$CD^2 = x^2 + y^2 + (z - 1)^2 = 2$ - Parker Core Knowledge

Understanding the Context

What Does $x^2 + y^2 + (z - 1)^2 = 2$ Mean?

The equation $x^2 + y^2 + (z - 1)^2 = 2$ describes a sphere centered at the point $(0, 0, 1)$ with radius $\sqrt{2}$. In general, the standard form of a sphere is:

$$
(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2
$$

Here,
- Center: $(a, b, c) = (0, 0, 1)$
- Radius: $r = \sqrt{2}$

Key Insights

This means every point $(x, y, z)$ lying on the surface of this sphere is exactly $\sqrt{2}$ units away from the center point $(0, 0, 1)$.

Why Is This Equation Illustrative in Geometry and Applications?

1. Distance Interpretation
   The left-hand side $x^2 + y^2 + (z - 1)^2$ is the squared Euclidean distance from the point $(x, y, z)$ to the center $(0, 0, 1)$. Thus, $CD^2 = 2$ expresses all points exactly $\sqrt{2}$ units from the center.

2. Geometric Visualization
   This equation simplifies visualizing a sphere translated along the $z$-axis. In 3D graphing software, it clearly shows a perfectly symmetrical sphere centered above the origin on the $z$-axis.

Final Thoughts

3. Use in Optimization and Machine Learning
   Such spherical equations appear in algorithms minimizing distances—like in clustering (k-means), where data points are grouped by proximity to centers satisfying similar equations.

4. Physical and Engineering Models
   In physics, radius-squared terms often relate to energy distributions or potential fields; the sphere models contours of constant value. Engineers use similar forms to define feasible regions or signal domains.

How to Plot and Analyze This Sphere

• Center: $(0, 0, 1)$ — located on the $z$-axis, one unit above the origin.
  - Radius: $\sqrt{2} pprox 1.414$ — a familiar irrational number suggesting precise geometric balance.
  - All points satisfying $x^2 + y^2 + (z - 1)^2 = 2$ lie on the surface; solving for specific $z$ values gives horizontal circular cross-sections, rotating around the center vertically.