每日彙整: 2017-10-03

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《變換‧F3 》

嬋娟篇‧唐‧孟郊

花嬋娟，泛春泉。竹嬋娟，籠曉煙。
妓嬋娟，不長妍。月嬋娟，真可憐。
夜半姮娥朝太一，人間本自無靈匹。
漢宮承寵不多時，飛燕婕妤相妒嫉。

 

數嬋娟

z^{'} = \frac{a \cdot z + b}{c \cdot z + d}

= \frac{a \cdot (z+\frac{d}{c}) - \frac{a \cdot d}{c} + b}{c \cdot (z + \frac{d}{c})}

= \frac{a}{c} - \frac{a \cdot d - b \cdot c}{c^2} \cdot \frac{1}{z + \frac{d}{c}}

至反演！

Inversive geometry

In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion. These transformations preserve angles and map generalized circles into generalized circles, where a generalized circle means either a circle or a line (loosely speaking, a circle with infinite radius). Many difficult problems in geometry become much more tractable when an inversion is applied.

The inverse, with respect to the red circle, of a circle going through O (blue) is a line not going through O (green), and vice versa.

The inverse, with respect to the red circle, of a circle not going through O (blue) is a circle not going through O (green), and vice versa.

Inversion with respect to a circle does not map the center of the circle to the center of its image

Relation to Erlangen program

According to Coxeter,[8] the transformation by inversion in circle was invented by L. I. Magnus in 1831. Since then this mapping has become an avenue to higher mathematics. Through some steps of application of the circle inversion map, a student of transformation geometry soon appreciates the significance of Felix Klein’s Erlangen program, an outgrowth of certain models of hyperbolic geometry

Dilations

The combination of two inversions in concentric circles results in a similarity, homothetic transformation, or dilation characterized by the ratio of the circle radii.

x \mapsto R^2 \frac {x} {|x|^2} = y \mapsto T^2 \frac {y} {|y|^2} = \left( \frac {T} {R} \right)^2 \ x.

Reciprocation

When a point in the plane is interpreted as a complex number  z=x+iy \,, with complex conjugate  \bar{z}=x-iy, then the reciprocal of z is  \scriptstyle \frac{1}{z} = \frac{\bar{z}}{|z|^2}. Consequently, the algebraic form of the inversion in a unit circle is given by  z \mapsto w where:

  w=\frac{1}{\bar z}=\overline{\left(\frac{1}{z}\right)}.

Reciprocation is key in transformation theory as a generator of the Möbius group. The other generators are translation and rotation, both familiar through physical manipulations in the ambient 3-space. Introduction of reciprocation (dependent upon circle inversion) is what produces the peculiar nature of Möbius geometry, which is sometimes identified with inversive geometry (of the Euclidean plane). However, inversive geometry is the larger study since it includes the raw inversion in a circle (not yet made, with conjugation, into reciprocation). Inversive geometry also includes the conjugation mapping. Neither conjugation nor inversion-in-a-circle are in the Möbius group since they are non-conformal (see below). Möbius group elements are analytic functions of the whole plane and so are necessarily conformal.

Transforming circles into circles

Consider, in the complex plane, the circle of radius {\displaystyle (z-a)(z-a)^{*}=r^{2}}

where without loss of generality,  a \in \mathbb{R} . Using the definition of inversion

  {\displaystyle w=1/z^{*}}

it is straightforward to show that  w obeys the equation

{\displaystyle ww^{*}-{\frac {a}{(a^{2}-r^{2})}}(w+w^{*})+{\frac {a^{2}}{(a^{2}-r^{2})^{2}}}={\frac {r^{2}}{(a^{2}-r^{2})^{2}}}}

Showing that the  w describes the circle of center  {\displaystyle {\frac {a}{(a^{2}-r^{2})}}} and radius {\displaystyle {\frac {r}{|a^{2}-r^{2}|}}}.

when  {\displaystyle a\rightarrow r}, the circle transforms into the line parallel to the imaginary axis  {\displaystyle w+w^{*}=1/a}.

 

月幾望，圖文見。

 

解析者，保其角

\frac{d}{dz} z^{'} = \frac{a \cdot d - b \cdot c}{c^2} \cdot \frac{1}{{(z+\frac{d}{c})}^2}, \ if \ z \neq - \frac{d}{c}

將自圓乎◎