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

**References**:**Bump Function***From:*Ajitkumar <ajitkumar@math.mu.ac.in>