GoPiGo 小汽車︰格點圖像算術《投影幾何》【一】

派生碼訊

丑 牛

宋‧陸游‧幽居歲暮五首‧其五

古井年年浚,荒疇日日犁。
刈茅苫鹿屋,插棘護鷄栖。
閑頼書遮眼,愁須酒到臍。
斜陽有常課,緩步上湖堤。

黑水北朱子治家有格言︰宜未雨而綢繆 ,毋臨渴而掘井。崔瑗行尚座右銘︰無使名過實,守愚聖所臧。學問一事,祇惟恐思之不精,念之不實。

派︰昔有『 Thue 』者,其言曰︰

阿克塞爾‧圖厄【挪威語 Axel Thue】一位數學家,以研究丟番圖用『有理數』逼近『實數』問題以及開拓『組合數學』之貢獻而聞名。他於一九一四發表了『詞之群論問題』Word problem for group 啟始了一個今天稱之為『字串改寫系統』SRS String Rewriting System 的先河,如從現今的研究和發現來看,它與圖靈機的『停機問題』密切相關。上個千禧年之時,John Colagioia 用『Semi-Thue System』寫了一個『奧秘的 esoteric 程式語言 Thue ,作者宣稱︰

Thue represents one of the simplest possible ways to construe 『constraint-based』基於約束 programming. It is to the constraint-based 『paradigm』典範 what languages like『 OISC 』── 單指令集電腦 One instruction set computer ── are to the imperative paradigm; in other words, it’s a 『tar pit』焦油坑.

,果然概念廣大能通天!!『 SRS 』卻是個『奧秘語言』??

生 ︰ 當真是︰ ☆★ 之火,可以燎原。能不精思實念乎?

三足鳥

180px-Processed_SAM_loki

斷頭台

人類的思維如果一旦不『審慎』,很容易邏輯『混亂』,以至於言論多所『謬誤』,有時或許『巧說詭辯』。比方說公孫龍子的『雞三足』之詭論︰…牛羊有毛,雞有羽。謂雞足一,數足二;二而一,故三。謂牛羊足一,數足四;四而一,故五。羊牛足五,雞足三,故曰:『牛合羊非雞』。非,有以非雞也。…裡頭的『謂雞足一』之『雞足』和『一』是什麼?『數足二』的『足』與『二』又是什麼?卻能相加,彷彿『一雞 + 二足』可以得到『三 \biguplus 』的一般?!已經完全不像他的『白馬非馬論』了。

洛基的賭注 Loki’s Wager
洛基乃北歐神話中以『詐騙』著名之神。傳說他曾與矮人打賭卻輸了。當矮人們依約來『提頭』時,洛基說︰沒問題』,但是必須依照『約定』,只能取走『他的頭』,而不能動著『他的脖子。於是彼此開始『爭論』該如何的『切割』:有哪些部分雙方同意『是頭』;又有哪些部分認同『是脖子只是脖子的『結束點』和頭之『開始點』究竟『是哪裡』,互相一直無法『取得共識』。於是洛基終於保住著了他的頭

── 摘自《M♪o 之學習筆記本《丑》控制︰【黑水北】當思恒念

 

當『數理』失卻了『參考系』確定『概念』之意義;宛如憑空地述說著『群論』可以將一元多次方程式可解性的問題連繫起來,豈非囫圇吞棗乎?不知滋味之物要怎樣講耶??

且思『近』之『視角大』、『遠』之『視角小』是『歐式幾何』之『性質』嗎?

300px-海岛算经

四庫全書海島算經

220px-Sea_island_survey

如果用《海島算經

三國時代魏國數學家劉徽所著的測量學著作,原為《劉徽九章算術注》第九卷勾股章內容的延續和發展,名為《九章重差圖》,附於《劉徽九章算術注》 之後作為第十章。唐代將《重差》從《九章》分離出來,單獨成書,按第一題今有望海島」,取名為《海島算經》,是《算經十書》之一。

劉徽《海島算經》「使中國測量學達到登峰造極的地步」,使「中國在數學測量學的成就,超越西方約一千年」(美國數學家弗蘭克·斯委特茲語)

之圖來作『三角測量』的計算︰

\overline{GH} = D
\overline{BG} = X
\overline{AB} = H
 \angle AHB = \alpha
 \angle AGB = \beta

\tan(\alpha) = \frac{H}{D + X}
\tan(\beta) = \frac{H}{X}

Sea_Island_Measurement

可以得到

 H = D \cdot \tan(\alpha) \cdot \frac{1}{1 - \frac{\tan(\alpha)}{\tan(\beta)}}

然而『天很高,日很遠』,因此 \angle \beta \approx \angle \alpha ,故而很難『度量』的『精準』,一點點『角度』之『誤差』就產生了那個

失之豪釐,差以千里

的吧!!

─── 摘自《失之豪釐,差以千里!!《上》

 

難到『有限高度』 H 之物,若位在『無窮遠』 \infty 處,竟然它的『視角』 能不趨近於『零』 \to 0  嗎??

就像『眼見之實』

消失點

如當你沿著鐵路線去看兩條鐵軌,沿著公路線去看兩邊排列整齊的樹木時,兩條平行的鐵軌或兩排樹木連線交與很遠很遠的某一點,這點在透視投影中叫做消失點。

藝術家和工程師在紙上表現立體圖時,常用一種透視法,這種方法源於人們的視覺經驗:大小相同的物體,離你較近的看起來比離你較遠的大。凡是平行的直線都消失於無窮遠處的同一個點,消失於視平線上的點的直線都是水平直線。

 

之『現象文字』及『理論內容』的陳述

Vanishing point

In graphical perspective, a vanishing point is an abstract point on the image plane where 2D projections (or drawings) of a set of parallel lines in 3D space appear to converge.[clarification needed] When the set of parallel lines is perpendicular to a picture plane, the construction is known as one-point perspective, and their vanishing point corresponds to the oculus, or “eye point”, from which the image should be viewed for correct perspective geometry.[1] Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points.

A 2D construction of perspective viewing, showing the formation of a vanishing point

 

需要『想像』呦!

The vanishing point may also be referred to as the “direction point”, as lines having the same directional vector, say D, will have the same vanishing point or converge at the same vanishing points. Mathematically, let q ≡ (x, y, f) be a point lying on the image plane, where f is the focal length (of the camera associated with the image), and let vq ≡ (x/h, y/h, f/h) be the unit vector associated with q, where h = x2 + y2 + f2. If we consider a straight line in space S with the unit vector ns ≡ (nx, ny, nz) and its vanishing point vs, the unit vector associated with vs is equal to ns, assuming both are assumed to point towards the image plane.[2]

When the image plane is parallel to two world-coordinate axes, lines parallel to the axis which is cut by this image plane will meet at infinity i.e. at the vanishing point. Lines parallel to the other two axes will not form vanishing points as they are parallel to the image plane. This is one-point perspective. Similarly, when the image plane intersects two world-coordinate axes, lines parallel to those planes will meet at infinity and form two vanishing points. This is called two-point perspective. In three-point perspective the image plane intersects the x, y, and z axes and therefore lines parallel to these axes intersect, resulting in three different vanishing points.

 

不知可否借假修真的哩◎