Injectivity Radius Calculator

Analyze geodesic limits with curvature, diameter, and loop inputs. See bounds, warnings, and derived metrics. Save clean outputs for classes, proofs, reviews, and projects.

Calculator Inputs

Use consistent units across all distance-like inputs. The tool provides a practical estimate, not a substitute for a formal proof.

Example Data Table

Scenario Upper Curvature Conjugate Radius Closed Geodesic Diameter Bound Convexity Radius Estimated Result
Compact manifold sample 1.00 3.14 8.40 5.10 1.85 3.141593
Lower curvature stress test 0.50 4.40 6.00 2.90 1.10 2.900000
Loop-constrained region 2.20 2.80 3.20 6.20 0.95 1.600000

Formula Used

The calculator uses a conservative geometric estimate:

Injectivity Radius Estimate = min(conjugate radius, half shortest closed geodesic, diameter cap, curvature cap)

For positive upper sectional curvature, the curvature cap is approximated by π / √K. The displayed scaled result then equals:

Scaled Radius = Estimate × Safety Factor × Metric Scale

This structure mirrors the common geometric idea that injectivity radius is restricted by conjugate points, short loops, and ambient distance limits.

How to Use This Calculator

  1. Enter an upper sectional curvature bound for your manifold or local model.
  2. Add a conjugate radius estimate from theory, simulation, or prior derivation.
  3. Provide the shortest closed geodesic length when a loop estimate is known.
  4. Enter diameter, convexity, and base-point limits using consistent units.
  5. Apply a safety factor below one for conservative teaching or engineering-style reporting.
  6. Submit the form and review the limiting source, interpretation, and export buttons.

Why These Inputs Matter

The injectivity radius describes how far the exponential map remains locally one-to-one around a point. In practical estimation, several quantities can become the active restriction. Conjugate radius captures where nearby geodesics begin to lose uniqueness due to focal behavior. Short closed geodesics indicate loop formation, and half their length often acts as a natural cap. Diameter and base-point bounds prevent unrealistic local estimates when the ambient region is small. Positive sectional curvature may impose another finite restriction through a curvature-based bound. This calculator collects these ingredients into one conservative estimate, then applies scaling and safety adjustments for teaching, numerical experimentation, and quick comparison across manifold scenarios.

FAQs

1. What does injectivity radius measure?

It measures the largest local distance where geodesics from a point remain unique and the exponential map stays one-to-one.

2. Is this tool a formal proof engine?

No. It is an estimation and teaching aid. Formal results still require rigorous hypotheses, manifold-specific arguments, and proof verification.

3. Why is half the shortest closed geodesic used?

A short closed loop can force geodesic non-uniqueness. Halving that loop length provides a common conservative cap in many geometric settings.

4. What happens when upper curvature is nonpositive?

The calculator removes the finite curvature cap, because the simple positive-curvature bound π divided by √K no longer applies.

5. Why include a safety factor?

It helps produce deliberately conservative outputs for reports, classroom examples, simulations, and sensitivity checks.

6. Can I mix different distance units?

No. Keep all radius, diameter, and geodesic length inputs in the same unit system to avoid distorted results.

7. What does the limiting source tell me?

It shows which geometric restriction produced the smallest cap, helping you identify the dominant reason the estimate stays small.