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

Title: Solving the 3D Geometric Equation: Understanding the Surface Defined by [x² + y² + (z − 1)²] − [(x − 1)² + y² + z²] = 0

Introduction
The equation [x² + y² + (z − 1)²] − [(x − 1)² + y² + z²] = 0 presents a compelling geometric object within three-dimensional space. Whether you're studying surfaces in computational geometry, analytical mechanics, or algebraic modeling, this equation reveals a meaningful shape defined by balancing two quadratic expressions. This article explores how to interpret and visualize this surface, derive its geometric properties, and understand its applications in mathematics and engineering.

Understanding the Context

Expanding and Simplifying the Equation

Start by expanding both cubic and squared terms:

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

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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:
\[
2x - 2z = 0 \quad \Rightarrow \quad x - z = 0
\]

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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.

Analytical Insights

From a coordinate geometry standpoint, this surface exemplifies how differences of quadratic forms yield linear constraints. The reduction from a quadratic difference to a linear equation illustrates the power of algebraic manipulation in uncovering simple geometric truths.