每日彙整: 2017-09-24

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】《導引八》觀察者《運動‧II 》

『透視』畢竟始於『眼』見『物像』也。『物』有形狀、遠近，且座落在世界中，『眼』感知為『觀念之像面』上的樣貌、景深也。

 

Rays of light travel from the object to the eye, intersecting with a notional picture plane.

 

Determining the geometry of a square floor tile on the perspective drawing

 

既然一條『視線』已經對應『像面』上的一點，一撇之見難以逆投『全貌』乎？況且『投影』可由幾個『透視』組成，故而終究專注『像面』與『像面』之『關係』哩。一如『投影線』和『投影線』之立論矣。

一張圖

 

借幾行文字

p Z_{{\infty}^{'}} Z_x {Z^{'}}_{\infty} 是平行四邊形， Z_x 是 l 、l^{'} 兩線交點。

三角形 \Delta p Z_{{\infty}^{'}} z 和 \Delta z' Z_x z 相似，

\frac{\overline{z z^{'}}}{\overline{p z}} = \frac{\overline{z Z_x}}{\overline{Z_{{\infty}^{'}} z}} \Rightarrow \frac{\overline{p z^{'}}}{\overline{p z}} = \frac{\overline{z Z_x} + \overline{Z_{{\infty}^{'}} z} }{\overline{Z_{{\infty}^{'}} z}} = \frac{\overline{p {Z^{'}}_{\infty}}}{\overline{Z_{{\infty}^{'}} z}}

\therefore \overline{p z^{'}} = \frac{\overline{p {Z^{'}}_{\infty}}}{\overline{Z_{{\infty}^{'}} z}} \overline{p z}

能否曲徑通幽處？身處高林觀日出！

z^{'} = \frac{\alpha \cdot \beta \cdot z \cdot (z_2-z_1) }{(\alpha - \beta) z + (\beta \cdot z_2 - \alpha \cdot z_1)}} = \frac{Z_{\infty}^{'} z}{z - Z_{{\infty}^{'}}}

得聞鐘磬聲◎

\overline{p z^{'}} e^{i \theta} = \overline{p {Z^{'}}_{\infty}} e^{ i {\theta}_l} \cdot \overline{p z} e^{i \theta} \cdot \frac{1}{\overline{Z_{{\infty}^{'}} z}} e^{ -i {\theta}_l}

※ 此處 {\theta}_l} 是 l 線之方向。

─── 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 VI‧XII 》》

 

如是同一『視野』下的景象，自可『辯證』耶！

pi@raspberrypi:~ $ ipython3
Python 3.4.2 (default, Oct 19 2014, 13:31:11) 
Type "copyright", "credits" or "license" for more information.

IPython 2.3.0 -- An enhanced Interactive Python.
?         -> Introduction and overview of IPython's features.
%quickref -> Quick reference.
help      -> Python's own help system.
object?   -> Details about 'object', use 'object??' for extra details.

In [1]: from sympy import *

In [2]: init_printing()

In [3]: zα,zβ,za,zb = symbols('zα,zβ,za,zb')

In [4]: M1 = Matrix([[zβ,0],[1,-zα]])

In [5]: M1
Out[5]: 
⎡zβ   0 ⎤
⎢       ⎥
⎣1   -zα⎦

In [6]: M2 = Matrix([[zb,0],[1,-za]])

In [7]: M2
Out[7]: 
⎡zb   0 ⎤
⎢       ⎥
⎣1   -za⎦

In [8]: M2*M1
Out[8]: 
⎡ zb⋅zβ      0  ⎤
⎢               ⎥
⎣-za + zβ  za⋅zα⎦

In [9]: M1*M2
Out[9]: 
⎡ zb⋅zβ     0  ⎤
⎢              ⎥
⎣zb - zα  za⋅zα⎦

In [10]: M2*M1.inv()
Out[10]: 
⎡     zb         ⎤
⎢     ──       0 ⎥
⎢     zβ         ⎥
⎢                ⎥
⎢    za    1   za⎥
⎢- ───── + ──  ──⎥
⎣  zα⋅zβ   zβ  zα⎦

In [11]: M1*M2.inv()
Out[11]: 
⎡    zβ        ⎤
⎢    ──      0 ⎥
⎢    zb        ⎥
⎢              ⎥
⎢1      zα   zα⎥
⎢── - ─────  ──⎥
⎣zb   za⋅zb  za⎦

In [12]: M2.inv()*M1.inv()
Out[12]: 
⎡          1                 ⎤
⎢        ─────            0  ⎥
⎢        zb⋅zβ               ⎥
⎢                            ⎥
⎢     1          1        1  ⎥
⎢- ──────── + ────────  ─────⎥
⎣  za⋅zα⋅zβ   za⋅zb⋅zβ  za⋅zα⎦

In [13]: M1.inv()*M2.inv()
Out[13]: 
⎡         1                ⎤
⎢       ─────           0  ⎥
⎢       zb⋅zβ              ⎥
⎢                          ⎥
⎢   1          1        1  ⎥
⎢──────── - ────────  ─────⎥
⎣zb⋅zα⋅zβ   za⋅zb⋅zα  za⋅zα⎦

In [14]:

 

故知『欲窮千里目，更上一層樓』的道理呦◎

Stereographic projections

Main article: Stereographic projection

It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is “above” the plane.

We can establish a one-to-one correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows. Given a point in the plane, draw a straight line connecting it with the north pole on the sphere. That line will intersect the surface of the sphere in exactly one other point. The point z = 0 will be projected onto the south pole of the sphere. Since the interior of the unit circle lies inside the sphere, that entire region (|z| < 1) will be mapped onto the southern hemisphere. The unit circle itself (|z| = 1) will be mapped onto the equator, and the exterior of the unit circle (|z| > 1) will be mapped onto the northern hemisphere, minus the north pole. Clearly this procedure is reversible – given any point on the surface of the sphere that is not the north pole, we can draw a straight line connecting that point to the north pole and intersecting the flat plane in exactly one point.

Under this stereographic projection the north pole itself is not associated with any point in the complex plane. We perfect the one-to-one correspondence by adding one more point to the complex plane – the so-called point at infinity – and identifying it with the north pole on the sphere. This topological space, the complex plane plus the point at infinity, is known as the extended complex plane. We speak of a single “point at infinity” when discussing complex analysis. There are two points at infinity (positive, and negative) on the real number line, but there is only one point at infinity (the north pole) in the extended complex plane.[6]

Imagine for a moment what will happen to the lines of latitude and longitude when they are projected from the sphere onto the flat plane. The lines of latitude are all parallel to the equator, so they will become perfect circles centered on the origin z = 0. And the lines of longitude will become straight lines passing through the origin (and also through the “point at infinity”, since they pass through both the north and south poles on the sphere).

This is not the only possible yet plausible stereographic situation of the projection of a sphere onto a plane consisting of two or more values. For instance, the north pole of the sphere might be placed on top of the origin z = −1 in a plane that is tangent to the circle. The details don’t really matter. Any stereographic projection of a sphere onto a plane will produce one “point at infinity”, and it will map the lines of latitude and longitude on the sphere into circles and straight lines, respectively, in the plane.

Riemann sphere which maps all points on a sphere except one to all points on the complex plane