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.

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: $r = \sqrt{2}$

Solved 1. y=x^2 - z2 2. x^2 - y^2 + z^2 = 1 3. y = 2x^2 + | Chegg.com

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Solved 1. y=x^2 - z2 2. x^2 - y^2 + z^2 = 1 3. y = 2x^2 + | Chegg.com
Solved x2+y2=1 x2+y2+z2=1 z=x+y z=x2−y2 z=x2 z2=x2+y2 | Chegg.com
Solved z^2 = x^2 + y^2 _____ z^2 = -1 + x^2 + y^2 _____ | Chegg.com
Solved 1. (x−1)2+(y−1)2+z2=1 2. x2+y2+z2=4 B 3. | Chegg.com
Solved | Chegg.com

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.

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📰 We can compute this by summing over all possible $7$-element subsets of the $10$ weights, then counting the number of permutations of those $7$ weights where the maximum is in the last position (feature $7$). 📰 Question: An epidemiologist is modeling disease spread using a population of $12$ individuals, where $5$ are initially infected and $7$ are susceptible. If the infection spreads only through random pairing—each day, one infected person transmits to one susceptible person—what is the probability that, after all possible transmissions until no new infections are possible, exactly $9$ individuals have been infected by the end? 📰 We begin with $5$ infected and $7$ susceptible individuals. The transmission rule: each day, an infected person infects one susceptible (via a random pairing, assume all infection pairs are equally likely over time, but the process stops when no new infections can occur). However, since only $5$

Final Thoughts

  1. 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.

  2. 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.

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: