賽德像差多項式是以物高和入射光瞳的極座標為冪級數展開而來，其中入光瞳的極座標又會是光源入射角度的函數，雖然使用光源入射角度比起使用光瞳的極座標，對於光線的描述更為複雜，但能更直覺且真實呈現入射光角度和光線像差之間的關係，因此本論文推導另一種形式的像差多項式，使用物高和光源入射角度作為獨立變數，對像差函數進行泰勒級數展開，再利用光線在旋轉對稱光學系統傳播的對稱性質，將展開式進行簡化，並與賽德像差多項式作對應，即可推得賽德像差多項式的系數表示式。接著使用此理論配合本研究室的數值計算軟體，以Petzval透鏡為範例，計算出賽德像差多項式係數和像差值，並與光學軟體Zemax模擬得到的像差值作比較，來驗證本論文所提出理論的正確性。
光學軟體中，賽德係數是由幾何光學的方式推導而來，過程中使用多次的近軸近似，然而賽德像差是三階的像差，因此使用一階的方式推導顯得不夠合理，本論文使用純數學的方式推導出賽德像差多項式係數，以光線的三階微分表示，並使用本研究室數值計算軟體計算出實際數值，來得到更加準確的三階像差值。

Seidel coefficients of Zemax are calculated by the method of geometric optics. Several paraxial equations were used in the process of calculation. However, Seidel aberrations are third order aberrations. It seems that it’s not proper to derive Seidel aberrations by using first order method. In this paper, we derived the ray aberration polynomials in terms of the partial derivatives of the ray. Moreover, using computer program to calculate the more accurate value of third-order ray aberrations.

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