find the furthest point

if i have an two arrays like that:
x=[  39  63  71  7  53  39  64  23  29  65  ];
y=[  20  11  22  78  61  71  9  20  78  94  ];

and

AA=[x ; y];

if p1 = (39,20) and p2 =(63,11)
how can i find the furthest point in AA than the points p1 and p2

Thanks, Hrabei

[Azzi Abdelmalek]

[Hassan commented]

Thanks for the answers, however, the question is"how can i find the furthest point in AA than the points p1 and p2"

i.e.

what is the point (only one point) in AA that is furthest from p1 and p2

[Azzi Abdelmalek]

Hassan, If you have two point p1 and p2, and two other points p3 and p4, how will you choose p3 rather than p4. What is your mathematics criterion?

[Image Analyst]

The phrase "how can i find the furthest point in AA than the points p1 and p2" does not make grammatical sense. Do you really mean "how can i find the furthest point in AA from the points p1 and p2"??? In other words, find the point in AA farthest from point p1, and then also find the furthest point in AA from point p2? And why are p1 and p2 already inside AA ? Do they need to be, or is that just a coincidence? Please clarify.

Answer 1

[Image Analyst]

Try this:

x=[ 39 63 71 7 53 39 64 23 29 65 ]; 
y=[ 20 11 22 78 61 71 9 20 78 94 ];
AA=[x ; y]

% Find point in AA closest to p1.
p1 = [39,20]
squaredDistance = sum((AA-repmat(p1', [1, size(AA, 2)])).^2, 1)
[maxSqDist1, indexOfMax1] = max(squaredDistance)
% Find point in AA closest to p2.
p2 =[63,11]
squaredDistance = sum((AA-repmat(p2', [1, size(AA, 2)])).^2, 1)
[maxSqDist2, indexOfMax2] = max(squaredDistance)

In the command window, you'll see:

AA =
  39    63    71     7    53    39    64    23    29    65
  20    11    22    78    61    71     9    20    78    94
p1 =
  39    20
squaredDistance =
         0         657        1028        4388        1877        2601         746         256        3464        6152
maxSqDist1 =
      6152
indexOfMax1 =
  10
p2 =
  63    11
squaredDistance =
       657           0         185        7625        2600        4176           5        1681        5645        6893
maxSqDist2 =
      7625
indexOfMax2 =
   4

[Jan]

@Image Analyst: "the furthest point" sounds like max() is wanted instead of min().

@Hassan: Notice that this solution uses the squared distance. The result is the same, because the maximal squared distance matches the same point as the maximal distance. But omitting the square root accelerates the program, because sqrt() is a very expensive function.

[Image Analyst]

You're right - corrected. (I wonder if it used to be closest, because I think Azzi's solution also does closest. I can't successfully run his because I don't have whatever toolbox fullfact() is in.)

[Azzi Abdelmalek]

What I've understood from his question is to find the two points (Pn,Pm) with distance Pn-Pm closest to the distance P1-P2. The points are not necessary the closest. The function fullfact (statitistic toolbox) can be replaced by ndgrid function

[Image Analyst]

I didn't think of that, but it could be. It's not really clear and specifically stated.

Answer 2

[Azzi Abdelmalek]

AA=[x ;y];
id=numel(x);
idx=unique(sort(fullfact([id id]),2),'rows');
idx(~diff(idx'),:)=[]
distance=sqrt((x(idx(:,1))-x(idx(:,2))).^2+(y(idx(:,1))-y(idx(:,2))).^2);
%----------------------------------------------------------------------
%p1 = (39,20) and p2 =(63,11)
x1=39;
y1=20;
x2=63;
y2=11;
dist1= sqrt((x1-x2)^2+(y1-y2)^2);
%----------------------------------------------------------------------
a=abs(distance-dist1);
[ii,jj]=sort(a);
id=jj(2);
point1=AA(:,idx(id,1))
point2=AA(:,idx(id,2))

%or

AA=[x ;y];
id=numel(x);
[ii,jj]=ndgrid(1:id,1:id);
idx=[jj(:) ii(:)]
idx=unique(sort(idx,2),'rows');
idx(~diff(idx'),:)=[]
distance=sqrt((x(idx(:,1))-x(idx(:,2))).^2+(y(idx(:,1))-y(idx(:,2))).^2);
%----------------------------------------------------------------------
%p1 = (39,20) and p2 =(63,11)
x1=39;
y1=20;
x2=63;
y2=11;
dist1= sqrt((x1-x2)^2+(y1-y2)^2);
%----------------------------------------------------------------------
a=abs(distance-dist1);
[ii,jj]=sort(a);
id=jj(2);
point1=AA(:,idx(id,1))
point2=AA(:,idx(id,2))