假使我們只說︰如果 $c = 0$ ，一個莫比烏斯變換

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

就是一種仿射變換

Affine transformation

In geometry, an affine transformation, affine map^[1] or an affinity (from the Latin, affinis, “connected with”) is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence.

If $X$ and $Y$ are affine spaces, then every affine transformation $f\colon X\to Y$ is of the form $x\mapsto Mx+b$ , where $M$ is a linear transformation on $X$ and $b$ is a vector in $Y$ . Unlike a purely linear transformation, an affine map need not preserve the zero point in a linear space. Thus, every linear transformation is affine, but not every affine transformation is linear.

All Euclidean spaces are affine, but there are affine spaces that are non-Euclidean. In affine coordinates, which include Cartesian coordinates in Euclidean spaces, each output coordinate of an affine map is a linear function (in the sense of calculus) of all input coordinates. Another way to deal with affine transformations systematically is to select a point as the origin; then, any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation.

。那麼我們能知道 $c = 0$ 的條件嗎？恐生疑惑乎？？

雖說『眼見為憑』，難保不是『錯覺』呢？

此事因平面國流傳一圖而起︰

卻納悶平面國似有『透鏡』，恍惚是『平行光轉換器』耶！然懷疑

無厚能有透鏡乎？！

─── 摘自《GoPiGo 小汽車︰格點圖像算術《投影幾何》【四‧平面國】《補丁》》

即使望圖聯想！何若解析思辨耶！！

或許『圖鑑』還有一張

表白文字幾行︰

依上圖，三角形 $\Delta {\gamma}_1 {\gamma}_2} z^{'}$ 與 $\Delta {\gamma}_1 {\gamma}_2} z$ 面積比可以兩算︰

‧ 以 $z-{\gamma}_1}$ 和 $z^{'}-{\gamma}_1}$ 為底 $= \frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}}$ ；

‧ 以 $z-{\gamma}_2}$ 和 $z^{'}-{\gamma}_2}$ 為底

$= \frac{|{\gamma}_1 {\gamma}_2| \cdot \sin(\angle \phi)}{|{\gamma}_1 {\gamma}_2| \cdot \sin(\angle \theta)} \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}} = k \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}}$

$\therefore k = \frac{ \sin(\angle \phi)}{ \sin(\angle \theta)}$

既『角不變』，『角比』能變嗎★

故而 $\frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}} = k \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}}$ ，

且將兩邊乘上 $\frac{z - {\gamma}_1}{z^{'} - {\gamma}_2}$

果非所求的嘛☆

─── 摘自《GoPiGo 小汽車︰格點圖像算術《投影幾何》【五‧線性代數】《導引七‧變換組合 VII‧B 》》

當逢 ${\gamma}_2 \to \infty$ ，光從無窮遠處來，平行同光照

$\lim \limits_{{\gamma}_2 \to \infty} \left[ \frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}} = k \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}} \right] \ \Rightarrow \ \frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}} = k$ ；

反思 ${\gamma}_1 \to \infty$ ，聚焦於無限，所見果非似焉

$\lim \limits_{{\gamma}_1 \to \infty} \left[ \frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}} = k \cdot \frac{z^{'} - {\gamma}_2}{z - {\gamma}_2}} \right] \ \Rightarrow \ k \cdot \frac{z^{'} - {\gamma}_1}{z - {\gamma}_1}} = 1$ 。

再以投影幾何之公設︰

相異兩線必交於一點。

故爾平行線不能例外也。所以代數計算求其意義一致性，亦無法分上下左右之 $\infty$ 矣。

Line extended by a point at infinity

The projective line may be identified with the line K extended by a point at infinity. More precisely, the line K may be identified with the subset of P¹(K) given by