Hyperbolic midpoint

Question

The hyperbolic line consists of positive real numbers with the hyperbolic distance d(a,b) defined as the absolute value of ln(a/b)

Find the hyperbolic midpoint X between P(63) and Q(175)

 The hyperbolic line consists of positive real numbers with the hyperbolic distance d(a,b) defined as the absolute value of ln(a/b)

Find the point X that is not between P(70) and Q(14) such that 2d_h(X,70)=d_h(X,14).

Is this like solving any other midpoint problem? How do we do this specific to points between a hyperbolic line? 

 

Answer 1

I'm a bit out on a limb here, but let's make an analogy with standard Euclidean geometry. We'll use d(a,b) to denote the distance between points a and b in a plane. Let ab be the line segment connecting both points. Then, the midpoint is the point x on ab such that d(a,x) = d(x,b).

 

Let's now turn our attention to the question at hand. Since the hyperbolic line "consists of all positive real numbers" with no other qualifiers, there seems to be no additional limitation on midpoint X (such as, it lies on the segment connecting two points). It looks like we can use any positive real number for X.  Maybe the fact that it ends up being between P and Q is equivalent to being on the segment connecting them.

 

Now, let's set up our equation, where P=63 and Q=175:

d(P,x) = d(x,Q)

ln(P/x) = ln(x/Q). Exponentiate both sides to clear the logarithms:

P/x = x/Q. Cross multiply:

x² = PQ = 63*175

x = √(63*175). 

 

We take the positive square root, since X must be greater than 0.  The answer is the geometric mean of P and Q.

 

Like I said, I could very well be wrong on this one, but hope it's helpful.