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The geometric visualization in this study concerns the three−dimensional (3D) polarimetric information of an arbitrary electromagnetic field. Based on previous research, a 3 × 3 coherency matrix <b>Φ</b> can be decomposed into an incoherent superposition of a totally 3D−polarized component <b>Φ</b>_{3D_p}, a specific partially 3D−polarized component <b>Φ</b>_{3D_pp} with a 3D degree of polarization (DoP) of 1/2, and a totally 3D−unpolarized component <b>Φ</b>_{3D_up}. Combining the physical meaning of this decomposition, we mathematically construct three polarization purities, namely, P_{3D_p}, P_{3D_pp}, and P_{3D_up}, for an arbitrary electromagnetic field to quantify the weight of the three 3D−polarized components. In order to show the proportion of the three polarized components of an electromagnetic field intuitively, we propose a geometric representation of a spatially quadric surface. Finally, two examples are cited to demonstrate the applicability of intuitively displaying the 3D polarimetric information of an arbitrary electromagnetic field.