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

Understanding the Equation $CD^2 = x^2 + y^2 + (z - 1)^2 = 2$ in Geometry and Applications

The equation $CD^2 = x^2 + y^2 + (z - 1)^2 = 2$ represents a key geometric concept in three-dimensional space. While $CD^2$ widely appears in distance and squared-distance notations, in this specific form it defines a precise geometric object: a sphere in ℝ³.

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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: $r = \sqrt{2}$

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

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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.

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.

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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.

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Mathematical Exploration: Parametric Representation

Parameterized form using spherical coordinates centered at $(0, 0, 1)$ offers deeper insight:

Let $\ heta$ be the azimuthal angle in the $xy$-plane, and $\phi$ the polar angle from the positive $z$-axis.

Then any point on the sphere can be written as:

$$x = \sqrt{2} \sin\phi \cos\ heta <br/>y = \sqrt{2} \sin\phi \sin\ heta <br/>z = 1 + \sqrt{2} \cos\phi$$

where $0 \leq \ heta < 2\pi$, $0 \leq \phi \leq \pi$.

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Summary

The equation$$x^2 + y^2 + (z - 1)^2 = 2$$is a compact and elegant representation of a sphere with center at $(0, 0, 1)$ and radius $\sqrt{2}$. It exemplifies how algebraic expressions encode geometric reality, serving fundamental roles in geometric modeling, distance calculations, and computational applications across science and technology.

Understanding such equations enhances depth in fields like computer graphics, mathematical modeling, and optimization.

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Keywords: $CD^2 = x^2 + y^2 + (z - 1)^2 = 2$, sphere equation, 3D geometry, distance formula, parametric representation, radius, center, geometric modeling, machine learning, optimization.