<![CDATA[This study develops, step by step, the geometric transformation that carries the conventional two-dimensional Cartesian plane into a parallel coordinate system. Cartesian coordinates, while foundational in geometry and mathematics, are inherently limited to three dimensions, restricting their effectiveness in representing multidimensional data. As a solution, the parallel coordinate system, originally introduced by Maurice d’Ocagne and later developed into a coordinate system for multidimensional geometry by Alfred Inselberg, is explored for its ability to visualize data with higher dimensions using equidistant, parallel axes. The transformation is carried out in two explicit steps: a 90° rotation of the X-axis about the origin, so that it comes to coincide with the Y-axis, followed by a translation that separates the two axes by a fixed distance d. The duality principle is then applied to illustrate the reversed roles of points and lines. Two results follow. Every point P(x1,y1) of the Cartesian plane is mapped to a line segment joining the two parallel axes; and every line y=mx+b with m≠1 is mapped to the single point (d/1-m, b/1-m), while in the singular case m = 1 the dual is an ideal point of the projective extension of the plane. Visual constructions and worked examples validate the model, showing that the transformation preserves collinearity and the ratios of segments along a line while enabling multidimensional insights. This research highlights the potential of parallel coordinates in mathematical visualization, particularly in representing relationships among multiple variables in two-dimensional spaces.]]>
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