透過相對於空間中一個固定的反演球面反演一個甜甜圈(Ring Torus)，我們可以得到四次圓紋曲面(Ring Cyclide)的幾何曲面；對於其他相關的四次圓紋曲面也將會被討論以及展示。
基於Steiner Porism以及三球定理，希望可以對於Steiner Porism的基本元素三個彼此相切的圓圈鏈(3-chain for Steiner)，推廣到三個彼此相切球面鏈，進而得到並展示類似於Steiner Porism的幾何性質。

By means of inversion with respect to a sphere in space, torus will be transformed to surface, called cyclide. The parabolic cyclide will be shown in this thesis. Also, the deformation of Dupin cyclide illustrated by a symmetric Dupin horn cyclide will be demonstrated, too. Based on Steiner porism, Steiner annulus 3-chain(sphere case) is studied in this thesis. To obtain the geometric properties similiar to Steiner porism on surface of cyclide, we intend to focus on three mutually tangential spheres in Dupin ring cyclide.
keywords: Inversion; Dupin cyclide; Steiner porism.

Bibliography
[1] http ://mathworld:wolfram:com/Cyclide:html .
[2] http ://mathworld:wolfram:com/ParabolicCyclide:html .
[3] http ://mathworld:wolfram:com/StandardTori:html .
[4] http ://mathworld:wolfram:com/Torus:html .
[5] http://poncelet.math.nthu.edu.tw/disk3/cabrijava/steiner-porisms.html .
[6] http ://steiner/math/nthu/edu/tw/d3/d2/3d/3spheres/ .
[7] http ://steiner/math/nthu/edu/tw/d3/d2/Porism/ .
[8] http://www.oz.nthu.edu.tw/g9621515/thesis.html .
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symmetric dupin horn cyclide by means of inversion. Computer Aided Geometric Design, pages
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[10] Walter J. Meyer. Geometry and its applications. Elsevier Academic Press, 2nd ed. edition, 2006.