Calculator Inputs
The page stays single-column, while inputs adapt to 3, 2, or 1 columns by screen size.
Formula Used
A level surface is the set of all points where a scalar field reaches a constant value.
F(x, y, z) = k
Included models
Sphere: F(x,y,z) = x² + y² + z²
Ellipsoid: F(x,y,z) = x²/a² + y²/b² + z²/c²
Plane: F(x,y,z) = Ax + By + Cz
Paraboloid: F(x,y,z) = x²/a² + y²/b² - z/h
Hyperboloid family: F(x,y,z) = x²/a² + y²/b² - z²/c²
Gradient and tangent plane
∇F(x₀,y₀,z₀) = (∂F/∂x, ∂F/∂y, ∂F/∂z)
∇F(x₀,y₀,z₀) · (x - x₀, y - y₀, z - z₀) = 0
How to Use This Calculator
- Select a surface family that matches your scalar field.
- Enter the level value k and the needed parameters.
- Add an optional test point to check whether it lies on the surface.
- Press the calculate button to generate classification, measures, tangent data, and graph.
- Use the export buttons to save a CSV summary or PDF report.
Example Data Table
| Model | Inputs | Resulting equation | Main output |
|---|---|---|---|
| Sphere | k = 25 | x² + y² + z² = 25 | Sphere with radius 5 |
| Ellipsoid | a = 3, b = 2, c = 1.5, k = 1 | x²/9 + y²/4 + z²/2.25 = 1 | Closed ellipsoid |
| Plane | A = 2, B = -1, C = 3, k = 6 | 2x - y + 3z = 6 | Plane, distance from origin = 1.6036 |
| Paraboloid | a = 2, b = 3, h = 4, k = 1 | x²/4 + y²/9 - z/4 = 1 | Elliptic paraboloid, vertex at z = -4 |
| Hyperboloid family | a = 2, b = 2, c = 3, k = -1 | x²/4 + y²/4 - z²/9 = -1 | Hyperboloid of two sheets |
FAQs
1. What is a level surface?
A level surface is the collection of all points where a scalar function F(x,y,z) equals the same constant k. It is the three-dimensional version of a contour line.
2. Why do some inputs return no real surface?
Some scalar fields never reach negative values. For example, x² + y² + z² cannot equal a negative number, so the corresponding level surface does not exist in real space.
3. What does the level value k control?
The value k shifts or scales the surface inside the same family. Changing k can enlarge a sphere, move a paraboloid vertex, or switch a hyperboloid family between different geometric types.
4. Why does the hyperboloid family become a cone at k = 0?
When k equals zero, the standard equation x²/a² + y²/b² - z²/c² = 0 rearranges into a double cone. That transition marks the boundary between one-sheet and two-sheet cases.
5. What is the optional test point used for?
The test point lets you evaluate F(x₀,y₀,z₀), compare it with k, and check whether the point lies on the level surface. It also supports the tangent plane calculation.
6. Is the graph exact for every possible case?
The graph is a high-quality visual model built from the selected family and parameters. It is accurate for interpretation, though resolution and display scaling may simplify extreme cases.
7. What can the export buttons save?
The CSV export saves a clean field-value table. The PDF export creates a compact report containing the equation, classification, gradient data, and key calculated measures.
8. When is the tangent plane unavailable?
A tangent plane needs a point on the surface and a nonzero gradient there. If the point is off the surface or the gradient vanishes, a unique tangent plane is not determined.