Every CAD and CAM system computes geometry using one of two approaches: analytic computation or numerical solving. The difference is fundamental — and for CNC machining, it is the difference between exact results and approximations.
Numerical Solvers: How They Work
A numerical solver finds a solution by iteration. Starting from an initial guess, it applies corrections repeatedly until the result is close enough to the target — within a specified tolerance.
This approach is powerful and general. It can handle complex constraint systems that have no closed-form solution. But it has three fundamental limitations:
- Tolerance accumulation — each iteration introduces a small error. Over many operations, these errors accumulate.
- Starting state dependency — the solver converges to the nearest valid solution from the starting state. Different starting states produce different solutions.
- Convergence failure — in some configurations, the solver fails to converge at all, producing an error or an incorrect result.
Analytic Computation: How It Works
An analytic computation finds a solution by applying a fixed mathematical formula directly. The result is computed in a single pass — no iteration, no convergence, no tolerance.
For example, the intersection of a line ax + by + c = 0 and a circle (x-cx)² + (y-cy)² = r² is computed by substituting the line equation into the circle equation, producing a quadratic with two exact roots. The direction selector picks which root to use.
The result is exact. Not approximate. Not within tolerance. Exact.
A Concrete Comparison
| Property | Numerical Solver | Analytic Computation |
|---|---|---|
| Result type | Approximate (within tolerance) | Exact |
| Computation method | Iterative convergence | Single-pass formula |
| Starting state dependency | Yes — different states, different results | No — always the same result |
| Tolerance accumulation | Yes — errors compound over operations | No — each result is exact |
| Ambiguity handling | Implicit — solver picks a solution | Explicit — direction selector required |
| Failure mode | Silent wrong answer or convergence failure | Explicit error — no silent wrong answer |
| Reproducibility | Depends on solver state and version | Always identical |
When Numerical Solvers Are Appropriate
Numerical solvers are appropriate for problems that have no closed-form solution — complex constraint systems with many interdependent parameters, freeform surface fitting, finite element analysis.
For these problems, an approximate solution is the only option. The tolerance is managed carefully, and the results are verified.
When Analytic Computation Is Required
For the geometry operations used in CNC machining programs — intersections, tangencies, parallel offsets, fillet arcs — closed-form analytic solutions exist. Every one of these operations can be computed exactly.
Using a numerical solver for these operations introduces unnecessary approximation, starting state dependency, and the risk of silent wrong answers. There is no benefit — only risk.
The XyzPilot Position
XyzPilot uses analytic computation for all 65 geometry operators. Every result is exact. Every operation is reproducible. The same inputs always produce the same outputs — on any machine, any version, any time.
This is not a technical limitation. It is a deliberate design choice. For CNC machining programs, exact geometry is not a nice-to-have. It is a requirement.
"The geometry remains entirely inside XyzPilot. The agent becomes a translator. The mathematics never changes."