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

Understanding the Context

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

The equation $x^2 + (y - 1)^2 + z^2 = 2$ defines a 3-dimensional sphere in the Cartesian coordinate system $(x, y, z)$, centered at $(0, 1, 0)$ with a radius of $\sqrt{2}$. The term $BD^2$ suggests a squared distance—specifically, it represents the squared distance from a point $(x, y, z)$ to the point $B = (0, 1, 0)$, scaled such that $BD^2 = 2$ describes all points at a fixed distance of $\sqrt{2}$ from $B$.

Breaking Down the Components

Key Insights

• Center at $(0, 1, 0)$: The $(y - 1)^2$ term shifts the center along the $y$-axis, positioning it one unit above the origin.
  - Spherical Surface: The standard form of a sphere’s equation is $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2$, matching perfectly here with center $(0, 1, 0)$ and radius $\sqrt{2}$.
  - Equal to 2: Restricting the distance to exactly $\sqrt{2}$ makes this a surface, not an entire volume—an important distinction in 3D modeling and physics.

Visualizing the Sphere

Imagine a sphere centered slightly off the origin, elevated along the $y$-axis to the point $(0, 1, 0)$. From every point precisely $\sqrt{2}$ units away from this center, the equation holds. This surface appears as a symmetrical, smooth balloon centered vertically in the space.

Final Thoughts

Mathematical Derivation: Starting from a Point-to-Point Distance

Let’s derive the equation step-by-step:

Let $P = (x, y, z)$ be any point in 3D space.
The distance $BD$ from $P$ to the fixed point $B = (0, 1, 0)$ is given by the Euclidean distance formula:

\[
BD = \sqrt{(x - 0)^2 + (y - 1)^2 + (z - 0)^2} = \sqrt{x^2 + (y - 1)^2 + z^2}
\]

Squaring both sides gives:

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