Andromeda
Note

Equations for Spheres

Definition

A sphere is the 3D surface consisting of all points in space that are a fixed distance aa (the radius) from a fixed point (x0,y0,z0)(x_0, y_0, z_0) (the center). It is the 3D analog of a circle.

Why It Matters

Spheres are the 3D signature of symmetry and invariant distance, serving as the essential “bounding box” for everything from collision detection in graphics to Gauss’s Law in physics. Mastering their equations is the prerequisite for navigating a curved planet and calculating the geometric clearances required in structural and mechanical engineering.

Core Concepts

  • Sphere at Origin x2+y2+z2=r2x^2 + y^2 + z^2 = r^2

    • How to read: “The sum x squared plus y squared plus z squared equals r squared.”
    • Meaning: All points whose distance from the origin is exactly r—the 3D circle generalized.

  • Standard Equation (x−x0)2+(y−y0)2+(z−z0)2=a2(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 = a^2

    • How to read: “The quantity x minus x zero squared, plus the quantity y minus y zero squared, plus the quantity z minus z zero squared, equals a squared.”
    • Meaning: Sphere centered at (x0,y0,z0)(x_0, y_0, z_0) with radius aa—shifted version of the origin form.

  • General Equation x2+y2+z2+Gx+Hy+Iz+J=0x^2 + y^2 + z^2 + Gx + Hy + Iz + J = 0

    • How to read: “The sum x squared plus y squared plus z squared plus G x plus H y plus I z plus J equals zero.”
    • Meaning: Expanded quadratic form—complete the square to recover center and radius.

  • Completing the Square: The technique used to convert a general quadratic equation of x,y,zx, y, z into the standard form to identify the center and radius.

  • Interior and Exterior:

    • Interior (ball): (x−x0)2+(y−y0)2+(z−z0)2<a2(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 < a^2.
      • How to read: “The sum of the quantities x minus x zero squared, y minus y zero squared, and z minus z zero squared is less than a squared.”
      • Meaning: Points strictly inside the sphere—filled ball, not just the surface.
    • Exterior: (x−x0)2+(y−y0)2+(z−z0)2>a2(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 > a^2.
      • How to read: “The sum of the quantities x minus x zero squared, y minus y zero squared, and z minus z zero squared is greater than a squared.”
      • Meaning: Points outside the sphere—all locations farther than radius aa from center.

Connected Concepts

Connected notes