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Re: Bump Function
*To*: mathgroup at smc.vnet.net
*Subject*: [mg31994] Re: [mg31967] Bump Function
*From*: Daniel Lichtblau <danl at wolfram.com>
*Date*: Sun, 16 Dec 2001 03:44:29 -0500 (EST)
*References*: <200112140921.EAA03644@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Ajitkumar wrote:
>
> Hi,
>
> Could somebody tell me how to plot the graph of bump functions. For
> example, if I want to draw the graph of the function say,
>
> f(x,y) = x^2+y^2 for x^2+y^2<1
> = 2 for x^2+y^2 >2
>
> Thanks
> Ajit Kumar
> Dept of Maths
> University of Mumbai
> Vidyanagari, Kalina
> Mumbai 400 098
> India
> Phone No. +91 652 6683
> Email- ajitkumar at math.mu.ac.in
> ajit_kumara at hotmail.com
> http://math.mu.ac.in/faculty/scholars/ajit.html
Below is, I think, a standard sort of "bump" function wherein we take
the value 1 for 0<=r<=1, 0 for r>=2, and smoothly move from the plateau
to 0 when 1<r<2. Here I use 'r' to indicate radial distance. We start
with a helper function that rises smoothly from 0 and falls back
smoothly to zero.
b[r_] := Exp[-1/(r*(1-r))]
normalization = NIntegrate[b[s], {s,0,1}];
By integration we use the helper to fall smoothly from 1 to 0 as r goes
from 1 to 2.
bigB[r_] /; r>=2 := 0
bigB[r_] /; r<=1 := 1
bigB[r_] /; 1<r<2 := NIntegrate[b[s], {s,r-1,1}] / normalization
Now define the bump function in {x,y} using radial distance
Sqrt[x^2+y^2].
bump[x_,y_] := bigB[Sqrt[x^2+y^2]]
You can plot it as below.
Plot3D[bump[x,y], {x,-3,3}, {y,-3,3}, PlotPoints->50]
Daniel Lichtblau
Wolfram Research
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