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Mathematics LibreTexts

20.9: Area in the neutral planes and spheres

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    Area can be defined in the neutral planes and spheres. In the definition, the solid unit square \(\mathcal{K}_1\) has to be exchanged to a fixed nondegenerate polygonal set \(\mathcal{U}\). One has to make such change for good reason — hyperbolic plane and sphere have no squares.

    The set \(\mathcal{U}\) in this case plays role of the unit measure for the area and changing \(\mathcal{U}\) will require conversion of area units.

    截屏2021-03-03 上午10.43.30.png

    According to the standard convention, the set \(\mathcal{U}\) is taken so that on small scales area behaves like area in the Euclidean plane. Say, if \(\mathcal{K}_a\) denotes the solid quadrangle \(\blacksquare ABCD\) with right angles at \(A\), \(B\), and \(C\) such that \(AB=BC=a\), then we may assume that

    \(\tfrac{1}{a^2}\cdot\text{area }\mathcal{K}_a\to 1 \quad \text{as} \quad a\to 0.\)

    This convention works equally well for spheres and neutral planes, including the Euclidean plane. In spherical geometry equivalently we may assume that if \(r\) is the radius of the sphere, then the area of whole sphere is \(4\cdot\pi\cdot r^2\).

    Recall that defect of triangle \(\triangle ABC\) is defined as

    \(\text{defect}(\triangle ABC) := \pi-|\measuredangle ABC|-|\measuredangle BCA|-|\measuredangle CAB|.\) 

    It turns out that any neutral plane or sphere there is a real number \(k\) such that

    \[k\cdot\text{area }(\blacktriangle ABC)+\text{defect}(\triangle ABC)=0 \]

    for any \(\triangle ABC\).

    This number \(k\) is called curvature; \(k=0\) for the Euclidean plane, \(k=-1\) for the h-plane, \(k=1\) for the unit sphere, and \(k=\tfrac1{r^2}\) for the sphere of radius \(r\).

    Since the angles of ideal triangle vanish, any ideal triangle in h-plane has area \(\pi\). Similarly in the unit sphere the area of equilateral triangle with right angles has area \(\tfrac\pi2\); since whole sphere can be subdivided in eight such triangles, we get that the area of unit sphere is \(4\cdot\pi\).

    The identity 20.9.1 can be used as an alternative way to introduce area function; it works on spheres and all neutral planes, except for the Euclidean plane.