每日彙整: 2017-08-14

樹莓派、樹莓派之學習、樹莓派之教育

GoPiGo 小汽車︰格點圖像算術《投影幾何》【五‧線性代數】《導引一》

假使有人說

相異兩點決定了一條線。

人們須得謹慎待之︰

‧ 何謂『點』？

‧ 是什麼樣的『線』？

‧ 『點』與『線』間有什麼『關係』？

若講歐基里德以『線』 line ，表達今日字義之『曲線』 curved line 而非『直線』 straight line 是否會驚訝耶！！

Curve

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that curvature is not necessarily zero.[a]

Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However, many of these meanings are special instances of the definition which follows. A curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation. A simple example of a curve is the parabola, shown to the right. A large number of other curves have been studied in multiple mathematical fields.

A closed curve is a curve that forms a path whose starting point is also its ending point—that is, a path from any of its points to the same point.

History

Interest in curves began long before they were the subject of mathematical study. This can be seen in numerous examples of their decorative use in art and on everyday objects dating back to prehistoric times.[1] Curves, or at least their graphical representations, are simple to create, for example by a stick in the sand on a beach.

Historically, the term “line” was used in place of the more modern term “curve”. Hence the phrases “straight line” and “right line” were used to distinguish what are today called lines from “curved lines”. For example, in Book I of Euclid’s Elements, a line is defined as a “breadthless length” (Def. 2), while a straight line is defined as “a line that lies evenly with the points on itself” (Def. 4). Euclid’s idea of a line is perhaps clarified by the statement “The extremities of a line are points,” (Def. 3).[2] Later commentators further classified lines according to various schemes. For example:[3]

  • Composite lines (lines forming an angle)
  • Incomposite lines
    • Determinate (lines that do not extend indefinitely, such as the circle)
    • Indeterminate (lines that extend indefinitely, such as the straight line and the parabola)

The Greek geometers had studied many other kinds of curves. One reason was their interest in solving geometrical problems that could not be solved using standard compass and straightedge construction. These curves include:

A fundamental advance in the theory of curves was the advent of analytic geometry in the seventeenth century. This enabled a curve to be described using an equation rather than an elaborate geometrical construction. This not only allowed new curves to be defined and studied, but it enabled a formal distinction to be made between curves that can be defined using algebraic equations, algebraic curves, and those that cannot, transcendental curves. Previously, curves had been described as “geometrical” or “mechanical” according to how they were, or supposedly could be, generated.[1]

Conic sections were applied in astronomy by Kepler. Newton also worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid). The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of differential calculus.

In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into ‘ovals’. The statement of Bézout’s theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions.

From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis. The era of the space-filling curves finally provoked the modern definitions of curve.

Analytic geometry allowed curves, such as the Folium of Descartes, to be defined using equations instead of geometrical construction.

 

就算用現下文意脈絡解讀為『直』『線』，那它的『維度』幾何乎 ？？可與『曲』『線』相同嗎？！

220px-Fractaldimensionexample

220px-KochFlake.svg

Onetwosix

人們『直覺』上都知道，『線』是一維的、『面』是二維的以及『體』是三維的，也許同意『點』是零維的。但是要怎麽定義『維度』 呢？假使設想用一根有刻度的 R 尺，量一條線段，得到 l \eqcirc ── 單位刻度 ──，如果有另一根 R^' 的尺，它的單位刻度 \eqcirc^' 是R 尺的 \lambda 倍，也就是說 \eqcirc^{'} \ = \lambda \cdot \eqcirc，那用這R^' 之尺來量該線段將會得到 l \cdot \lambda^{-1} \eqcirc^'。同樣的如果 R 尺量一個正方形得到數值 l^2，那用R^' 之尺來量就會得到數值 l^2 \cdot \lambda^{-2}，這樣那R^{' }, R 兩尺的度量之數值比 N = \lambda^{-2}。
於是德國數學家費利克斯‧豪斯多夫 Felix Hausdorff 是這樣子定義『維度』D 的︰

