23 | 8 月 | 2016 | FreeSandal

王之渙‧登鸛雀樓

白日依山盡，黃河入海流；
欲窮千里目，更上一層樓。

黑水智︰天地如風箱，開關司啟閉，陰 ䷁ 陽 ䷀ 之情見矣。伏羲氏之大易理則，孤虛者的邏輯宇宙，布林代數邏輯電路的數位設計之國度。

派︰未知何年何月，有一

孤虛者言︰

物有無者，非真假也。苟日新，日日新，又日新。真假者，物之論也。論也者，當或不當而已矣。故世有孤虛者，言有孤虛論。孤虛何謂也？甲乙孤虛，言不得全真也，索其孤其虛而已矣。天地孤虛，去其上下也，善惡孤虛，何得善惡並真乎？是故孤虛論全矣！

其法曰︰物物孤虛，言物之非也；孤虛之孤虛，此孤虛之非也。使甲與乙並，此甲乙辜虛之非也，強使之或，乃非甲非乙之孤虛也。若云由此及彼，雖言之鑿鑿，若非彼與此之孤虛，无能以斷疑是也！！

假使依據孤虛 ── Sheffer 豎線 ──所說則︰

非 $P$ ︰ $\sim P = P \mid P$

$P$ 且 $Q$ ︰ $P \cdot Q = P \wedge Q = (P \mid Q) \mid (P \mid Q)$

$P$ 或 $Q$ ︰ $P + Q = P \vee Q = (P \mid P) \mid (Q \mid Q)$

若 $P$ 則 $Q$ ︰ $P \rightarrow Q = P \mid (Q \mid Q)$

── 摘自《M♪o 之學習筆記本《子》開關︰【黑水智】數位之源》

若說生物擁有眼睛，所以得見天地萬象。人類已有出類拔萃之眼睛，而且能知『成像定律』︰

‧ 空間‧透鏡‧空間，成像也！

‧ 透鏡‧空間‧透鏡，等效於透鏡乎？？

pi@raspberrypi:~ $ipython3 Python 3.4.2 (default, Oct 19 2014, 13:31:11) Type "copyright", "credits" or "license" for more information. IPython 2.3.0 -- An enhanced Interactive Python. ? -> Introduction and overview of IPython's features. %quickref -> Quick reference. help -> Python's own help system. object? -> Details about 'object', use 'object??' for extra details. In [1]: from sympy import * In [2]: from sympy.physics.optics import FreeSpace, FlatRefraction, ThinLens, GeometricRay, CurvedRefraction, RayTransferMatrix In [3]: init_printing() In [4]: f1, L, f2 = symbols('f1, L, f2') In [5]: 相距L之兩薄透鏡組合 = ThinLens(f2) * FreeSpace(L) * ThinLens(f1) In [6]: 相距L之兩薄透鏡組合 Out[6]: ⎡ L ⎤ ⎢ - ── + 1 L ⎥ ⎢ f₁ ⎥ ⎢ ⎥ ⎢ L ⎥ ⎢ - ── + 1 ⎥ ⎢ 1 f₂ L ⎥ ⎢- ── - ──────── - ── + 1⎥ ⎣ f₂ f₁ f₂ ⎦ In [7]: 相距L之兩薄透鏡組合.C.expand() Out[7]: L 1 1 ───── - ── - ── f₁⋅f₂ f₂ f₁ In [8]: 前主平面 = (1 - 相距L之兩薄透鏡組合.D) / 相距L之兩薄透鏡組合.C In [9]: 前主平面.expand() Out[9]: L ─────────── L f₂ ── - 1 - ── f₁ f₁ In [10]: 後主平面 = (1 - 相距L之兩薄透鏡組合.A) / 相距L之兩薄透鏡組合.C In [11]: 後主平面.expand() Out[11]: L ─────────── L f₁ ── - ── - 1 f₂ f₂ In [12]: 等效薄透鏡 = FreeSpace(後主平面.expand()) * 相距L之兩薄透鏡組合 * FreeSpace(前主平面.expand()) In [13]: 等效薄透鏡.A.simplify() Out[13]: 1 In [14]: 等效薄透鏡.B.simplify() Out[14]: 0 In [15]: 等效薄透鏡.C.simplify() Out[15]: L - f₁ - f₂ ─────────── f₁⋅f₂ In [16]: 等效薄透鏡.D.simplify() Out[16]: 1 In [17]: </pre> 豈非是大自然『得色』者耶！！？？然而雖想更上一層樓，也欲窮『千里目』，無奈遠方『視角』太小 <h1 id="firstHeading" class="firstHeading" lang="en"><a style="color: #003300;" href="https://en.wikipedia.org/wiki/Angle_of_view">Angle of view</a></h1> In <a style="color: #808080;" title="Photography" href="https://en.wikipedia.org/wiki/Photography">photography</a>, angle of view (AOV)<a style="color: #808080;" href="https://en.wikipedia.org/wiki/Angle_of_view#cite_note-1">[1]</a> describes the <a style="color: #808080;" title="Angle" href="https://en.wikipedia.org/wiki/Angle">angular</a> extent of a given scene that is imaged by a <a style="color: #808080;" title="Camera" href="https://en.wikipedia.org/wiki/Camera">camera</a>. It is used interchangeably with the more general term <a style="color: #808080;" title="Field of view" href="https://en.wikipedia.org/wiki/Field_of_view">field of view</a>. <img class="alignnone size-full wp-image-58491" src="http://www.freesandal.org/wp-content/uploads/425px-Angle_of_view.svg.png" alt="425px-Angle_of_view.svg" width="425" height="351" /> A camera's angle of view can be measured horizontally, vertically, or diagonally. It is important to distinguish the angle of view from the angle of coverage, which describes the angle range that a lens can image. Typically the <a style="color: #808080;" title="Image circle" href="https://en.wikipedia.org/wiki/Image_circle">image