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I see in the Wikipedia article on the hyperboloid model and also in this other Math.SE question about the hyperboloid model that this is how you calculate distance on the hyperboloid model:

Let $u = (x_0,x_1,x_2)$ and $v = (y_0,y_1,y_2)$ be two points on the positive hyperbolic sheet so $x_0 = 1 + x_1^2 + x_2^2$ and $y_0 = 1 + y_1^2 + y_2^2$. Then the distance $d(u,v) = \cosh^{-1}(B(u,v))$, where $B(u,v) = x_0y_0 - x_1y_1 - x_2y_2$.

Okay. Let's say that $u = (17,4,0)$ and $v = (17,4,0)$. Yes. Yes indeed, these are exactly the same point. Now $B(u,v) = 17 \cdot 17 - 4 \cdot 4 - 0 \cdot 0 = 289-16-0 = 273$. Now, $\cosh^{-1}(x)$ is positive and strictly increasing when $x > 1$, so already I know the distance will be positive, but just to drive the point home, $\cosh^{-1}(273) \approx 6.303$.

That's right. The distance between two points in the exact same spot is significantly greater than $0$. I don't understand why this makes sense or why this is the formula for distance between two points on the hyperboloid model. It seems very odd to me that no one would have noticed this strange fact, so I can't help but think that there must be a mistake in my thinking. Where and what is my error?

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I figured it out. My error was thinking that $x_0 = 1+x_1^2+x_2^2$. In fact, $x_0 = \sqrt{1+x_1^2+x_2^2}$. Thus, $u = v = (\sqrt{17},4,0)$, so $B(u,v) = 17 - 16 = 1$, and $\cosh^{-1}(1) = 0$. The distance between two points at the exact same coordinate is indeed $0$.

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