[x^2 + y^2 + (z - 1)^2] - [(x - 1)^2 + y^2 + z^2] = 0 - Deep Underground Poetry

Understanding the Context

Expanding and Simplifying the Equation

Start by expanding both cubic and squared terms:

Left side:
\[ x^2 + y^2 + (z - 1)^2 = x^2 + y^2 + (z^2 - 2z + 1) = x^2 + y^2 + z^2 - 2z + 1 \]

Key Insights

Right side:
\[ (x - 1)^2 + y^2 + z^2 = (x^2 - 2x + 1) + y^2 + z^2 = x^2 - 2x + 1 + y^2 + z^2 \]

Now subtract the right side from the left:

\[
\begin{align}
&(x^2 + y^2 + z^2 - 2z + 1) - (x^2 - 2x + 1 + y^2 + z^2) \
&= x^2 + y^2 + z^2 - 2z + 1 - x^2 + 2x - 1 - y^2 - z^2 \
&= 2x - 2z
\end{align}
\]

Thus, the equation simplifies to:
\[
2x - 2z = 0 \quad \Rightarrow \quad x - z = 0
\]

Final Thoughts

Geometric Interpretation

The simplified equation \( x - z = 0 \) represents a plane in 3D space. Specifically, it is a flat surface where the x-coordinate equals the z-coordinate. This plane passes through the origin (0,0,0) and cuts diagonally across the symmetric axes, with a slope of 1 in the xz-plane, and where x and z increase or decrease in tandem.

• Normal vector: The vector [1, 0, -1] is normal to the plane.
  - Orientation: The plane is diagonal relative to the coordinate axes, tilted equally between x and z directions.
  - Intersection with axes:
  - x-z plane (y = 0): traces the line x = z
  - x-axis (y = z = 0): x = 0 ⇒ z = 0 (only the origin)
  - z-axis (x = 0): z = 0 ⇒ only the origin

Visualizing the Surface

Although algebraically simplified, the original equation represents a plane—often easier to sketch by plotting key points or using symmetry. The relationship \( x = z \) constrains all points so that moving equally in x and z directions keeps you on the plane.