circle</a> produced by a lens is large enough to cover the film or sensor completely, possibly including some <a style="color: #808080;" title="Vignetting" href="https://en.wikipedia.org/wiki/Vignetting">vignetting</a> toward the edge. If the angle of coverage of the lens does not fill the sensor, the image circle will be visible, typically with strong vignetting toward the edge, and the effective angle of view will be limited to the angle of coverage. A camera's angle of view depends not only on the lens, but also on the sensor. Digital sensors are usually smaller than <a class="mw-redirect" style="color: #808080;" title="35mm film" href="https://en.wikipedia.org/wiki/35mm_film">35mm film</a>, and this causes the lens to have a narrower angle of view than with 35mm film, by a constant factor for each sensor (called the <a style="color: #808080;" title="Crop factor" href="https://en.wikipedia.org/wiki/Crop_factor">crop factor</a>). In everyday digital cameras, the crop factor can range from around 1 (professional <a class="mw-redirect" style="color: #808080;" title="Digital SLR" href="https://en.wikipedia.org/wiki/Digital_SLR">digital SLRs</a>), to 1.6 (consumer SLR), to 2 (<a class="mw-redirect" style="color: #808080;" title="Micro Four Thirds" href="https://en.wikipedia.org/wiki/Micro_Four_Thirds">Micro Four Thirds</a> ILC) to 4 (<a class="new" style="color: #808080;" title="Enthusiast compact camera (page does not exist)" href="https://en.wikipedia.org/w/index.php?title=Enthusiast_compact_camera&action=edit&redlink=1">enthusiast compact cameras</a>) to 6 (most <a class="mw-redirect" style="color: #808080;" title="Compact camera" href="https://en.wikipedia.org/wiki/Compact_camera">compact cameras</a>). So a standard 50mm lens for 35mm photography acts like a 50mm standard "film" lens even on a professional digital SLR, but would act closer to an 80mm lens (1.6 x 50mm) on many mid-market DSLRs, and the 40 degree angle of view of a standard 50mm lens on a film camera is equivalent to a 28 - 35mm lens on many digital SLRs. <img class="alignnone size-full wp-image-58490" src="http://www.freesandal.org/wp-content/uploads/Angle_of_View_F_V_Chambers_1916.png" alt="Angle_of_View_F_V_Chambers_1916" width="759" height="714" /> In 1916, Northey showed how to calculate the angle of view using ordinary carpenter's tools.<a style="color: #999999;" href="https://en.wikipedia.org/wiki/Angle_of_view#cite_note-2">[2]</a> The angle that he labels as the angle of view is the half-angle or "the angle that a straight line would take from the extreme outside of the field of view to the center of the lens;" he notes that manufacturers of lenses use twice this angle. <img class="alignnone size-full wp-image-58489" src="http://www.freesandal.org/wp-content/uploads/Camera_focal_length_distance_house_animation.gif" alt="Camera_focal_length_distance_house_animation" width="512" height="512" /> In this simulation, adjusting the angle of view and distance of the camera while keeping the object in frame results in vastly differing images. At distances approaching infinity, the light rays are nearly parallel to each other, resulting in a "flattened" image. At low distances and high angles of view objects appear "foreshortened". <h3>Derivation of the angle-of-view formula</h3> Consider a rectilinear lens in a camera used to photograph an object at a distance <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" alt="S_{1}" />, and forming an image