I'm a little rusty on my algebra so I'm hoping this is fairly easy. I'm trying to write some equations describing the Euclidean distance between two points but are subject to the constraints that the distance must be greater than zero and must also not be equal to one. Here is what I have so far:

Let $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$ be the two points, Let the $\displaystyle d$ be the distance between them.

Then I should be able to use the following equation to require that the distance be anything greater than zero:

$\displaystyle ((x_1-x_2)^2+(y_1-y_2)^2)^\frac{1}{2} > 0$

could be written as:

$\displaystyle t_1((x_1-x_2)^2+(y_1-y_2)^2)^\frac{1}{2}-1=0$

I think this would work because $\displaystyle \frac{1}{t_1}$ can never be zero. I don't think $\displaystyle t_1$ needs to be squared because if it's negative, there won't be a solution.

So where I'm stuck is: how would I write the distance equation subject to the constraint that:

$\displaystyle ((x_1-x_2)^2+(y_1-y_2)^2)^\frac{1}{2} \neq 1$

Any suggestions to get me on the right foot?