派︰Einstein 1933

Pure logical thinking cannot yield us any knowledge of the empirical world; all knowledge starts from experience and ends in it. Propositions arrived at by purely logical means are completely empty as regards reality.

……

☿ 行︰《小狐狸》之書曾寫過︰

就讓我們透過『 Sysfs 』── 是 Linux 2.6 所提供的一種『虛擬檔案系統』 VFS virtual files system 。這個檔案系統不僅可以把『裝置』 devices 和『驅動程式』 device drivers 的資訊從『核心』 kernel 內部輸出到『使用者空間』 user space ，也可以用來對裝置和驅動程式做『設定』。── 來控制『系統 ACT LED』，與 GPIO 『建立聯繫』吧！

# 為什麼一定得是 root 呢？可以只用 sudo 嗎？？
sudo -s

echo none > /sys/class/leds/led0/trigger
echo 1 >/sys/class/leds/led0/brightness
echo 0 >/sys/class/leds/led0/brightness
echo 1 >/sys/class/leds/led0/brightness
echo 0 >/sys/class/leds/led0/brightness
exit

 

Ah Ha！！這樣想要有顆『閃爍』的『小星星』，怕是不可得的了 ，何不閱讀《 Advanced Bash-Scripting Guide 》一下，寫個『 Shell 』程式呢？？

。雖說曾聞『早鳥氏』實證時否定了，然『盡信書，不如無書』，況此一時彼一時已經不同，何不自己格之！！

root@raspberrypi:/sys/class/gpio# cd /sys/class/leds/
root@raspberrypi:/sys/class/leds# ls
led0  led1
root@raspberrypi:/sys/class/leds# cd led0
root@raspberrypi:/sys/class/leds/led0# ls
brightness  device  max_brightness  subsystem  trigger	uevent
echo none > /sys/class/leds/led0/trigger
root@raspberrypi:/sys/class/leds/led0# echo 1 >/sys/class/leds/led0/brightness
root@raspberrypi:/sys/class/leds/led0# echo 0 >/sys/class/leds/led0/brightness
root@raspberrypi:/sys/class/leds/led0# 

 

果然大功告成。噫！怎有個『 led1 』哩？何妨依樣畫葫蘆！… ☿☺

☆ 編者言

此處『…』並非編者刻意不譯，實在是 M♪o 沒寫，從她的笑臉推斷 ， M♪o 似了然於心的了，讀者何不自己嘗試嘗試的呢？？

訊︰☿☺ 天下無難事，只怕有心人。

─── 《M♪o 之學習筆記本《子》開關︰【青木仁】格物致知》

 

也許星星知我心，她暗夜裡眨眼睛︰

《一閃一閃亮晶晶》（英語：Twinkle, Twinkle, Little Star），又譯《一閃一閃小星星》或稱《小星星》《閃爍的小星》，是一首相當著名的英國兒歌，旋律出自於法國民謠《媽媽請聽我說》（法語：Ah! vous dirai-je, Maman），歌詞則出自於珍·泰勒（Jane Taylor）的英文詩《小星星》（The Star）。後莫扎特依此創作了著名的《小星星變奏曲》。此詩第一次出版於1806年，收錄在珍與其姊安·泰勒（Ann Taylor）的合選集《Rhymes for the Nursery》中。

 

所以 M♪o 曾為歡欣，球面國有人證信︰

為何天邊放光明？地平之上誰傳訊！

Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere. It is an example of a geometry that is not Euclidean. Two practical applications of the principles of spherical geometry are navigation and astronomy.

In plane geometry, the basic concepts are points and (straight) lines. On a sphere, points are defined in the usual sense. The equivalents of lines are not defined in the usual sense of “straight line” in Euclidean geometry, but in the sense of “the shortest paths between points”, which are called geodesics. On a sphere, the geodesics are the great circles; other geometric concepts are defined as in plane geometry, but with straight lines replaced by great circles. Thus, in spherical geometry, angles are defined between great circles, resulting in a spherical trigonometry that differs from ordinary trigonometry in many respects; for example, the sum of the interior angles of a triangle exceeds 180 degrees.