-D =\lim\limits_{\lambda \to 0} \frac{log N} {log \lambda}

，使它也能適用於『分形』的『分維』。

在科赫雪花的建構過程中，從最初 n=0 的 △，到第 n 步時︰

總邊數︰ N_n = 3 \cdot 4^n
單一邊長︰ L_n = (\frac {1}{3})^n
總周長︰l_n = 3 \cdot (\frac {4}{3})^n
圍繞面積︰A_n = \frac {1}{5} (8 - 3 (\frac {4}{9})^n) \triangle

因此科赫雪花的分維是

-\lim\limits_{n \to \infty} \frac{log N_n} {log L_n} = \frac{log 4}{log 3} = 1.261859507\cdots

，它所圍繞的極限面積 A_{\infty} = \frac{8}{5} \triangle，那時的總周長 l_{\infty} = \infty。

─── 摘自《思想實驗！！》

 

或將思『單位長度』如何來？☆也可知『比』之概念遠古矣！ ◎

Ratio

In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second.[1] For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7).

The numbers in a ratio may be quantities of any kind, such as quantities of persons, objects, lengths, weights, etc.

A ratio may be either a whole number or a fraction.

A ratio may be written as “a to b” or a:b, or it may be expressed as a quotient of “a and b“.[2]

When the two quantities are measured with the same unit, as is often the case, their ratio is a dimensionless number. A quotient of two quantities that are measured with different units is called a rate .[3]

The ratio of width to height of standard-definition television.

History and etymology

It is impossible to trace the origin of the concept of ratio because the ideas from which it developed would have been familiar to preliterate cultures. For example, the idea of one village being twice as large as another is so basic that it would have been understood in prehistoric society.[9] However, it is possible to trace the origin of the word “ratio” to the Ancient Greek λόγος (logos). Early translators rendered this into Latin as ratio (“reason”; as in the word “rational”). (A rational number may be expressed as the quotient of two integers.) A more modern interpretation of Euclid’s meaning is more akin to computation or reckoning.[10] Medieval writers used the word (“proportion”) to indicate ratio and proportionalitas (“proportionality”) for the equality of ratios.[11]

Euclid collected the results appearing in the Elements from earlier sources. The Pythagoreans developed a theory of ratio and proportion as applied to numbers.[12] The Pythagoreans’ conception of number included only what would today be called rational numbers, casting doubt on the validity of the theory in geometry where, as the Pythagoreans also discovered, incommensurable ratios (corresponding to irrational numbers) exist. The discovery of a theory of ratios that does not assume commensurability is probably due to Eudoxus of Cnidus. The exposition of the theory of proportions that appears in Book VII of The Elements reflects the earlier theory of ratios of commensurables.[13]

The existence of multiple theories seems unnecessarily complex to modern sensibility since ratios are, to a large extent, identified with quotients. This is a comparatively recent development however, as can be seen from the fact that modern geometry textbooks still use distinct terminology and notation for ratios and quotients. The reasons for this are twofold. First, there was the previously mentioned reluctance to accept irrational numbers as true numbers. Second, the lack of a widely used symbolism to replace the already established terminology of ratios delayed the full acceptance of fractions as alternative until the 16th century.[14]

Euclid’s definitions

Book V of Euclid’s Elements has 18 definitions, all of which relate to ratios.[15] In addition, Euclid uses ideas that were in such common usage that he did not include definitions for them. The first two definitions say that a part of a quantity is another quantity that “measures” it and conversely, a multiple of a quantity is another quantity that it measures. In modern terminology, this means that a multiple of a quantity is that quantity multiplied by an integer greater than one—and a part of a quantity (meaning aliquot part) is a part that, when multiplied by an integer greater than one, gives the quantity.

Euclid does not define the term “measure” as used here, However, one may infer that if a quantity is taken as a unit of measurement, and a second quantity is given as an integral number of these units, then the first quantity measures the second. Note that these definitions are repeated, nearly word for word, as definitions 3 and 5 in book VII.

