Thursday, June 23, 2022

[wbpjdinu] spherical cap height, surface area, and volume

a spherical cap has a height, surface area, and volume, and each of these quantities can be normalized by the corresponding value for the whole sphere (something involving pi).  pairwise, these normalized values define monotonic functions on [0,1] * [0,1].  what do these functions look like?  this should be easy.

these functions can be generalized to hyperspheres of any number of dimensions.  how do these functions change with dimension?

the 2D analogue of a spherical cap is a called circular segment, and its height is called its sagitta.

in 1D, the height and volume are equal, and "surface area" is always constant, the area of one point (whatever that means), until the spherical cap becomes the whole sphere, when its surface area becomes the area of two points (whatever that means).

a spherical cap also has a perimeter, but it cannot be normalized by the "perimeter" of the whole sphere.  it could be normalized by the maximum, the circumference of a great circle.  there are more things in higher dimensions.

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