements of one point, two points, and a line. This thesis advances the related research by investigating the line geometry of the incompletely specified displacements of plane elements. In addition, the line geometry of screw systems arising in the instantaneous kinematics of incompletely specified displacements is also explored in this thesis.
Line geometry provides geometric foundations to screw systems in finite and instantaneous kinematics. A screw system is represented by the linear combination of basis screws, while the linear line set corresponding to the screw system is obtained by the intersection of linear line congruences corresponding to the basis screws. This thesis utilizes the intersection and sum operations of linear line sets to find the linear line sets corresponding to certain screw systems.
This thesis gives detailed derivations of the linear line sets pertaining to the displacements of a plane, a directed plane, and a line on a plane. In instantaneous kinematics of incompletely-specified displacements, we give explicit expressions for screw systems, and the corresponding linear line sets are illustrated. Furthermore, physical meanings of the linear line sets in instantaneous kinematics of incompletely-specified displacements are also discussed in this thesis.
This thesis provides a thorough study in the line geometry of incompletely specified displacements, including finitely and infinitesimally separated positions. In addition to their significances in theoretical kinematics, the results in this thesis give detailed insights into the screw systems arising in incompletely specified displacements, which can be applied to practical cases when the displacement of only a portion of a rigid body is of interest.

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