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Simulation of Uniform Distribution on Surfaces. Giuseppe Melfi Université de Neuchâtel Espace de l’Europe, 4 2002 Neuchâtel. Introduction. Random distributions are quite usual in nature. In particular: Environmental sciences Geology Botanics Meteorology are concerned .

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### Simulation of Uniform Distribution on Surfaces

Giuseppe Melfi

Université de Neuchâtel

Espace de l’Europe, 4

2002 Neuchâtel

Random distributions are quite usual in nature. In particular:

• Environmental sciences

• Geology

• Botanics

• Meteorology

are concerned

Distribution of trees in a typical cultivated field.

Distribution of trees in a typical intensive production. For the same surface and the same minimal distance, there are 15% more trees.

Distribution of trees in a plane forest. Uniform random distribution on a plane.

• In a nonplanar surface

• Such that points are distributed according to a random uniform distribution, namely the quantity of points for distinct unities of surface area (independently of gradient) follows a Poisson distribution X

• The input of such a problem is a function

D compact, f supposed to be differentiable. This function describes the surface

• The basic tool is a (pseudo-) random number generator.

Algorithm 1Step 1:Generation of N points inD

• D is bounded, so

• Random points in the box

can be partly inbedded in D.

• This procedure allows us to simulate an arbitrary number of uniformily distributed points in D, say N, denoted

• We assign to each point in D a random number w in (0,1).

• We have that w1, w2, …,wNare drawn according to a uniform distribution.

• This will be employed to select points on the basis of a suitable probability of selection.

• The region corresponds into the surface S to a region whose area can be approximated by

• We compute

• The probability of (xi, yi, f(xi, yi)) to be selected must be proportional to the quantity

• The point (xi, yi, f(xi, yi)) is selected if

• If S does not come from a bivariate function, but is simply a compact surface (e.g., a sphere), this approach is possible by Dini’s theorem.

• If D is bounded but not necessarily compact, it suffices that

is bounded.

• Let

f(x,y)=6exp{-(x2+y2)}

• Let

D=(-3,3)x(-3,3)

• We apply the preceding algorithm. We have 1000 points in D. A selection of these points will appear in simulation.

• Let

f(x,y)=x2-y2

• Let

D=(-1,1)x(-1,1)

Again, 1000 points have been used.

Under another perspective S={(x,y,6arctan x)}

Density can depend on

• slope

• orientation

• punctual function

These factors correspond to a positive function z(x,y) describing their punctual influence.

• Step 1: Generation of random points in D

• Step 2: Random assignment

• Step 3: Compute

• Step 4: (xi,yi,f(xi,yi)) is selected if

Non uniform distribution: an example

• Let f(x,y)=6 exp{-(x2+y2)}

It is the surface considered in first example

• Let z1(x,y)=3-|3-f(x,y)|

This corresponds to give more probability to points for which f(x,y)=3

• Let z2(x,y)=exp{-f(x,y)2}

In this case we give a probability of Gaussian type, depending on value of f(x,y)

A non uniform distribution onS={(x,y,6 exp{-x2-y2})} usingz1

A non uniform distribution onS={(x,y,6 exp{-x2-y2})} usingz2

Non uniform distribution onS = {(x,y, x2-y2)}

Non uniform distribution onS={(x,y,6arctan x)}

Non uniform distribution onS={(x,y,(x2+y2)/2)}

• A quantity of interest Q can depend on reciprocal distance of points

• on disposition of points in a neighbourood of each point

• A suitable model for an estimation of Q by Monte Carlo methods could be imagined.