Definition 3 describes what a ratio is in a general way. It is not rigorous in a mathematical sense and some have ascribed it to Euclid’s editors rather than Euclid himself.[16] Euclid defines a ratio as between two quantities of the same type, so by this definition the ratios of two lengths or of two areas are defined, but not the ratio of a length and an area. Definition 4 makes this more rigorous. It states that a ratio of two quantities exists when there is a multiple of each that exceeds the other. In modern notation, a ratio exists between quantities p and q if there exist integers m and n so that mp>q and nq>p. This condition is known as the Archimedes property.

Definition 5 is the most complex and difficult. It defines what it means for two ratios to be equal. Today, this can be done by simply stating that ratios are equal when the quotients of the terms are equal, but Euclid did not accept the existence of the quotients of incommensurate, so such a definition would have been meaningless to him. Thus, a more subtle definition is needed where quantities involved are not measured directly to one another. Though it may not be possible to assign a rational value to a ratio, it is possible to compare a ratio with a rational number. Specifically, given two quantities, p and q, and a rational number m/n we can say that the ratio of p to q is less than, equal to, or greater than m/n when np is less than, equal to, or greater than mq respectively. Euclid’s definition of equality can be stated as that two ratios are equal when they behave identically with respect to being less than, equal to, or greater than any rational number. In modern notation this says that given quantities p, q, r and s, then p:q::r:s if for any positive integers m and n, np<mq, np=mq, np>mq according as nr<ms, nr=ms, nr>ms respectively. There is a remarkable similarity between this definition and the theory of Dedekind cuts used in the modern definition of irrational numbers.[17]

Definition 6 says that quantities that have the same ratio are proportional or in proportion. Euclid uses the Greek ἀναλόγον (analogon), this has the same root as λόγος and is related to the English word “analog”.

Definition 7 defines what it means for one ratio to be less than or greater than another and is based on the ideas present in definition 5. In modern notation it says that given quantities p, q, r and s, then p:q>r:s if there are positive integers m and n so that np>mq and nr≤ms.

As with definition 3, definition 8 is regarded by some as being a later insertion by Euclid’s editors. It defines three terms p, q and r to be in proportion when p:q::q:r. This is extended to 4 terms p, q, r and s as p:q::q:r::r:s, and so on. Sequences that have the property that the ratios of consecutive terms are equal are called geometric progressions. Definitions 9 and 10 apply this, saying that if p, q and r are in proportion then p:r is the duplicate ratio of p:q and if p, q, r and s are in proportion then p:s is the triplicate ratio of p:q. If p, q and r are in proportion then q is called a mean proportional to (or the geometric mean of) p and r. Similarly, if p, q, r and s are in proportion then q and r are called two mean proportionals to p and s.

 

要是我們無法建構『座標系』將『透視』數值化︰

Projective geometry

In projective geometry the points of a line are called a projective range, and the set of lines in a plane on a point is called a pencil.

Given two lines  \ell and  m in a plane and a point P of that plane on neither line, the bijective mapping between the points of the range of  \ell and the range of  m determined by the lines of the pencil on P is called a perspectivity (or more precisely, a central perspectivity with center P).[4] A special symbol has been used to show that points X and Y are related by a perspectivity;  X \doublebarwedge Y . In this notation, to show that the center of perspectivity is P, write  X \ \overset {P}{\doublebarwedge} \ Y. Using the language of functions, a central perspectivity with center P is a function  f_P \colon [\ell] \mapsto [m] (where the square brackets indicate the projective range of the line) defined by  f_P (X) = Y \text{ whenever } P \in XY.[5] This map is an involution, that is, f_P (f_P (X)) = X \text{ for all }X \in [\ell].

The existence of a perspectivity means that corresponding points are in perspective. The dual concept, axial perspectivity, is the correspondence between the lines of two pencils determined by a projective range.

 

該如何談『投影解析』呀★