Constructing a Torus: Logical Justification

[Wendel]

Homework Statement

Suppose we wish to construct a torus {(x,y,z)∈ℝ³:(R-√(x²+y²))²+z²=r²} whose axis of symmetry is the z-axis and the distance from the center of the tube to the center of the torus is R. Let r be the radius of the tube.

Homework Equations

The large circle in the xy-plane: x²+y²=R².
The smaller circle which revolves around the larger one: t²+z²=r² where t is the component of the distance from the point on the torus to the center of the tube that lies in the xy-plane.

The Attempt at a Solution

(R-√(x²+y²))²+z²=r² consists of all the points in ℝ³ a distance of r from the circle {(x,y)∈ℝ²:x²+y²=R²}. And t=R-√(x²+y²). Substituting this in the place of t in the equation t²+z²=r² gives us the equation of the torus. My question is why does this work; What is the logical justification for doing so? I don't understand the exact process by which we go from a one-dimensional object in a plane, to a surface in ℝ³.

 

[Orodruin]

You have one constraint equation for a three-dimensional space. This describes a two-dimensional surface. I do not understand your reference to a one-dimensional object in the plane.