說起眼睛之『光學模型』，焉能不提及亥姆霍茲

赫爾曼·馮·亥姆霍茲（Hermann von Helmholtz，1821年8月31日－1894年9月8日），德國物理學家、醫生。

生平

赫爾曼·馮·亥姆霍茲1821年出生於德國的波茨坦，父親為當地文法中學的教師。從小愛好自然科學，但為生活計，在柏林的醫學和外科研究所諗了醫科，由於該研究所的畢業生必須參加8年的兵役，亥姆霍茲1843年起在波茨坦擔任軍醫。1848年在亞歷山大·馮·洪堡的推薦下，提前結束兵役，開始了漫長的教學生涯，先是在柏林藝術學院教解剖學，1849年前往柯尼斯堡（時屬普魯士王國的東普魯士省，今為俄羅斯的加里寧格勒）擔任生理學和病理學教授，1855年接手波恩的解剖學和生理學教席，1858年轉去海德堡的生理學教席 ，1870年成為普魯士科學學會的會員。1871年亥姆霍茲任柏林大學物理學教授，1888年成為新成立的夏洛特堡帝國物理學工程研究所的第一任主席。

物理學研究

1847年，亥姆霍茲出版了《力量的保存》（Erhaltung der Kraft）一書，闡明了能量守恆的原理，亥姆霍茲自由能即以他來命名。他也研究過電磁學 ，他的研究預測了麥克斯韋方程組中的電磁輻射，相關的方程式以他來命名。

除了物理，亥姆霍茲也對感知的研究作出貢獻。他發明了檢眼鏡，以及以他命名的共鳴器（Helmholtz-Resonator），他兩部光學和聲學的著作，《作為樂理的生理學基礎的音調感受的研究》（Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik）、《生理光學手冊》（Handbuch der Physiologischen Optik），對後世影響很大。

生理學研究

Helmholz是第一位將物理方法運用到神經傳導速度測量的人。1860年，他測量出神經傳導速率是90 m/s，然後他開始測量生物的反應速度，發現神經傳導到大腦後還要許久才會有反應，於是他推測，在感官資訊變成有意識的知覺之前，大腦必然在我們意識不到 的範圍裡先做了許多事去處理感官資訊，對神經訊號做評估、轉換、與重新導向。這就是認知心理學和認知神經科學中所稱｢無意識認知歷程｣的最初發現。^[2]

 

《生理光學》巨著的呢！！難到百年前的論文，至今仍能有大用嗎 ？？只講科學研究之風範、治學之嚴謹、敘述的條理分明、……就很值得一讀的了。感謝知識開放之潮流，可以有英文翻譯本閱讀的哩☆☆

Hermann von Helmholtz

Treatise on Physiological Optics

The Optical Society of America’s Southall (1924) translation of Hermann von Helmholtz’s Treatise on Physiological Optics (1910) is offered here for free download from the Graduate Center for Vision Research at the SUNY College of Optometry. The pages were originally scanned for Professor Benjamin Backus in 2001 by the University of Pennsylvania.

The page images in the PDF files are of excellent quality, but you may find they are too large to view using your web browser. We suggest you download them (e.g. right click and “save as”) and then view them on your computer. The smaller DjVu version may be more convenient: download the zip archive, extract it to a folder, and then open the file “directory.djvu” using a DjVu viewer.

The optical character recognition (OCR) in the DjVu and PDF files are useful for searching. Alas, the OCR is not of high quality and you may not find all instances of your target word(s). We would be delighted should you see fit to make and share a cleaner copy of the text.

Volume III begins with a discussion of perceptual inference. This is where most students of perception will want to start.

Happy reading!

The Backus Lab
September 2011

 

Helmholtz’s Treatise on Physiological Optics

Permission to use these files is granted for nonprofit purposes only.
Please see first page of the document for additional copyright information.

Volume I: Anatomy, physiology, and dioptrics of the eye; Gullstrand appendix

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Volume II: Visual sensation (color, contrast, adaptation, etc.)

