非球面後視鏡曲面之最優化設計II

The Optimal Design of Non-Spherical Side Mirrors and Rear-View Mirrors II

本文討論如何設計非球面車輛側鏡曲面，使其有較佳之反射影像品質。非球面側鏡曲面的外側彎曲程度通常比內側大，藉此增加外側視角，使駕駛員得以感知較大範圍的後方車況。不過，這類側鏡易使反射影像非均向放大，造成影像扭曲。基於提升其反射影像品質的考慮，我們說明非球面側鏡曲面的設計可表述為一最優化曲面問題：使鏡面上的兩個主要曲率差的均方根值有最小值。針對此最優化曲面的計算，文中建議一種可行的數值計算策略。這個方法是先決定側鏡中心線的形狀，再依此建構最優化曲面。此計算過程是一複雜的計算過程，現階段筆者先針對側鏡中心線的形狀做分析。這樣的分析使我們得以估算側鏡的水平視角。同時，筆者設計一種非球面側鏡中心線形式，其曲率變化率沿著鏡面長度為常數，以維持鏡面中心線上的反射影像品質。對常見的側鏡大小而言，若其長度為W且外側一端及內側一端的曲率分別為ês1及ês2。其相對於平面鏡增加的水平視角約等於鏡面的長度再乘以鏡面兩端曲率的和，也就是W（ ês1 + ês2）。因此，一個長度為W = 200 mm的典型非球面側鏡；若外側取一般大型車輛球面側鏡的曲率值 ês1 =1250mm-1，且內側取小型車輛球面側鏡曲率值ês2 =11400mm-1，它增加的視角約為55°。

In this paper, we discuss how to design the shape of a non-sphericalside mirror with high image quality. For a non-spherical side mirror,respective to the driver, its outer-side is more curved than the inner-side.Because of the design of the mirror’s shape, it provides a wide angle imagewhen viewed from the driver’s perspective and lets the driver know theconditions behind in a wide range. A non-spherical side mirror alwayscauses distortion of the reflected image. We point out that, in order toreduce the image distortion on the non-spherical side mirror, the optimiza tion of the mirror’s profile must take the root-mean-square of the differencebetween two principle curvatures as the minimum. Based on this idea, apossible method to find the optimal profile is proposed in this paper. Thecentral line of the mirror must be fixed first, and then the profile constructedby the optimization. Mathematically, it is complicated to findthis profile, even when computer-aid numerical calculation is applied.Here the optimal shape of the central line of a non-spherical side mirror isanalyzed. According to this study, the horizontal viewing angle, whichcan be provided to the driver, is calculated. Let us consider a side mirrorwith length W and the outer-end and inner-end exhibit the curvature ês1 andês2 respectively. A new design of the central line is proposed. Its variationof the curvature, along the arc length of the curve on the central line, isa constant. The functional form of the central line can be numericallyobtained. For a non-spherical side mirror equipped with this central line,the horizontal viewing angle is W(ês1 + ês2) larger than the planar mirrorwith the same length W. As a result, for a typical one, W = 200 mm, ês1 =1250mm-1 and ês2 = 11400mm-1, the increased viewing angle is about 55°.