Thread: 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).

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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


How do I solve these two analytically?

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