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Math Help - Euclidean and Taxicab problem

  1. #1
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    Joined
    Feb 2011
    Posts
    23

    Euclidean and Taxicab problem

    Hi guys. I have a homework problem that I don't know how to approach.

    Find graphically (using MATLAB) for each of the following cases, in both Euclidean geometry, and Taxicab geometry the loci of points that satisfy the following properties:

    (a) The points whose distance from the axes origin is 8.
    (b) The points which have the sum of their distances to the points A(-3,-4), and B(4, 3) equals to 16.
    (c) The points which have the difference of their distances to the points A(-3,-4) and B(4,3) equals to 2.
    (d) The point S such that d(S,P) + d(S,Q) = d(P,Q), where P(-2,3) and Q(1,-4)
    The instructor gives hints to each of the cases:
    a) a circle centered at the origin and whose radius is 8,
    b) an ellipse with A and B as foci,
    c) a hyperbola branch with A and B as foci, and
    d) a point R(-1/2, -1/2). The solution for this case, in Taxicab geometry, is the set of all points inside the rectangle shown(not shaded in order to show point R).

    -----------

    I didn't get part c and part b. I know what Taxicab geometry is. I know how to compute the distance. I can use MATLAB to generate such graphical representation for part a and b.

    Code:
    % code for part a
    x = -10:0.1:10;
    y = -10:0.1:10;
    Ax = 0;
    Ay = 0;
    Bx = -1;
    By = 0;
    [X,Y] = meshgrid(x,y);
    Z1=(abs(X)+abs(Y))+(abs(X)+abs(Y));    % computes Taxicab-G
    contour(X,Y,Z1,[16 16],'r')
    hold on
    Z2 = sqrt((X).^2+(Y).^2)+sqrt((X).^2+(Y).^2);  % computes Eu-G
    contour(X,Y,Z2,[16 16],'b--')
    plot(Ax,Ay,'k*',Bx,By,'ko')
    axis square
    set(gca,'Xtick',[-10:1:10],'Ytick',[-10:1:10])
    I think I can handle the MATLAB part, but I just didn't know how to solve for c and d.

    The answer to part c is shown below
    Euclidean and Taxicab problem-110317_120622.jpg

    How do I solve these two analytically?
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  2. #2
    Newbie
    Joined
    Feb 2011
    Posts
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    For D, i tried to solve the absolute value equation
    (d(S,P) + d(S,Q) = d(P,Q)) where d(P,Q) is 10

    [ |Sx - Px | + |Sy - Py| ] + [ | Sx - Qx| + | Sy - Qy| ] = |Px - Qx| + |Py - Qy|
    I would get two unknowns and two equations. The two equations come from the absolute value equation, that is negative of the left expression, and the negative of the right expression

    I get y = 7, x = -3
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