在二維(2-Dimension)卡氏座標系統(Cartesian coordinate system)中，座標系統經過平移及旋轉之變換後，其相對於原座標系統之轉換關係，可經由幾何圖形推算兩座標系統間之轉換關係，同理；三維(3-Dimension)卡氏座標系統經過平移及旋轉之變換後，相對於原座標系統之轉換關係，亦可經幾何圖形推算兩座標系統間之轉換關係。
然而，N維(N 4)座標系統無經由幾何圖形來推算兩座標系統間的轉換關係，因此，藉由二維旋轉轉換公式作為理論基礎，配合以矩陣表示平面間之交集，推導N維座標之旋轉。此外齊次座標系統中之齊次轉換矩陣，將平移及旋轉視為仿射轉換(Affine transformations)，且其矩陣的維度與齊次座標維度相同，因此可將旋轉及平移之兩種變化，同時以齊次轉換矩陣表示，進而推算N維座標系統轉換後之座標位置，即將原座標系統之座標位置以齊次座標系統表示後，乘上齊次轉換矩陣後可得。
此外，推導出之N維轉換矩陣可應用於幾何圖形上之旋轉與平移，本研究以橢圓為例，將橢圓以參數式表示法，配合轉換矩陣，可得經平移及旋轉後之橢圓參數式。

In the two-dimensional space, a Cartesian coordinate system might be transformed into another Cartesian coordinate system by rotations and translations, and their relation can be calculated via the geometric graph. Similarly, that kind of relationship in the three-dimensional space can also be calculated via the geometric graph. On the other hand, the geometric graph in the N-dimensional space cannot be visual as long as N is greater than three. Consequently, transformation relations among N-dimensional coordinate systems cannot be calculated via the geometric graph. For this reason, we set out to derive the relation between two coordinate systems by utilizing the two-dimensional transformation matrices as the basic component. When the transformation matrices in homogeneous coordinate system, the translations and rotations can be taken for an affine transformation where the dimensional size of those matrices is equal to the size of the homogeneous coordinate system. Therefore, the translations and rotations can be expressed by homogeneous matrices. Additionally, the N-dimensional homogeneous transformation matrix can be readily applied to represent 2nd-order (N-1)-dimensional space curves in the N-dimension space such as hyper-ellipses. geometric graphs which are after rotation and translation. For illustration, parametric expressions of an ellipse in various transformed coordinate systems are presented in this thesis, where the final parametric expressions of ellipses are obtained via multiplying the rotation matrix and followed by adding the position vector.

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