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In algebraic geometry, ${\displaystyle \mu }$ is said to be a multiplicative distance function over a field if it satisfies

• ${\displaystyle \mu (AB)>1.\,}$
• AB is congruent to A'B' iff ${\displaystyle \mu (AB)=\mu (A'B').\,}$
• AB < A'B' iff ${\displaystyle \mu (AB)<\mu (A'B').\,}$
• ${\displaystyle \mu (AB+CD)=\mu (AB)\mu (CD).\,}$

See also

• Algebraic geometry
• Hyperbolic geometry
• Poincaré disc model
• Hilbert's arithmetic of ends

References

1. Hartshorne, Robin (2000), Geometry: Euclid and beyond, Undergraduate Texts in Mathematics, New York: Springer-Verlag, p. 363, doi:10.1007/978-0-387-22676-7, ISBN 0-387-98650-2, MR 1761093.

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