Chordal metric

The chordal metric is a metric on the Riemann sphere that is defined with the help of stereographic projection .

definition

The sphere embedded in Euclidean space is denoted by. Now let the inverse image of the stereographic projection by the North Pole with . For two points on the Riemann sphere, the chordal metric is defined by ${\displaystyle \mathbb {S} ^{2}\subset \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle P_{N}^{-1}\colon \mathbb {C} \cup \{\infty \}\to \mathbb {S} ^{2}}$ ${\displaystyle N}$ ${\displaystyle P_{N}^{-1}(\infty )=N}$ ${\displaystyle z,w\in \mathbb {C} \cup \{\infty \}}$ ${\displaystyle \chi }$

${\displaystyle \chi (z,w):=\|P_{N}^{-1}(z)-P_{N}^{-1}(w)\|_{2}}$,

where denotes the Euclidean norm . ${\displaystyle \|\cdot \|_{2}}$

The representation results explicitly for points ${\displaystyle w,z\in \mathbb {C} }$

${\displaystyle \chi (z,w)=\|P_{N}^{-1}(z)-P_{N}^{-1}(w)\|_{2}={\frac {2\cdot \|w-z\|}{{\sqrt {1+\|w\|^{2}}}{\sqrt {1+\|z\|^{2}}}}}}$.

For and can the representation ${\displaystyle w=\infty }$ ${\displaystyle z\in \mathbb {C} }$

${\displaystyle d_{c}(z,\infty )=\|P_{N}^{-1}(z)-N\|_{2}={\frac {2}{\sqrt {1+\|z\|^{2}}}}}$

can be determined and applies to ${\displaystyle w=z=\infty }$

${\displaystyle \chi (\infty ,\infty )=\|N-N\|_{2}=0}$.

properties

The Riemann number sphere is a compact metric space with regard to the chordal metric. Since the chordal metric and the Euclidean metric are equivalent for any one , properties such as openness or closure of bounded subsets of are identical for the two metrics. ${\displaystyle \mathbb {C} \cup \{\infty \}}$ ${\displaystyle B_{R}(0)}$ ${\displaystyle R>0}$ ${\displaystyle \mathbb {C} }$

generalization

Since there is also a stereographic projection from the - sphere into the one-point compactification of , the above definition can be generalized and one obtains that with regard to this metric there is also a compact metric space. ${\displaystyle P_{N}\colon S^{n}\to {\widehat {\mathbb {R} ^{n}}}}$ ${\displaystyle n}$ ${\displaystyle {\widehat {\mathbb {R} ^{n}}}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle {\widehat {\mathbb {R} ^{n}}}}$

Individual evidence

1. ^ Rolf Walter: Introduction to Analysis 1 . Walter de Gruyter 2007, ISBN 9783110195392 , pp. 354-355.