1. The problem is to identify and solve the surface defined by the equation $$x^2 - y^2 + z^2 = 1$$. 2. This is a quadratic surface equation involving three variables $x$, $y$, and $z$. 3. The general form of a quadratic surface is $$Ax^2 + By^2 + Cz^2 + D = 0$$. Here, $A=1$, $B=-1$, $C=1$, and $D=-1$. 4. Since the equation can be rewritten as $$x^2 + z^2 - y^2 = 1$$, it matches the form of a hyperboloid of one sheet or two sheets depending on signs. 5. Because the positive terms are $x^2$ and $z^2$ and the negative term is $-y^2$, and the right side is positive, this is a hyperboloid of one sheet. 6. To visualize, for fixed $y$, the cross-section is $$x^2 + z^2 = 1 + y^2$$, which is a circle with radius $$\sqrt{1 + y^2}$$. 7. This means the surface expands as $|y|$ increases. 8. The surface is symmetric about the $x$ and $z$ axes. 9. The solution is the set of all points $(x,y,z)$ satisfying $$x^2 - y^2 + z^2 = 1$$, which is a hyperboloid of one sheet. Final answer: The surface is a hyperboloid of one sheet defined by $$x^2 - y^2 + z^2 = 1$$.