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  Volume III: Visual perception (depth, motion, etc.)

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雖然此處眼睛之古典球面近似『模型參數』並非出自該巨著，實則道理相通，不過援引新『典型』數據而已。也許現今科技已經追求『個人化』之模型的了。

 

 

角膜→房水【前室】→虹膜【瞳孔】→晶狀體→玻璃體【後部】→視網膜

【角膜】 cornea
非球面
折射率︰1.376
前緣半徑︰ 7.8 mm
後緣半徑︰6.4 mm
厚度︰ 0.6 mm

【房水】 aqueous humor
折射率︰1.336
厚度︰3.0 mm

【瞳孔】 pupil
直徑變化︰ 1.5 mm ~ 8.0 mm

【晶狀體】 crystalline lens
折射率︰梯度， 1.386 ~ 1.406
前緣半徑︰10.1 mm
後緣半徑︰6.1 mm
厚度︰4.0 mm

【玻璃體】 vitreous humor
厚度︰16.9 mm

 

 

 

 

 

 

 

 

 

 

 

詩經‧國風‧齊風‧猗嗟

猗嗟昌兮，頎而長兮。
抑若颺兮，美目颺兮。
巧趨蹌兮，射則臧兮。

猗嗟名兮，美目清兮。
儀既成兮，終日射侯，
不出正兮，展我甥兮。

猗嗟孌兮，清颺婉兮。
舞則選兮，射則貫兮，
四矢反兮，以禦亂兮。

 

魯莊公清颺美目來自寬容敦厚乎！所以能放管仲，因此方有

論語‧《憲問》

子貢曰：管仲非仁者與？桓公殺公子糾，不能死，又相之。

子曰：管仲相桓公，霸諸侯，一匡天下，民到于今受其賜。微管仲 ，吾其被髮左衽矣。豈若匹夫匹婦之為諒也，自經於溝瀆，而莫之知也。

論語褒貶耶？

之前我們曾經宣說 Michael Nielsen 之《神經網絡與深度學習》 Neural Networks and Deep Learning 大作。乃今談談『辨物識人』眼睛的光學原理，也算前後完整的吧。

尚未講『眼睛』之□○︰

眼（亦稱眼睛、目、招子）是視覺的器官，可以感知光線，轉換為神經中電化學的脈衝。比較複雜的眼睛是一個光學系統，可以收集周遭環境的光線，藉由虹膜調整進入眼睛的強度，利用可調整的晶狀體來聚焦，投射到對光敏感的視網膜產生影像，將影像轉換為電的訊號，透過視神經傳遞到大腦的視覺系統及其他部份。眼睛依其辨色能力可以分為十種不同的種類，有96%的動物其眼睛都是複雜的光學系統^[1]。其中軟體動物、脊索動物及節肢動物的眼睛有成像的功能^[2]。

微生物的「眼睛」構造最簡單，只偵測環境的暗或是亮，這對於晝夜節律的牽引有關^[3]。若是更複雜的眼睛，視網膜上的感光神經節細胞沿著視網膜下視丘路徑傳送信號到視叉上核來影響影響生理調節，也送到頂蓋前核控制瞳孔光反射。

 

且先提

屈光度

屈光度，或稱焦度，英語用「Dioptre」表示，是量度透鏡或曲面鏡屈光能力的單位。

焦距f的長短標誌著折光能力的大小，焦距越短，其折光能力就越大，近視的原因就是眼睛折光能力太大，遠視的人則折光能力太弱。

焦距的倒數叫做透鏡焦度，或屈光度，用φ表示，即: φ=  ，如：焦距是15m，那麼φ= 
凸透鏡（如：遠視鏡片）的度數是正數(+)，凹透鏡（如：近視鏡片）的度數是負數(-)。

