The answer to the question posed can be received by means of replacement of a system of coordinates. As their choice is not stipulated, and ways can be a little. Anyway, it is about a sphere form in new space.
Instruction
1. In order that further it was more clear, begin with a flat case. Of course "we will turn out" a word it is necessary to quote. Consider x^2+y^2=R^2 circle. Apply curvilinear coordinates. For this purpose make replacements of the variables u=R/x, v=R/y, respectively the return transformation x=R/u, y=R/v. Substitute it in the equation of a circle and receive [(1/u) ^2+(1/v) ^2] *R^2=R^2 or (1/u) ^2+(1/v) ^2=1. Further (u^2+v^2)/(u^2) (v^2) =1, or u^2+v^2=(u^2) (v^2). Schedules of such functions do not keep within a framework of curves of the second order (the fourth order here).
2. In order that the type of a curve in coordinates of u0v considered as Cartesian became clear pass to polar coordinates ρ=ρ (φ). At this u=ρcosφ, v=ρsinφ. Then (ρcosφ)^2+ (ρsinφ)^2= [(ρcosφ)^2] [(ρsinφ)^2]. (ρ^2) [(cosφ)^2+(sinφ)^2]= (ρ^4) [(cosφ)^2] [(sinφ)^2], 1= (ρ^2) [(cosφ) (sinφ)]^2. Apply a formula of a sine of a double corner and receive ρ^2=4 / (sin2φ)^2 or ρ=2 / |(sin2φ)|. Branches of this curve are very similar to hyperbole branches (see fig. 1).
3. Now you should pass to the sphere of x^2+y^2+z^2=R^2. By analogy with a circle make replacements of u=R/x, v=R/y, w=R/z. Then x=R/u, y=R/v, z=R/w. Further receive [(1/u) ^2+(1/v) ^2+(1/w) ^2] *R^2=R^2, (1/u) ^2+(1/v) ^2+(1/w) ^2=1 or (u^2)(v^2)+ (u^2)(w^2)+ (v^2)(w^2)= (u^2) (v^2) (w^2). To spherical coordinates in limits 0uvw, considered as Cartesian, it is not necessary to pass as it will not bring simplification in search of the sketch of the received surface.
4. Nevertheless, this sketch was already designated from preliminary data of a flat case. Besides, it is obvious that it is the surface consisting of separate fragments and that these fragments do not cross coordinate planes u=0, v=0, w=0. They can approach them asymptotically. In general the figure consists of eight fragments similar to hyperboloids. If to give them the name "conditional hyperboloid", then it is possible to speak about four couples of dvupolostny conditional hyperboloids which axis of symmetry are straight lines with the directing cosines {1 / √ 3, 1 / √ 3, 1 / √ 3 }, {-1 / √ 3, 1 / √ 3, 1 / √ 3 }, {1 / √ 3,-1 / √ 3, 1 / √ 3 }, {-1 / √ 3,-1 / √ 3, 1 / √ 3 }. To give an illustration rather difficult. Nevertheless, the provided description can be considered rather complete.