Skip to main content

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


Comments

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...