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Plotting a section of an hemisphere


I'm writing a little package in Mathematica for geology where a particular stone may be approximated as an hemisphere. Anyway this is a rough estimation because a real hemisphere has its height as loong as its radius. Instead, a reservoir stone (for an hydrocarbon) has often a form of a section of an hemisphere, its height is lower than the radius. For example, I can have an hemisphere with radius long 5 km and height of only 3 km and I can plot it like that:


semisfera[x_, y_, raggio_] := Sqrt[raggio^2 - (x - raggio)^2 - (y - raggio)^2];

 plotsemisfera = Plot3D[semisfera[x, y, raggioSfera], {x, 0, 2  raggioSfera}, {y, 0, 2  raggioSfera}, PlotRange -> {0, 3}, AxesLabel -> {"lunghezza km" , "larghezza km","profondità km"}, PlotLabel ->  Style[Framed["Referenced Theorical Hemisphere"], 22, Black]]

and I get the following graphic: enter image description here


you'll agree with me that is a section of ah hemisphere without the top part, won't you?


Sometimes it may happen that the height is << radius. In my case, my geology student worked on a stone with radius of 5km and an height of only 0.2 km. If I try to plot this as I've done before, I get a very awful graphic, here:


enter image description here


So, I'd just like to know if there is a way to plot a more precise graphic, without all that irregular part at the base of the hemisphere.


The centre of the "hemisphere" should be in = <0,0>


Maybe it could be something like that: http://uploadpie.com/eAVvq


but I really don't understand why for low values of the height the base of the hemisphere is so jagged!



How can I plot that? Thank you



Answer



Edit


I now have a better understanding of what you are looking for.


To get plot centered at the origin defined in terms of the radius and height, then you can use SphericalPlot3D as Kuba suggested. It would go like this.


theta[r_, h_] /; 0 < h < r := π/2. - ArcTan[Sqrt[r^2 - h^2], h]
With[{r = 5, h = 3, zScale = .3}, 
  SphericalPlot3D[r, {θ, theta[r, h], π/2}, {ϕ, 0, 2 π}, BoxRatios -> {1, 1, zScale}]]

spherical-sect-1



Note the use of a parameter to scale z-axis. It is set to h/(2 r) in the above plot. This gives true proportions.


In the extreme of r = 5 and h = .2, zScale will need to be adjusted to give a reasonable looking plot, which is going to look very much like a cylinder.


With[{r = 5, h = .2, zScale = .25}, 
  SphericalPlot3D[r, {θ, theta[r, h], π/2.}, {ϕ, 0, 2 π}, BoxRatios -> {1, 1, zScale}]]

spherical-sect-2


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