線幾何學之研究，自於蒲律克 (Plücker) 提出了六參數之直線坐標式開始發展。空間中之直線，經由線性組合可形成特殊幾何外觀之線集合。另一方面，使用六參數坐標式定義螺旋為一帶有螺距的直線，並用於剛體的瞬時運動與受力狀態之研究，發展出螺旋理論。其主要優點為，瞬時運動螺旋與扳鉗具有線性性質，可進行線性運算進而形成螺旋系統。這使得螺旋理論成為空間機構之極佳分析工具，例如應用於串列式及並列式機器手臂之奇異構型與剛性分析。線集合與螺旋系統在組成的方式上，非常類似，其各自由獨立直線與基底螺旋經線性組合而得。經由剛體質點於瞬時運動之速度指向，可建構出瞬時運動螺旋所對應的線集合Linear Line Complex (LLC)。同時，應用互逆運算與線集合的交集原理，可以得到螺旋系統所對應之各式線集合。所以，直線與線集合可視為螺旋之建構元素。
已有學者針對剛體之有限位移，佐以新的螺距定義經代數運算而得到具有線性性質的有限位移螺旋。然而，鮮少研究是以線幾何學之觀點來求證有限位移具有線性性質。本論文採用已於瞬時螺旋運動中被証實的線幾何理論，研究點位移與線位移情況，得到了法向面(Nullplane) 與螺旋向量場 (Helicoidal Vector Field) 之存在定義。藉由法向面上之線叢 (Pencil of Lines)，建構出有限位移螺旋所對應之LLC，也証實其具有線性性質。我們也應用線幾何學的交集理論，研究不完全指定位移下之有限位移螺旋系統所對應的線集合。
在點的有限位移方面，法向面位於對應點之中點，其指向相同於對應點之指向。空間中所有法向面上的線叢形成了LLC，此螺旋向量場之螺距為沿螺旋軸的二分之一位移量除以二分之一旋轉量的正切值。二組對應點與一組對應點之不完全指定位移所對應的線集合分別是Congruence和線叢。在線的有限位移方面，法向面位於對應線之交會點，其指向相同於對應線之內角平分線方向。空間中所有法向面上的線叢形成了LLC，此螺旋向量場之螺距為沿螺旋軸位移量除以旋轉量的正弦值。單一組對應線之不完全指定位移所對應的線集合是為一個Regulus。
將線幾何學應用於剛體位移之求解問題，三組對應點所對應到的三個線集合，提供了五條線性獨立直線，可合成出位移螺旋及其大小。二組對應線所對應到的二個線集合，同樣地提供了五條線性獨立直線以合成出位移螺旋。在引用了最佳化LLC求解法及誤差評估準則後，以直線為位置參考之過指定位移亦得以求解，極具實務應用價值。
經由本論文的証明，點與線之有限位移於其特定的螺距定義下，可使其位移螺旋具有線性性質，進而可由交集法得到有限位移螺旋系統對應之線集合。本論文提出的論點，擴展了線幾何學的運用範圍，使得今後在研究瞬時運動螺旋與有限位移螺旋之時，具有統一的觀點與方法。

The research of line geometry stems from Plücker’s unique 6-tuple vector representation of a line in the three-dimensional space. In line geometry, we are concerned with line varieties, which are composed of linearly dependent lines of certain ranks. In kinematics, a screw can be thought of as a line with an associated pitch, and a screw is also represented by a 6-tuple vector form similar to the Plücker coordinates of a line. Screw theory has been shown to be a great tool in studying the instantaneous kinematics and statics of rigid-body systems because instantaneous screws and wrenches have linear properties and form screw systems. These linear features provide efficient methods for investigating spatial mechanisms, such as analyzing instantaneous motion, singularities and stiffness of serial and parallel manipulators. A line variety and a screw system are constructed similarly by the linear combinations of linearly independent lines and screws, respectively. Furthermore, there is a one-to-one correspondence between a screw and a line variety. In the context of instantaneous kinematics, all the lines normal to the velocity of every point of a rigid body undergoing a screw motion constitute a line variety called linear line complex. By using the intersections of line varieties and the notion of reciprocity, we can also obtain the correspondence between line varieties and screw systems. Therefore lines and line varieties are the basic entities in the research of screw geometry. In this dissertation, line geometry will be employed to investigate the finite kinematics of a rigid body.
Research in the past two decades had introduced new definition of pitch and obtained screw systems pertaining to finite displacements. However, there was a lack of investigation into the linearity of finite screws using line geometry. This dissertation extends the research of line geometry in instantaneous screws to that in finite screws. As a result, the nullplane and helicoidal vector field are found in the displacements of point and line elements of a rigid body. The linear line complex associated with a finite displacement screw, which is composed of the pencils of lines on all the nullplanes in the three-dimensional space, is also found. Furthermore, line varieties corresponding to the finite screw systems associated with point and line displacements are obtained by the intersections of linear line complexes.
For the finite displacement of a point, the nullplane passes through the midpoint of the pair of homologous points. The nullplane is perpendicular to the line segment connecting the homologous points. All the pencils of lines on the nullplanes constitute the linear line complex of the finite screw of the point displacement. According to the helicoidal vector field associated with the screw, the pitch is half-translation divided by the tangent of half-rotation. Moreover, the line varieties correspond to the two-point and one-point incompletely specified displacements are, respectively, a congruence and a pencil of lines.
For the finite displacement of a line, the nullplane passes through the intersection of a pair of intersecting homologous lines. The direction of nullplane is the same as the internal bisector the pair of homologous lines. All the pencils of lines on the nullplanes constitute the linear line complex of the finite screw of the line displacement. According to the helicoidal vector field of the screw, the pitch is the translation divided by the sine of rotation. Furthermore, the line variety corresponds to the one-line incompletely specified displacement is a regulus.
In this dissertation, the developed line geometry was also used to solve rigid body registration problems. It is well known that the displacement of a rigid body can be determined by three pairs of homologous points. Three pairs of homologous points of a finite displacement provide three pencils of lines that uniquely determine a linear line complex. We select five lines from the three pencils of lines to calculate the linear line complex, which in term gives a unique screw. Another way to specify a displacement is by using two pairs of homologous lines. Two pairs of homologous lines of a finite displacement provide two reguli that uniquely determine a linear line complex. We select five lines from the two reguli to calculate the linear line complex and thus the screw. In addition to exact specifications, we also introduce an approximate algorithm to determine the linear line complex based on the specifications of more than three points or two lines.
This dissertation has proved, based on line geometry, that the new definition of pitch of finite displacement screws provides linear properties of finite displacements. It gives geometric insight into line varieties which correspond to the finite screw systems associated with point and line displacements. The research presented in this dissertation also helps unify the study of instantaneous kinematics and finite kinemtics by using screws.

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