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

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

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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.

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Expanding and Simplifying the Equation

Start by expanding both cubic and squared terms:

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

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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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

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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.

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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.

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