一個+3屈光度的透鏡，會把平行的光線聚焦在鏡片的1/3米外。

屈光度的單位簡寫是D，國際單位制的單位是 m^-1。

一般眼鏡常使用度數來表示屈光度，以屈光度 D 的數值乘以 100 就是度數^[1] ，例如 -1.0D 等於近視眼鏡（凹透鏡）的 100度。

Dioptre

A dioptre (uk), or diopter (us), is a unit of measurement of the optical power of a lens or curved mirror, which is equal to the reciprocal of the focal length measured in metres (that is, 1/metres). It is thus a unit of reciprocal length. For example, a 3-dioptre lens brings parallel rays of light to focus at ¹⁄₃ metre. A flat window has an optical power of zero dioptres, and does not converge or diverge light.

Dioptres are also sometimes used for other reciprocals of distance, particularly radii of curvature and the vergence of optical beams. The usage was proposed by French ophthalmologist Ferdinand Monoyer in 1872, based on earlier use of the term dioptrice by Johannes Kepler.^[1][2][3]

The main benefit of using optical power rather than focal length is that the lensmaker’s equation has the object distance, image distance, and focal length all as reciprocals. A further benefit is that when relatively thin lenses are placed close together their powers approximately add. Thus, a thin 2-dioptre lens placed close to a thin 0.5-dioptre lens yields almost the same focal length as a 2.5-dioptre lens would have.

Though the dioptre is based on the SI–metric system it has not been included in the standard so that there is no international name or abbreviation for this unit of measurement—within the international system of units, this unit for optical power would need to be specified explicitly as the inverse metre (m⁻¹). However most languages have borrowed the original name and some national standardization bodies like DIN specify a unit name (dioptrie, dioptria, etc.) and derived unit symbol “dpt”.

In vision correction

The fact that optical powers are approximately additive enables an eye care professional to prescribe corrective lenses as a simple correction to the eye’s optical power, rather than doing a detailed analysis of the entire optical system (the eye and the lens). Optical power can also be used to adjust a basic prescription for reading. Thus an eye care professional, having determined that a myopic (nearsighted) person requires a basic correction of, say, −2 dioptres to restore normal distance vision, might then make a further prescription of ‘add 1’ for reading, to make up for lack of accommodation (ability to alter focus). This is the same as saying that −1 dioptre lenses are prescribed for reading.

In humans, the total optical power of the relaxed eye is approximately 60 dioptres.^[4] The cornea accounts for approximately two-thirds of this refractive power (about 40 dioptres) and the crystalline lens contributes the remaining one-third (about 20 dioptres).^[4]) In focusing, the ciliary muscle contracts to reduce the tension or stress transferred to the lens by the suspensory ligaments. This results in increased convexity of the lens which in turn increases the optical power of the eye. As humans age, the amplitude of accommodation reduces from approximately 15 to 20 dioptres in the very young, to about 10 dioptres at age 25, to around 1 dioptre at 50 and over.

Convex lenses have positive dioptric value and are generally used to correct hyperopia (farsightedness) or to allow people with presbyopia (the limited accommodation of advancing age) to read at close range. Concave lenses have negative dioptric value and generally correct myopia (nearsightedness). Typical glasses for mild myopia will have a power of −1.00 to −3.00 dioptres, while over the counter reading glasses will be rated at +1.00 to +3.00 dioptres. Optometrists usually measure refractive error using lenses graded in steps of 0.25 dioptres.

 

這個度量單位，不只是因為實務上的使用，假使簡單考察『相距 L 之兩薄透鏡組合』

pi@raspberrypi:~ f\phi = \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{L}{f_1 \cdot f_2}\phi = {\phi}_{f_1} + {\phi}_{f_2} - L \cdot {\phi}_{f_1} \cdot {\phi}_{f_2}$

或能知其原由的了！！
 
 
 
 
 
 
 
 
 
 
 
 
	

何不讓我們再次享受 Justin Peatross 和 Michael Ware 兩位先生精彩的文筆︰

 

 

 

 

 

 

隨著書香、滿懷興致，飄進光學裝置之天地也！！

 

 

 

 

 

 

