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GoPiGo 小汽車︰格點圖像算術《投影幾何》【五‧線性代數】《導引八》觀察者《變換‧B 》

有人說︰

天下『一切』事情，都不過是個『分類』。

『有人』就『有事』，天下何得『无事』？天下能得『無人』乎？

有人想︰

酸葡萄『沒有』想『要』，甜檸檬『有』卻想『不要』；

都是一種『心理』。

有人分︰

有所謂『好』與『壞』？ 分別著『對』或『錯』，

『是』其所『非』，『非』其所『是』。

有人問︰

『蜜蜂』為何不見了？

只因『益』和『害』！！

─── 《『蜜蜂』為何不見了？》

 

即便講︰識物辨名始於分門別類。重要的還是門類之界定性徵。若從完備無餘來看 A \cdot z + B 變換形式，僅止於平面上的縮放、旋轉 、平移爾，實未得仿射變換之全也。

In the plane

Affine transformations in two real dimensions include:

  • pure translations,
  • scaling in a given direction, with respect to a line in another direction (not necessarily perpendicular), combined with translation that is not purely in the direction of scaling; taking “scaling” in a generalized sense it includes the cases that the scale factor is zero (projection) or negative; the latter includes reflection, and combined with translation it includes glide reflection,
  • rotation combined with a homothety and a translation,
  • shear mapping combined with a homothety and a translation, or
  • squeeze mapping combined with a homothety and a translation.

To visualise the general affine transformation of the Euclidean plane, take labelled parallelograms ABCD and A′B′C′D′. Whatever the choices of points, there is an affine transformation T of the plane taking A to A′, and each vertex similarly. Supposing we exclude the degenerate case where ABCD has zero area, there is a unique such affine transformation T. Drawing out a whole grid of parallelograms based on ABCD, the image T(P) of any point P is determined by noting that T(A) = A′, T applied to the line segment AB is A′B′, T applied to the line segment AC is A′C′, and T respects scalar multiples of vectors based at A. [If A, E, F are collinear then the ratio length(AF)/length(AE) is equal to length(A′F′)/length(A′E′).] Geometrically T transforms the grid based on ABCD to that based in A′B′C′D′.

Affine transformations do not respect lengths or angles; they multiply area by a constant factor

area of A′B′C′D′ / area of ABCD.

A given T may either be direct (respect orientation), or indirect (reverse orientation), and this may be determined by its effect on signed areas (as defined, for example, by the cross product of vectors).

A central dilation. The triangles A1B1Z, A1C1Z, and B1C1Z get mapped to A2B2Z, A2C2Z, and B2C2Z, respectively.

 

Effect of applying various 2D affine transformation matrices on a unit square. Note that the reflection matrices are special cases of the scaling matrix.

 

然以立論例釋而言足矣哉。何況此本乎莫比烏斯變換

\left( \begin{array}{cc} z^{'} \\ 1 \end{array} \right) = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \left( \begin{array}{cc} z \\ 1 \end{array} \right)

，在 c = 0 條件下所得。

恰宜說點分類基本也。

為什麼 A \cdot z + B 是個平移，必得 A = 1 呢？因既為平移，平面上無點能不動。假使 A \neq 1 ，那麼 z = A \cdot z + B 有定點 z = \frac{B}{1-A} ，故語意矛盾也。再由複數的極座標表示法 r \cdot e^{i \theta} ，可知 r=1 為旋轉， 而 r 就是縮放比的了。

自可解讀定點的重要性︰

Determining the fixed points

The fixed points of the transformation

f(z)={\frac {az+b}{cz+d}}

are obtained by solving the fixed point equation f(γ) = γ. For c ≠ 0, this has two roots obtained by expanding this equation to

  c\gamma ^{2}-(a-d)\gamma -b=0\ ,

and applying the quadratic formula. The roots are

\gamma _{{1,2}}={\frac {(a-d)\pm {\sqrt {(a-d)^{2}+4bc}}}{2c}}={\frac {(a-d)\pm {\sqrt {(a+d)^{2}-4(ad-bc)}}}{2c}}.

Note that for parabolic transformations, which satisfy (a+d)2 = 4(ad−bc), the fixed points coincide. Note also that the discriminant is

(a-d)^{2}+4cb=(a-d)^{2}+4ad-4=(a+d)^{2}-4=\operatorname {tr}^{2}{\mathfrak {H}}-4.

When c = 0, the quadratic equation degenerates into a linear equation. This corresponds to the situation that one of the fixed points is the point at infinity. When a ≠ d the second fixed point is finite and is given by

\gamma =-{\frac {b}{a-d}}.

In this case the transformation will be a simple transformation composed of translations, rotations, and dilations:

z\mapsto \alpha z+\beta .\,

If c = 0 and a = d, then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation:

  z\mapsto z+\beta .

 

明白投影線會定

x + \infty = \infty, if \ x \neq \infty 的吧◎