Question

# In the upper-plane plane model for hyperbolic geometry, calculate the distance between the points A(0, 4) text{and} B(3, 5). Give your answer accurate to three decimals. Hint: Recall the definition of distance in the upper-half plane model.

Decimals
In the upper-plane plane model for hyperbolic geometry, calculate the distance between the points $$A(0,\ 4)\ \text{and}\ B(3,\ 5).$$ Give your answer accurate to three decimals. Hint: Recall the definition of distance in the upper-half plane model.

2021-02-11
Step 1
We have the two points $$A (0,\ 4)\ \text{and}\ B (3,\ 5)$$ let
$$\displaystyle{A}={\left({0},{4}\right)}\to{\left({x}_{{1}},{x}_{{2}}\right)}$$
$$\displaystyle\therefore{x}_{{1}}={0},{x}_{{2}}={4}$$
$$\displaystyle{B}={\left({3},{5}\right)}\to{\left({y}_{{1}},{y}_{{2}}\right)}$$
$$\displaystyle\therefore{y}_{{1}}={3},{y}_{{2}}={5}$$
We have distance,
$$\displaystyle{d}{i}{s}{\left(\begin{matrix}{x}_{{1}}&{y}_{{1}}\\{x}_{{2}}&{y}_{{2}}\end{matrix}\right)}$$
$$\displaystyle={2} \ln{{\left(\frac{{\sqrt{{{\left({x}_{{2}}-{x}_{{1}}\right)}^{2}+{\left({y}_{{2}}-{y}_{{1}}\right)}^{2}}}+\sqrt{{{\left({x}_{{2}}-{x}_{{1}}\right)}^{2}+{\left({y}_{{2}}-{y}_{{1}}\right)}^{2}}}}}{{{2}\sqrt{{{y}_{{1}}{y}_{{2}}}}}}\right)}}$$
Substitute the values
$$\displaystyle={2} \ln{{\left(\frac{{\sqrt{{{\left({4}-{0}\right)}^{2}+{\left({5}-{3}\right)}^{2}}}+\sqrt{{{\left({4}-{0}\right)}^{2}+{\left({5}-{3}\right)}^{2}}}}}{{{2}\sqrt{{{3}\times{5}}}}}\right.}}$$
Step 2
$$\displaystyle={2} \ln{{\left(\frac{{\sqrt{{{4}^{2}+{3}^{2}}}+\sqrt{{{4}^{2}+{8}^{2}}}}}{{{2}\sqrt{{15}}}}\right)}}$$
$$\displaystyle={2} \ln{{\left(\frac{{\sqrt{{{16}+{9}}}+\sqrt{{{16}+{64}}}}}{{{2}\sqrt{{15}}}}\right)}}$$
$$\displaystyle={2} \ln{{\left(\frac{{\sqrt{{{25}}}+\sqrt{{{80}}}}}{{{2}\sqrt{{15}}}}\right)}}$$
$$\displaystyle={2} \ln{{\left(\frac{{{5}+\sqrt{{{80}}}}}{{{2}\sqrt{{15}}}}\right)}}$$
$$\displaystyle\sqrt{{80}}=\sqrt{{{16}\times{5}}}$$
$$\displaystyle=\sqrt{{16}}\times\sqrt{{5}}$$
$$\displaystyle={4}\sqrt{{5}}$$
$$\displaystyle={2} \ln{{\left(\frac{{{5}+{4}\sqrt{{5}}}}{{{2}\sqrt{{15}}}}\right)}}$$
$$\displaystyle={2} \ln{{\left(\frac{13.94407191}{{7.7459666924}}\right)}}$$
$$= 1.18355406$$
$$= 1.184$$
So the answer is $$1.184$$