 

 

 

 

 

孫子兵法

《兵勢》

孫子曰：凡治眾如治寡，分數是也。鬥眾如鬥寡，形名是也。三軍之眾，可使必受敵而無敗者，奇正是也。兵之所加，如以碬投卵者 ，虛實是也。

凡戰者，以正合，以奇勝。故善出奇者，無窮如天地，不竭如江河 ，終而復始，日月是也；死而復生，四時是也。聲不過五，五聲之變，不可勝聽也。色不過五，五色之變，不可勝觀也。味不過五，五味之變，不可勝嘗也。戰勢不過奇正，奇正之變，不可勝窮也。奇正相生，如循環之無端，孰能窮之哉！

激水之疾，至于漂石者，勢也。鷙鳥之擊，至于毀折者，節也。是故善戰者，其勢險，其節短，勢如張弩，節如機發。

紛紛紜紜，鬥亂，而不可亂也。渾渾沌沌，形圓，而不可敗也。亂生于治，怯生于勇，弱生于強。治亂，數也。勇怯，勢也。強弱，形也。故善動敵者，形之，敵必從之；予之，敵必取之；以利動之 ，以實待之。

故善戰者，求之于勢，不責于人，故能擇人任勢；任勢者，其戰人也，如轉木石，木石之性，安則靜，危則動，方則止，圓則行。故善戰人之勢，如轉圓石于千仞之山者，勢也。

 

就幾何光學理論來說，透鏡、材質、虛空之矩陣的組合亦無窮矣！不知能否對治各種

像差

像差（英語：Optical aberration）是光學中，實際像與根據單透鏡理論確定的理想像的偏離。這些偏離是折射作用造成的。像差是由透鏡對色光的不同彎曲能力所致，並造成帶有色暈的像。單色像差與是與色無關的像差，包括使畸變、像場彎曲等變形像差和面像、形像、散光等使像模糊的像差。像差在照相機、望遠鏡和其他光學儀器中可以通過透鏡的組合減小到最低限度。面鏡也有與透鏡一樣的單色像差，沒有像差。

初階像差分為五種：球面像差、 彗形像差、 散光、 場曲、 畸變。

Optical aberration

An optical aberration is a departure of the performance of an optical system from the predictions of paraxial optics.^[1] In an imaging system, it occurs when light from one point of an object does not converge into (or does not diverge from) a single point after transmission through the system. Aberrations occur because the simple paraxial theory is not a completely accurate model of the effect of an optical system on light (due to the wave nature of light), rather than due to flaws in the optical elements.^[2]

Aberration leads to blurring of the image produced by an image-forming optical system. Makers of optical instruments need to correct optical systems to compensate for aberration.

The articles on reflection, refraction and caustics discuss the general features of reflected and refracted rays.

 

的呢？且先知目前科技總結乎！！

Laser guide stars are used to eliminate optical aberrations.^[8]

Practical elimination of aberrations

The classical imaging problem is to reproduce perfectly a finite plane (the object) onto another plane (the image) through a finite aperture. It is impossible to do so perfectly for more than one such pairs of planes (this was proven with increasing generality by Maxwell in 1858, by Bruns in 1895, and by Carathéodory in 1926, see summary in Walther, A., J. Opt. Soc. Am. A 6, 415–422 (1989)). For a single pair of planes (e.g. for a single focus setting of an objective), however, the problem can in principle be solved perfectly. Examples of such a theoretically perfect system include the Luneburg lens and the Maxwell fish-eye.

