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？？