Sphere Focus Calculator
Example Data Table
| Case | Input | Center | Radius | Classical Foci | Cap Approximation |
|---|---|---|---|---|---|
| Unit sphere | x² + y² + z² = 1 | (0, 0, 0) | 1 | None | 0.5 from cap vertex |
| Standard form | (x - 2)² + (y + 3)² + (z - 4)² = 25 | (2, -3, 4) | 5 | None | 2.5 from cap vertex |
| General form | x² + y² + z² - 4x + 6y - 8z - 11 = 0 | (2, -3, 4) | 6.3249 | None | 3.1625 from cap vertex |
Formula Used
A sphere in standard form is written as
(x - h)² + (y - k)² + (z - l)² = r².
Its center is (h, k, l). Its radius is r.
A general sphere equation can be written as
A(x² + y² + z²) + Dx + Ey + Fz + G = 0.
The center is
(-D / 2A, -E / 2A, -F / 2A).
The radius squared is
(D² + E² + F²) / 4A² - G / A.
A real sphere needs a positive radius squared.
A perfect sphere has no unique foci in ordinary three dimensional geometry.
This calculator also shows the paraxial spherical cap estimate:
f = R / 2.
That value applies to a small spherical mirror cap, not to the whole sphere.
How to Use This Calculator
- Select the input method.
- Enter center and radius values, or enter general equation coefficients.
- Choose a unit label and decimal precision.
- Select a reference axis for diameter endpoints.
- Select the spherical cap side for the optional mirror estimate.
- Click calculate.
- Read the result above the form.
- Use the CSV or PDF buttons to save the output.
Article: Understanding Sphere Foci
A Sphere Is Not an Ellipse
A sphere is the set of all points at one fixed distance from one center. That fixed distance is the radius. This shape is highly symmetric. Every direction from the center behaves the same way. Because of this symmetry, a sphere does not need two special points to define it. An ellipse needs foci. A sphere does not.
Why the Focus Question Appears
Many learners meet foci while studying ellipses, hyperbolas, and parabolas. These curves are conic sections. Their foci help define distances and reflections. A sphere is a quadric surface, but it is not defined by focal distance rules. Its defining rule is simpler. Every surface point stays the same distance from the center.
What This Tool Calculates
This calculator confirms the sphere equation. It finds the center, radius, diameter, surface area, and volume. It also states the correct focus result. The classical focus count is zero. That result may feel unusual, but it is the correct geometric answer for a complete perfect sphere.
General Equation Support
The tool also accepts a general equation. It completes the square through coefficient formulas. This is useful when the sphere is not written in standard form. The calculator checks if the equation creates a real sphere. If the radius squared is not positive, the input cannot describe a normal real sphere.
Spherical Cap Approximation
Some optical discussions use a spherical mirror. A small spherical mirror cap has an approximate paraxial focal length equal to half the radius. This is not the same as a true sphere focus. It is only an optical approximation near one selected cap. The calculator labels this clearly to prevent confusion.
Practical Use
Use this page for geometry checks, study notes, and report preparation. It gives clean values and export options. It is also helpful when explaining why sphere foci are not part of standard geometry.
FAQs
1. Does a sphere have foci?
No. A perfect sphere has no classical foci. It is defined by one center and one radius.
2. Why do ellipses have foci but spheres do not?
Ellipses use distance relationships from two fixed points. Spheres use equal distance from one center in every direction.
3. What does this calculator return as sphere foci?
It returns no classical foci and shows a focus count of zero. It also gives useful sphere measurements.
4. What is the spherical cap focus value?
It is an optical approximation for a small spherical mirror cap. The estimated focal length is radius divided by two.
5. Is the cap focus a real sphere focus?
No. It is only a paraxial mirror approximation. It should not be treated as a true focus of a full sphere.
6. Can I use a general sphere equation?
Yes. Enter the shared quadratic coefficient, linear coefficients, and constant term. The tool finds center and radius.
7. What happens if the equation is invalid?
The calculator shows an error. A valid real sphere needs a nonzero quadratic coefficient and positive radius squared.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple printable report.