Practical methods solve this problem with an accuracy which mostly suffices for the special purpose of each species of instrument. The problem of finding a system which reproduces a given object upon a given plane with given magnification (insofar as aberrations must be taken into account) could be dealt with by means of the approximation theory; in most cases, however, the analytical difficulties were too great for older calculation methods but may be ameliorated by application of modern computer systems. Solutions, however, have been obtained in special cases (see A. Konig in M. von Rohr’s Die Bilderzeugung, p. 373; K. Schwarzschild, Göttingen. Akad. Abhandl., 1905, 4, Nos. 2 and 3). At the present time constructors almost always employ the inverse method: they compose a system from certain, often quite personal experiences, and test, by the trigonometrical calculation of the paths of several rays, whether the system gives the desired reproduction (examples are given in A. Gleichen, Lehrbuch der geometrischen Optik, Leipzig and Berlin, 1902). The radii, thicknesses and distances are continually altered until the errors of the image become sufficiently small. By this method only certain errors of reproduction are investigated, especially individual members, or all, of those named above. The analytical approximation theory is often employed provisionally, since its accuracy does not generally suffice.

In order to render spherical aberration and the deviation from the sine condition small throughout the whole aperture, there is given to a ray with a finite angle of aperture u* (width infinitely distant objects: with a finite height of incidence h*) the same distance of intersection, and the same sine ratio as to one neighboring the axis (u* or h* may not be much smaller than the largest aperture U or H to be used in the system). The rays with an angle of aperture smaller than u* would not have the same distance of intersection and the same sine ratio; these deviations are called zones, and the constructor endeavors to reduce these to a minimum. The same holds for the errors depending upon the angle of the field of view, w: astigmatism, curvature of field and distortion are eliminated for a definite value, w*, zones of astigmatism, curvature of field and distortion, attend smaller values of w. The practical optician names such systems: corrected for the angle of aperture u* (the height of incidence h*) or the angle of field of view w*. Spherical aberration and changes of the sine ratios are often represented graphically as functions of the aperture, in the same way as the deviations of two astigmatic image surfaces of the image plane of the axis point are represented as functions of the angles of the field of view.

The final form of a practical system consequently rests on compromise; enlargement of the aperture results in a diminution of the available field of view, and vice versa. But the larger aperture will give the larger resolution. The following may be regarded as typical:

(1) Largest aperture; necessary corrections are — for the axis point, and sine condition; errors of the field of view are almost disregarded; example — high-power microscope objectives.

(2) Wide angle lens; necessary corrections are — for astigmatism, curvature of field and distortion; errors of the aperture only slightly regarded; examples — photographic widest angle objectives and oculars.

Between these extreme examples stands the normal lens: this is corrected more with regard to aperture; objectives for groups more with regard to the field of view.

(3) Long focus lenses have small fields of view and aberrations on axis are very important. Therefore zones will be kept as small as possible and design should emphasize simplicity. Because of this these lenses are the best for analytical computation.

然後了此處摘要之偏差名義︰

【球面像差】 Spherical aberration

On top is a depiction of a perfect lens without spherical aberration: all incoming rays are focused in the focal point.

The bottom example depicts a real lens with spherical surfaces, which produces spherical aberration: The different rays do not meet after the lens in one focal point. The further the rays are from the optical axis, the closer to the lens they intersect the optical axis (positive spherical aberration).

(Drawing is exaggerated.)

 

【彗形像差】 comatic aberration

Coma of a single lens

 

【像散】 Astigmatism

Page explaining and illustrating astigmatism^[2]

 

【佩茲瓦爾像場彎曲】 Petzval field curvature

Field curvature: the image “plane” (the arc) deviates from a flat surface (the vertical line).

 

【色差】 Chromatic aberration

Photographic example showing high quality lens (top) compared to lower quality model exhibiting lateral chromatic aberration (seen as a blur and a rainbow edge in areas of contrast.)

