给定两焦点坐标与距离差的单叶双曲面：是否存在更简洁表达式？

Great question! The radical form you have is the fundamental geometric definition, but we can absolutely simplify it to a cleaner algebraic form (no square roots) and even map it to the classic standard hyperboloid equation with a bit of coordinate adjustment. Let's break this down:

Step 1: Eliminate Square Roots for a Cleaner Algebraic Form #

Start with your original definition:

$\sqrt{(x-x_1)^2+(y-y_1)2+(z-z_1)^2} - \sqrt{(x-x_2)^2+(y-y_2)2+(z-z_2)^2} = d$

First, rearrange one radical to the right-hand side to isolate it:
$$\sqrt{(x-x_1)^2+(y-y_1)2+(z-z_1)^2} = d + \sqrt{(x-x_2)^2+(y-y_2)2+(z-z_2)^2}$$

Square both sides to remove the left radical:
$$(x-x_1)^2+(y-y_1)2+(z-z_1)^2 = d^2 + 2d\sqrt{(x-x_2)^2+(y-y_2)2+(z-z_2)^2} + (x-x_2)^2+(y-y_2)2+(z-z_2)^2$$

Now expand and simplify the non-radical terms. For example, $(x-x_1)^2 - (x-x_2)^2$ simplifies to $2(x_2-x_1)x + x_1^2 - x_2^2$ (the same pattern applies to y and z terms). Combine all these simplified terms:
$$2(x_2-x_1)x + 2(y_2-y_1)y + 2(z_2-z_1)z + (x_1^2+y_12+z_1^2 - x_2^2-y_22-z_2^2) - d^2 = 2d\sqrt{(x-x_2)^2+(y-y_2)2+(z-z_2)^2}$$

Square both sides again to eliminate the remaining radical. After expanding and combining like terms, you'll end up with a quadratic equation (no square roots) in the form:
$$Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$$

All coefficients here can be directly computed using the coordinates of $F_1$, $F_2$, and the distance difference $d$. This is already a much more practical form than the radical definition.

Step 2: Standard Hyperboloid Form via Coordinate Transformation #

For an even more intuitive and simplified form, we can shift and rotate the coordinate system to align with the hyperboloid's natural axes:

1. Translate the origin to the midpoint $O$ of $F_1F_2$: $O\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$
2. Rotate the coordinate system so the x-axis runs along the line connecting $F_1$ and $F_2$.

In this standard coordinate system:

• Let $2c = |F_1F_2|$ (the full distance between the two focuses)
• Let $a = \frac{|d|}{2}$ (half the absolute value of your distance difference)
• Let $b^2 = c^2 - a^2$ (a parameter defining the hyperboloid's "spread")

The equation reduces to the classic single-sheeted hyperboloid form:
$$\frac{x^2}{a2} - \frac{y^2 + z^2}{b2} = 1$$

Key Notes: #

• This form only applies when $|d| < 2c$: if $|d| = 2c$, the equation reduces to two rays; if $|d| > 2c$, there are no real points satisfying the equation.
• As you noted, when $d=0$, the equation collapses to the perpendicular bisecting plane of $F_1F_2$—this makes sense because $|PF_1| = |PF_2|$ defines all points equidistant to both focuses.

内容的提问来源于stack exchange，提问作者Guglie