that just barely fits in the dimension, <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" alt="d" />, of the frame (the <a style="color: #808080;" title="Photographic film" href="https://en.wikipedia.org/wiki/Photographic_film">film</a> or <a style="color: #808080;" title="Image sensor" href="https://en.wikipedia.org/wiki/Image_sensor">image sensor</a>). Treat the lens as if it were a <a style="color: #808080;" title="Pinhole camera model" href="https://en.wikipedia.org/wiki/Pinhole_camera_model">pinhole</a> at distance <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" alt="S_{2}" /> from the image plane (technically, the <a class="new" style="color: #808080;" title="Center of perspective (page does not exist)" href="https://en.wikipedia.org/w/index.php?title=Center_of_perspective&action=edit&redlink=1">center of perspective</a> of a <a style="color: #808080;" title="Rectilinear lens" href="https://en.wikipedia.org/wiki/Rectilinear_lens">rectilinear lens</a> is at the center of its <a style="color: #808080;" title="Entrance pupil" href="https://en.wikipedia.org/wiki/Entrance_pupil">entrance pupil</a>):<a style="color: #808080;" href="https://en.wikipedia.org/wiki/Angle_of_view#cite_note-7">[7]</a> <img class="alignnone size-full wp-image-58488" src="http://www.freesandal.org/wp-content/uploads/Lens_angle_of_view.svg.png" alt="Lens_angle_of_view.svg" width="535" height="340" /> Now <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc94aaeb93d142252c471643dea5826c78797fdd" alt="\alpha /2" /> is the angle between the <a style="color: #808080;" title="Optical axis" href="https://en.wikipedia.org/wiki/Optical_axis">optical axis</a> of the lens and the ray joining its optical center to the edge of the film. Here <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" alt="\alpha " /> is defined to be the angle-of-view, since it is the angle enclosing the largest object whose image can fit on the film. We want to find the relationship between: <dl><dd><dl><dd>the angle <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" alt="\alpha " /></dd><dd>the "opposite" side of the right triangle, <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/582b6455b1ff5f4fb027024a8b1458687dc8ed74" alt="d/2" /> (half the film-format dimension)</dd><dd>the "adjacent" side, <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" alt="S_{2}" /> (distance from the lens to the image plane)</dd></dl></dd></dl>Using basic trigonometry, we find: <dl><dd><dl><dd> <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61a28fda5183c97596d8ab9849e2e1f465dc9fc6" alt="\tan(\alpha /2)={\frac {d/2}{S_{2}}}." /></dd></dl></dd></dl>which we can solve for α, giving: <dl><dd><dl><dd><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba196939e86f14fd3436381db8408d87c1e5221" alt="\alpha =2\arctan {\frac {d}{2S_{2}}}" /></dd></dl></dd></dl>To project a sharp image of distant objects, S 2 {\displaystyle S_{2}} <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" alt="S_{2}" /> needs to be equal to the <a style="color: #808080;" title="Focal length" href="https://en.wikipedia.org/wiki/Focal_length">focal length</a>, F {\displaystyle F} <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" alt="F" />, which is attained by setting the lens for <a style="color: #808080;" title="Infinity focus" href="https://en.wikipedia.org/wiki/Infinity_focus">infinity focus</a>. Then the angle of view is given by: <dl><dd><dl><dd><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e141c70c4127fcf2d2d59ac41c7f3c5f82307f33" alt="\alpha =2\arctan {\frac {d}{2f}}" /> where <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7383360bd488f252a2dc801771d612354e922d43" alt="f=F" /></dd></dl></dd></dl>Note that the angle of view varies slightly when the focus is not at infinity (See <a style="color: #808080;" title="Breathing (lens)" href="https://en.wikipedia.org/wiki/Breathing_%28lens%29">breathing (lens)</a>), given by <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a31381c48040e54461d782092f627f2a2f69b02" alt="S_{2}={\frac {S_{1}f}{S_{1}-f}}" /> rearranging the lens equation. <h4>Macro photography</h4> For macro photography, we cannot neglect the difference between <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" alt="S_{2}" /> and <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" alt="F" />. From the <a style="color: #808080;" title="Lens (optics)" href="https://en.wikipedia.org/wiki/Lens_%28optics%29#Imaging_properties">thin lens formula</a>, <dl><dd><dl><dd><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c06e062467b759014f591ce6c1b25448a4fe12f" alt="{\frac {1}{F}}={\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}" />.</dd></dl></dd></dl>From the definition of <a style="color: #808080;" title="Magnification" href="https://en.wikipedia.org/wiki/Magnification">magnification</a>, <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dd92ab3eac5d8b98d6a8bbae9d96c5f6bba41ae" alt="m=S_{2}/S_{1}" />, we can substitute <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" alt="S_{1}" /> and with some algebra find: <dl><dd><dl><dd><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f497867326b3486b5dee64b743486682f6c89bf3" alt="S_{2}=F\cdot (1+m)" /></dd></dl></dd></dl>Defining <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa764b3e6e147a38405ed027f28465136fa7b3a" alt="f=S_{2}" /> as the "effective focal length", we get the formula presented above: <dl><dd><dl><dd><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e141c70c4127fcf2d2d59ac41c7f3c5f82307f33" alt="\alpha =2\arctan {\frac {d}{2f}}" /> where <img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b5bad6d59188c6f93c1b7dfa359d82cdf36c23" alt="f=F\cdot (1+m)" />.</dd></dl></dd></dl>A second effect which comes into play in macro photography is lens asymmetry (an asymmetric lens is a lens where the aperture appears to have different dimensions when viewed from the front and from the back). The lens asymmetry causes an offset between the nodal plane and pupil positions. The effect can be quantified using the ratio (P) between apparent exit pupil diameter and entrance pupil diameter. The full formula for angle of view now becomes:<a style="color: #808080;" href="https://en.wikipedia.org/wiki/Angle_of_view#cite_note-Paul_van_Walree_2009-5">[5]</a> <dl><dd><dl><dd><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f8a08d31928d73dfb0bc437bbd904a286ba2a05" alt="\alpha =2\arctan {\frac {d}{2F\cdot (1+m/P)}}" /></dd></dl></dd></dl>東西又太遙，焉能得 <h1 id="firstHeading" class="firstHeading" lang="zh-TW"><a style="color: #003300;" href="https://zh.wikipedia.org/zh-tw/%E5%AA%BD%E7%A5%96">媽祖</a></h1> 媽祖（<a style="color: #808080;" title="莆仙語" href="https://zh.wikipedia.org/wiki/%E8%8E%86%E4%BB%99%E8%AA%9E">莆仙語</a>：Mâ-cô；<a class="mw-redirect" style="color: #808080;" title="閩南語" href="https://zh.wikipedia.org/wiki/%E9%96%A9%E5%8D%97%E8%AA%9E">閩南語</a>：Má-chó͘；<a style="color: #808080;" title="閩東語" href="https://zh.wikipedia.org/wiki/%E9%96%A9%E6%9D%B1%E8%AA%9E">閩東語</a>：Mā-cū）是以<a style="color: #808080;" title="中國" href="https://zh.wikipedia.org/wiki/%E4%B8%AD%E5%9C%8B">中國</a>東南沿海為中心、包括<a class="mw-redirect" style="color: #808080;" title="東亞" href="https://zh.wikipedia.org/wiki/%E6%9D%B1%E4%BA%9E">東亞</a>（<a style="color: #808080;" title="琉球" href="https://zh.wikipedia.org/wiki/%E7%90%89%E7%90%83">琉球</a>、<a style="color: #808080;" title="日本" href="https://zh.wikipedia.org/wiki/%E6%97%A5%E6%9C%AC">日本</a>及<a class="mw-redirect" style="color: #808080;" title="東南亞" href="https://zh.wikipedia.org/wiki/%E6%9D%B1%E5%8D%97%E4%BA%9E">東南亞</a>）沿海地區<a style="color: #808080;" href="https://zh.wikipedia.org/wiki/%E5%AA%BD%E7%A5%96#cite_note-.E6.BF.B1.E4.B8.8B.E6.AD.A6.E5.BF.972009-1">[1]</a>的<a class="mw-redirect mw-disambig" style="color: #808080;" title="海神" href="https://zh.wikipedia.org/wiki/%E6%B5%B7%E7%A5%9E">海神</a>信仰，又稱天上聖母 、天后聖母、天后、天后娘娘、天妃、天妃娘娘、湄洲娘媽等<a style="color: #808080;" href="https://zh.wikipedia.org/wiki/%E5%AA%BD%E7%A5%96#cite_note-.E8.81.96.E6.AD.8C-2">[2]</a>。媽祖的影響力由福建<a class="mw-redirect" style="color: #808080;" title="湄洲" href="https://zh.wikipedia.org/wiki/%E6%B9%84%E6%B4%B2">湄洲</a>傳播開來，歷經千百年，對於<a class="mw-redirect" style="color: #808080;" title="東亞" href="https://zh.wikipedia.org/wiki/%E6%9D%B1%E4%BA%9E">東亞</a>海洋文化及<a style="color: #808080;" title="中國" href="https://zh.wikipedia.org/wiki/%E4%B8%AD%E5%9C%8B">中國</a>沿海文化產生重大的影響，被學者們稱為媽祖文化。2009年10月，媽祖信仰入選<a class="mw-redirect" style="color: #808080;" title="聯合國教科文組織" href="https://zh.wikipedia.org/wiki/%E8%81%AF%E5%90%88%E5%9C%8B%E6%95%99%E7%A7%91%E6%96%87%E7%B5%84%E7%B9%94">聯合國教科文組織</a><a style="color: #808080;" title="人類非物質文化遺產代表作名錄" href="https://zh.wikipedia.org/wiki/%E4%BA%BA%E7%B1%BB%E9%9D%9E%E7%89%A9%E8%B4%A8%E6%96%87%E5%8C%96%E9%81%97%E4%BA%A7%E4%BB%A3%E8%A1%A8%E4%BD%9C%E5%90%8D%E5%BD%95">人類非物質文化遺產代表作名錄</a>。 <img class="alignnone size-full wp-image-58492" src="http://www.freesandal.org/wp-content/uploads/1280px-Main_statue_of_Datianhou_Temple.jpg" alt="1280px-Main_statue_of_Datianhou_Temple" width="1280" height="960" /> <h2>相關傳說</h2> <h3>千里眼順風耳</h3> 在一般的傳說裡，替媽祖察、聽世情的兩大駕前護衛神分別為左手持<a style="color: #808080;" title="方天畫戟" href="https://zh.wikipedia.org/wiki/%E6%96%B9%E5%A4%A9%E7%95%AB%E6%88%9F">方天畫戟</a>，右手舉至額前做遠視狀的<a style="color: #808080;" title="千里眼" href="https://zh.wikipedia.org/wiki/%E5%8D%83%E9%87%8C%E7%9C%BC">千里眼</a>（又稱<a class="new" style="color: #808080;" title="金精將軍（頁面不存在）" href="https://zh.wikipedia.org/w/index.php?title=%E9%87%91%E7%B2%BE%E5%B0%87%E8%BB%8D&action=edit&redlink=1">金精將軍</a>），以及左手持<a class="new" style="color: #808080;" title="月眉斧頭（頁面不存在）" href="https://zh.wikipedia.org/w/index.php?title=%E6%9C%88%E7%9C%89%E6%96%A7%E9%A0%AD&action=edit&redlink=1">月眉斧頭</a>，右手舉至側耳作聽音狀的<a style="color: #808080;" title="順風耳" href="https://zh.wikipedia.org/wiki/%E9%A0%86%E9%A2%A8%E8%80%B3">順風耳</a>（又稱<a class="new" style="color: #808080;" title="水精將軍（頁面不存在）" href="https://zh.wikipedia.org/w/index.php?title=%E6%B0%B4%E7%B2%BE%E5%B0%87%E8%BB%8D&action=edit&redlink=1">水精將軍</a>）。 的千里眼與順風耳，效法其視聽分明救苦救難之精神乎？？！！ 其實這『縮地』之術、『近天』之法，古來早已知之 當$ L = f_1 + f_2 $之時，相距$ L $之兩薄透鏡組合，參數$ C$ 為零。

不知為何鮮少闡述此『平行光分解原理』的哩？？？或可參照

Christoph U. Keller

Professor of Experimental Astrophysics

‧ Current Courses
‧ Course Outline
QAI-waves2.pdf

教授講解的了！！！

FreeSandal

每日彙整: 2016-08-23

光的世界︰矩陣光學六辛

輕。鬆。學。部落客

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