Re: overlapping area of two ellipses
- From: Valeri Astanoff <astanoff@xxxxxxxxx>
- Date: Mon, 8 Sep 2008 05:03:05 -0700 (PDT)
On 3 sep, 10:13, fatih.arslan....@xxxxxxxxx wrote:
hello guys,
I have twoellipses, which I know their centers, axis a and axis b and
rotations for each of them. Is there any analytical formula for
calculatingoverlappingregion?
Like Ellips1: center(x,y)=(10,20) axis a=5 axis b=4 rotation=30 degree
Ellps2: center(x,y)=(20,30) axis a=4 axis b=5 rotation=40
degree
(rotation is clockwise)
PS1: Even if I could not find the exact area, a solution which
approximates theoverlappingarea is ok. Because I am writing a C++
code, and I just have to compare theoverlappingareas of differentellipses...So if I can find a reasonable approximation, it is ok...
PS2:If you do not have time or patience to write, you can give me
references...It will be very good...
Thank you
Good day,
Just for the fun, I wrote my own solution with
help of Mathematica's MonteCarlo integration:
overlap[
{a1_,b1_,x1_,y1_,t1_ /; -Pi/2 <= t1 <= Pi/2} /; a1 >= b1,
{a2_,b2_,x2_,y2_,t2_ /; -Pi/2 <= t2 <= Pi/2} /; a2 >= b2]:=
Module[{},
If[x1+a1 <= x2-a2 || y1+a1 <= y2-a2 ||
x2+a2 <= x1-a1 || y2+a2 <= y1-a1, Return[0]];
f[{a_,b_,xc_,yc_,t_},x_,y_]=
(c = Sqrt[a^2 - b^2];
Sqrt[Abs[x - xc - c*Cos[t]]^2 + Abs[y - yc - c*Sin[t]]^2] +
Sqrt[Abs[x - xc + c*Cos[t]]^2 + Abs[y - yc + c*Sin[t]]^2] - 2*a);
f1=f[{a1, b1, x1, y1,t1}, x, y]//N;
f2=f[{a2, b2, x2, y2,t2}, x, y]//N;
in1=f1 <= 0 && x1-a1 <= x <= x1+a1 && y1-a1 <= y <= y1+a1;
in2=f2 <= 0 && x2-a2 <= x <= x2+a2 && y2-a2 <= y <= y2+a2;
xmin = NMinimize[{x,in1 && in2},{x,y}][[1]];
xmax = NMaximize[{x,in1 && in2},{x,y}][[1]];
ymin = NMinimize[{y,in1 && in2},{x,y}][[1]];
ymax = NMaximize[{y,in1 && in2},{x,y}][[1]];
If[xmin >= xmax || ymin >= ymax,0,
NIntegrate[Boole[f1 <=0 && f2 <= 0], {x,xmin, xmax}, {y,ymin, ymax},
Method -> "MonteCarlo", MaxPoints -> 10000]
]
];
Examples:
Identical ellipses:
overlap[{3.,2.,1.,2.,30.Degree},{3.,2.,1.,2.,30.Degree}] //Timing
{1.062,19.091}
Check:
6.Pi
18.8496
No overlapping ellipses:
overlap[{3.,2.,1.,2.,10.Degree},{5.,4.,10.,20.,30.Degree}]//Timing
{0.,0}
Unspecified case:
overlap[{4.,3.,7.,5.,10.Degree},{6.,3.,11.,8.,30.Degree}]//Timing
{1.063,17.196}
V.Astanoff
.
- References:
- overlapping area of two ellipses
- From: fatih . arslan . koc
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