 

【鏡頭眩光】 Lens flare

Lens flare on Borobudur stairs to enhance the sense of ascending

 

也可思複雜光學系統之原由吧？？將能分解閱讀器物耶！！？？

 

 

 

 

 

 

 

 

 

 

 

 

思創之心自然生起，遊戲精神自在無跡。

器物發明文載史記，十之八九需求第一。

 

偶讀俞欣豪博士轉載的

《笛卡爾與三葉蟲有什麼關係？》

文章，不禁讚嘆造物之神奇，進化的奧義！假使要問

三葉蟲

三葉蟲（學名：trilobite）是節肢動物門中已經滅絕的三葉蟲綱中的動物。它們最早出現於寒武紀，在古生代早期達到頂峰，此後逐漸減少至滅絕。最晚的三葉蟲於二億五千萬年前二疊紀結束時的生物集群滅絕中消失。三葉蟲是非常知名的化石動物，其知名度可能僅次於恐龍。在所有的化石動物中三葉蟲是種類最豐富的，至今已經確定的有九（或者十）個目，一萬五千多個物種。大多數三葉蟲是比較簡單的、小的海生動物，它們在海底爬行，通過過濾泥沙來吸取營養。它們身體分節，有帶溝將身體分為三個垂直的葉。在世界各地都有發現過其化石。

Koneprusia brutoni

……

雖然三葉蟲只在背部有盔甲，但是它們的外骨骼還是相當重的，它們的外骨骼是由甲殼素為主的蛋白質聯合方解石和磷化鈣等礦物組成的。不像其他節肢動物那樣能夠在蛻皮前重新吸收外骨骼中的大部分礦物，三葉蟲在蛻皮是將所有盔甲中的礦物全部拋棄，因此一個三葉蟲可以留下多個良好地礦物化的外骨骼，這提高了三葉蟲化石的數量。在蛻皮時 外骨骼首先在頭部和胸部之間分開，這是為什麼許多三葉蟲的化石不是缺少頭部就是缺少胸部，其實許多化石是三葉蟲蛻掉的皮，而不是死去的三葉蟲形成的。大多 數三葉蟲的頭部有兩個面部縫合來簡化蛻皮過程。頭部的兩側有一對複眼，有些種的複眼相當先進。事實上約5.43億年前三葉蟲是第一批進化出真正的眼睛的動物。有人認為眼睛的出現是寒武紀生命大爆發的導致原因。

……

許多三葉蟲有眼睛，它們還有可能用來作味覺和嗅覺器官的觸角。有些三葉蟲是瞎的，可能它們居住在非常深的海底，那裡沒有光，因此用不著眼睛。有些（比如蛙形鏡眼蟲Phacops rana）有很大的眼睛。

三葉蟲的眼睛是由方解石（碳酸鈣，CaCO₃）組成的。純的方解石是透明的，有些三葉蟲使用單晶的、透明的方解石來組成其每隻眼睛的透鏡。這與大多數其他節肢動物不同，大多數節肢動物使用軟透鏡的、由甲殼素組成的眼睛。三葉蟲堅固的方解石透鏡無法像人的軟晶狀體的眼睛那樣來調節焦距。但是有些三葉蟲的方解石組成一個內部的、複合結構，這個結構可以降低球差，同時提供極好的景深。在今天生存的動物中蛇尾海星Ophiocoma wendtii使用類似的透鏡。

典型的三葉蟲眼睛是複眼，每個透鏡都是一個拉長的稜鏡。每隻複眼內的透鏡數不等，有些只有一個，有些可達上千。在這樣的複眼中其透鏡一般排列為六邊形。

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有何特異？難到想有眼睛而已！生命突破雖是不易，自始唯堅定一心矣！！笛卡爾已曉像差之義﹐豈不補天窮盡其理

Cartesian oval

In geometry, a Cartesian oval, named after René Descartes, is a plane curve, the set of points that have the same linear combination of distances from two fixed points.

Example of Cartesian ovals.

Definition

Let P and Q be fixed points in the plane, and let d(P,S) and d(Q,S) denote the Euclidean distances from these points to a third variable point S. Let m and a be arbitrary real numbers. Then the Cartesian oval is the locus of points S satisfying d(P,S) + m d(Q,S) = a. The two ovals formed by the four equations d(P,S) + m d(Q,S) = ± a and d(P,S) − m d(Q,S) = ± a are closely related; together they form a quartic plane curve called the ovals of Descartes.^[1]

Special cases

In the equation d(P,S) + m d(Q,S) = a, when m = 1 and a > d(P,Q) the resulting shape is an ellipse. In the limiting case in which P and Q coincide, the ellipse becomes a circle. When  it is a limaçon of Pascal. If  and  the equation gives a branch of a hyperbola and thus is not a closed oval.

