Choose a surface type, enter values, then submit. Fields update automatically for your selection.
Try these sample inputs to see different surfaces and forms.
| Surface | Inputs | Expected equation |
|---|---|---|
| Plane from 3 points | (0,0,0), (1,0,0), (0,1,0) | z = 0 |
| Plane from point + normal | Point (0,0,2), normal (0,0,1) | z - 2 = 0 |
| Sphere from center + radius | Center (1,-1,0), r = 3 | (x-1)² + (y+1)² + z² = 9 |
| Ellipsoid | Center (0,0,0), a=3, b=2, c=4 | x²/9 + y²/4 + z²/16 = 1 |
| Elliptic cylinder | Center (0,0), a=2, b=1 | x²/4 + y² = 1 |
Plane from three points: Use direction vectors u = P₂ − P₁ and v = P₃ − P₁. The normal is n = u × v. Then n · (r − r₀) = 0, which expands to ax + by + cz + d = 0.
Plane from point and normal: With normal n = ⟨a,b,c⟩ and point r₀ = ⟨x₀,y₀,z₀⟩, the plane is a(x − x₀) + b(y − y₀) + c(z − z₀) = 0.
Sphere from four points: Subtract squared-distance equations to eliminate r² and solve a 3×3 linear system for the center. Radius is distance to any point.
Quadrics: Standard parameter forms define ellipsoids, cylinders, cones, paraboloids, and hyperboloids using shifted coordinates (x−h, y−k, z−l) and positive scales a, b, c.
- Select a surface type from the dropdown.
- Enter the required coordinates or parameters shown in the form.
- Press Calculate to generate the equation and checks.
- Use the export buttons to save a CSV table or PDF report.
- If you get a degeneracy note, adjust inputs and retry.
What the calculator produces
This tool outputs a clean surface equation from geometric inputs. It reports canonical form, optional expanded form, and diagnostic notes such as degeneracy. For planes it also reports the normal magnitude and distance from the origin, supporting fast verification during modeling. Typical use cases include fitting analytic surfaces to CAD references, validating homework results, and generating constraints for solvers. Because the output is symbolic, you can copy equations directly into graphing tools or simulation scripts. The notes section flags nonphysical parameter choices, such as nonpositive scales, and helps you correct inputs before exporting. for consistent, audit-ready documentation across teams.
Plane construction from measured points
When you enter three points, the calculator forms vectors u = P2 − P1 and v = P3 − P1, then computes n = u × v. The standard plane is ax + by + cz + d = 0 with d = −n·P1. If |n| is near zero, the points are nearly collinear.
Point–normal planes for constraints
Engineering constraints often specify a location plus a direction. With point r0 and normal n, the calculator builds a(x − x0) + b(y − y0) + c(z − z0) = 0 and converts it to standard form. This is useful for collision planes, clipping, and optimization constraints.
Sphere estimation and quality checks
For a sphere with known center and radius, it reports (x − h)² + (y − k)² + (z − l)² = r² and an expanded polynomial. For four points, it solves a 3×3 linear system to recover the center, then computes r as the distance to any point. Coplanar or noisy points can reduce stability.
Quadric families and parameter meaning
Ellipsoids, cylinders, cones, paraboloids, and hyperboloids are expressed using shifted coordinates and positive scales a, b, c. For example, an ellipsoid uses x²/a² + y²/b² + z²/c² = 1 after shifting by (h,k,l). Increasing a stretches along x, while changing signs distinguishes one‑sheet versus two‑sheet hyperboloids.
Visualization, export, and reporting
The interactive Plotly view samples the surface on a grid and renders a 3D plot. Planes show a patch plus reference points; spheres and quadrics show parametric meshes. Use CSV for spreadsheets and PDF for sharing results. Keep units consistent and prefer moderate magnitudes to avoid numeric overflow.
1) What is a surface equation in 3D?
It is a relation among x, y, and z that defines a set of points forming a geometric surface, such as a plane, sphere, or quadric.
2) Why does the plane from three points sometimes fail?
If the points are collinear or nearly collinear, the cross product normal becomes close to zero, so no unique plane exists. Adjust points or improve measurement precision.
3) How is the sphere from four points computed?
The calculator subtracts squared-distance equations to eliminate r², solves a 3×3 linear system for the center, then finds the radius using the distance to any input point.
4) What do a, b, and c represent for quadrics?
They are scale parameters that set curvature or stretching along axes. Larger values widen the surface along the corresponding direction after applying the center shift (h,k,l).
5) Why are positive scales required in several modes?
Positive scales keep denominators meaningful and preserve standard geometric interpretations. Nonpositive values can flip signs, create invalid divisions, or represent a different surface family.
6) How should I choose plotting ranges?
Use ranges centered on your input points or center parameters. If the graph looks flat or clipped, expand ranges modestly and avoid extremely large magnitudes that can amplify rounding.