Spherical geometry is not elliptic geometry, but is rather a subset of elliptic geometry. For example, it shares with that geometry the property that a line has no parallels through a given point. Contrast this with Euclidean geometry, in which a line has one parallel through a given point, and hyperbolic geometry, in which a line has two parallels and an infinite number of ultraparallels through a given point.

An important geometry related to that of the sphere is that of the real projective plane; it is obtained by identifying antipodal points (pairs of opposite points) on the sphere. (This is elliptic geometry.) Locally, the projective plane has all the properties of spherical geometry, but it has different global properties. In particular, it is non-orientable, or one-sided.

Concepts of spherical geometry may also be applied to the oblong sphere, though minor modifications must be implemented on certain formulas.

Higher-dimensional spherical geometries exist; see elliptic geometry.

On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. The surface of a sphere can be represented by a collection of two dimensional maps. Therefore, it is a two dimensional manifold.

Properties

With points defined as the points on a sphere and lines as the great circles of that sphere, a spherical geometry has the following properties:^[7]

• Any two lines intersect in two diametrically opposite points, called antipodal points.
• Any two points that are not antipodal points determine a unique line.
• There is a natural unit of angle measurement (based on a revolution), a natural unit of length (based on the circumference of a great circle) and a natural unit of area (based on the area of the sphere).
• Each line is associated with a pair of antipodal points, called the poles of the line, which are the common intersections of the set of lines perpendicular to the given line.
• Each point is associated with a unique line, called the polar line of the point, which is the line on the plane through the center of the sphere and perpendicular to the diameter of the sphere through the given point.

As there are two arcs (line segments) determined by a pair of points, which are not antipodal, on the line they determine, three non-collinear points do not determine a unique triangle. However, if we only consider triangles whose sides are minor arcs of great circles, we have the following properties:

• The angle sum of a triangle is greater than 180° and less than 540°.
• The area of a triangle is proportional to the excess of its angle sum over 180°.
• Two triangles with the same angle sum are equal in area.
• There is an upper bound for the area of triangles.
• The composition (product) of two (orthogonal) line reflections may be considered as a rotation about either of the points of intersection of their axes.
• Two triangles are congruent if and only if they correspond under a finite product of line reflections.
• Two triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent).

 

早知光線直行徑？？

光在真空裡與均勻的介質中走直線，直到遭遇不同折射率之界面，或者反射、然而通常多折射，它的軌跡就像『折線』 line chart

也許這正是『屈光學』名義之由來也。雖說此『折線圖』看來平淡無奇，那要如何描述光之行徑呢？怎樣選取『座標系』才能定位的耶？？光學書本一般將此當成不說自明之理！於是直接展開論述，若干圖例後，難免彷彷彿彿似懂非懂的乎！！

茲舉一圖示為例

In ray transfer (ABCD) matrix analysis, an optical element (here, a thick lens) gives a transformation between at the input plane and  when the ray arrives at the output plane.

， 不說那條『折線』？不知『透鏡』厚薄？卻講『輸入面』 input plane 以及『輸出面』 output plane ，圖中強調『光入點』 與『光出點』 之『座標』！宛如『透鏡』在哪無關也！！

假 使分解光之『折線圖』的構成，可得『線段』和『轉折點』，那『線段』表示光在『彼介質』中『直行』也，這『轉折點』說明光『此處』發生『反射』或『折 射』，而後在『此介質』裡『直行』矣。一個『光學元件』、『光學系統』通常有其物體的『邊界』，如果知道『輸入面』與『輸出面』間 之光線的『行徑關係』，如是這個『光學元件』、『光學系統』的行為就確定了。由於參照自身『邊界面』的原故，因此與其座落『光軸』之何處無關耶？？！！事 實上『光軸座標系』乃是一系列『面』與『面』間『相對』位置關係，甚至和『光學元件』、『光學系統』之物體大小不必相涉矣 ！！？？

知此而後知 Justin Peatross 與 Michael Ware 先生們之大哉論也︰

三種基本『光學元素』︰均勻介質裡直線行、球面反射、球面折射的『效應矩陣』足以組成任意複雜之光學成像系統，能夠構造整體『矩陣光學』的了！！！

─── 摘自《光的世界︰矩陣光學三‧下》

 

江山含笑道情真！！

到底幾眼幾顆方才定耶？★☆