Polynomial equation

The set of points (x,y) satisfying the quartic polynomial equation^[1][2]

[(1 – m²)(x² + y²) + 2m²cx + a² − m²c²]² = 4a²(x² + y²),

where c is the distance between the two fixed foci P = (0, 0) and Q = (c, 0), forms two ovals, the sets of points satisfying the two of the four equations

d(P,S) ± m d(Q,S) = a,

d(P,S) ± m d(Q,S) = −a^[2]

that have real solutions. The two ovals are generally disjoint, except in the case that P or Q belongs to them. At least one of the two perpendiculars to PQ through points P and Q cuts this quartic curve in four real points; it follows from this that they are necessarily nested, with at least one of the two points P and Q contained in the interiors of both of them.^[2] For a different parametrization and resulting quartic, see Lawrence.^[3]

Applications in optics

As Descartes discovered, Cartesian ovals may be used in lens design. By choosing the ratio of distances from P and Q to match the ratio of sines in Snell’s law, and using the surface of revolution of one of these ovals, it is possible to design a so-called aplanatic lens, that has no spherical aberration.^[4]

Additionally, if a spherical wavefront is refracted through a spherical lens, or reflected from a concave spherical surface, the refracted or reflected wavefront takes on the shape of a Cartesian oval. The caustic formed by spherical aberration in this case may therefore be described as the evolute of a Cartesian oval.^[5]

History

The ovals of Descartes were first studied by René Descartes in 1637, in connection with their applications in optics.

These curves were also studied by Newton beginning in 1664. One method of drawing certain specific Cartesian ovals, already used by Descartes, is analogous to a standard construction of an ellipse by stretched thread. If one stretches a thread from a pin at one focus to wrap around a pin at a second focus, and ties the free end of the thread to a pen, the path taken by the pen, when the thread is stretched tight, forms a Cartesian oval with a 2:1 ratio between the distances from the two foci.^[6] However, Newton rejected such constructions as insufficiently rigorous.^[7] He defined the oval as the solution to a differential equation, constructed its subnormals, and again investigated its optical properties.^[8]

The French mathematician Michel Chasles discovered in the 19th century that, if a Cartesian oval is defined by two points P and Q, then there is in general a third point R on the same line such that the same oval is also defined by any pair of these three points.^[2]

James Clerk Maxwell rediscovered these curves, generalized them to curves defined by keeping constant the weighted sum of distances from three or more foci, and wrote a paper titled Observations on Circumscribed Figures Having a Plurality of Foci, and Radii of Various Proportions. An account of his results, titled On the description of oval curves, and those having a plurality of foci, was written by J.D. Forbes and presented to the Royal Society of Edinburgh in 1846, when Maxwell was at the young age of 14 (almost 15).^[6][9][10]

 

或因不解彩虹情意

色差

色差是指光學上透鏡無法將各種波長的色光都聚焦在同一點上的現象^[1]。它的產生是因為透鏡對不同波長的色光有不同的折射率（色散現象）。對於波長較長的色光，透鏡的折射率較低。在成像上，色差表現為高光區與低光區交界上呈現出帶有顏色的「邊緣」，這是由於透鏡的焦距與折射率有關，從而光譜上的每一種顏色無法聚焦在光軸上的同一點。色差可以是縱向的，由於不同波長的色光的焦距各不相同，從而它們各自聚焦在距離透鏡遠近不同的點上；色差也可以是橫向或平行排列的，由於透鏡的放大倍數也與折射率有關，此時它們會各自聚焦在焦平面上不同的位置。

對於消色差雙合透鏡而言，可見光的波長近似具有相等的焦距

 

且將留待后人傳奇？？