Chapter map

1. 1. Function
2. 2. Inside
3. 3. Outside
4. 4. Normal
5. 5. Curvature
6. 6. Speed
7. 7. Time
8. 8. Updated sign
9. 9. Grid contour
10. 10. Upwinding
11. 11. Reinitialise
12. 12. Volume
13. 13. Refine
14. 14. Conclusion

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

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1. Function

The useful question is not only how to calculate, but what the calculation assumes. Set phi=sqrt(x^2+y^2)-1.

A concrete situation

Zero contour is unit circle.

What the mathematics does not license

Declare sign. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record phi. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. Inside

A small example makes the mechanism visible. Evaluate at origin conceptually.

A concrete situation

phi=-1.

What the mathematics does not license

Gradient singular there but interface is away. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record sign. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Outside

This is where modelling choices begin to affect evidence. Evaluate at (1.05,0).

A concrete situation

phi=.05.

What the mathematics does not license

Point is outside initially. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record value. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Normal

The central mathematical idea is concrete. At (1,0), compute grad phi.

A concrete situation

Normal=(1,0).

What the mathematics does not license

Magnitude one. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record normal. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Curvature

Here the geometry becomes operational. Compute divergence of normal on circle.

A concrete situation

Curvature=1.

What the mathematics does not license

Sign convention matters. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record value. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. Speed

This step deserves a careful pause. Set outward F=.2.

A concrete situation

Radius rate=.2.

What the mathematics does not license

Constant normal speed. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record law. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Time

The useful question is not only how to calculate, but what the calculation assumes. Set t=.5.

A concrete situation

Radius becomes 1.1.

What the mathematics does not license

Exact benchmark. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record radius. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Updated sign

A small example makes the mechanism visible. Re-evaluate (1.05,0).

A concrete situation

phi=-.05.

What the mathematics does not license

Point is now inside. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record value. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Grid contour

This is where modelling choices begin to affect evidence. Sample phi on nodes.

A concrete situation

Sign changes bracket interface.

What the mathematics does not license

Interpolate crossings. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record contour. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Upwinding

The central mathematical idea is concrete. Choose derivatives by speed direction.

A concrete situation

Controls Hamilton-Jacobi stability.

What the mathematics does not license

Avoid blind centring. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record stencil. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Reinitialise

Here the geometry becomes operational. Restore distance-like gradient.

A concrete situation

Zero contour should barely move.

What the mathematics does not license

Measure displacement. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Record change. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Volume

This step deserves a careful pause. Calculate enclosed area.

A concrete situation

Exact pi R squared.

What the mathematics does not license

Track numerical loss. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record error. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Refine

The useful question is not only how to calculate, but what the calculation assumes. Halve grid and time step.

A concrete situation

Compare position and curvature.

What the mathematics does not license

Separate errors. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record table. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Conclusion

A small example makes the mechanism visible. State sign,speed,scheme,reinitialisation and checks.

A concrete situation

Worked radius at .5 is 1.1.

What the mathematics does not license

Topology ability is not accuracy proof. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Write conclusion. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

The level set method represents a moving interface as the zero contour of a function defined on a surrounding grid. The interface can split, merge or change topology without explicit reconnection rules.

For a circle of radius one centred at the origin, choose phi(x,y)=sqrt(x squared+y squared)-1. Points outside have positive phi, points inside have negative phi and the interface has phi=0.

Away from the centre, the gradient of this signed-distance function is (x/r,y/r) and has magnitude one. On the unit circle it is the outward normal.

The divergence of the normal gives curvature 1/r, so the unit circle has curvature one under this sign convention. A different sign convention reverses signs and must be declared.

For constant outward normal speed F=0.2, the level set equation phi_t+F magnitude grad phi=0 moves the radius from 1 to 1+0.2t. At t=0.5 the exact radius is 1.1.

The point (1.05,0) begins outside with phi=0.05. After half a time unit it lies inside the expanded circle with updated signed distance 1.05-1.1=-0.05.

Did You Know? The grid function is not the physical front. Only its zero contour is the interface, while nearby values carry normals, curvature and transport information.

Numerical advection can make magnitude grad phi drift away from one. Reinitialisation restores a distance-like function, but careless reinitialisation can move the zero contour and lose mass.

Hamilton-Jacobi upwinding chooses derivatives according to propagation direction. Centred differences can create oscillations or instability near sharp gradient structures.

Narrow-band methods update only a neighbourhood of the interface to reduce work. The band must remain wide enough for stencils and front motion before rebuilding.

Curvature is a second-derivative quantity and amplifies grid noise. Smoothing may help numerically while also changing the intended geometric motion.

Students can evaluate phi on a small grid, mark sign changes, interpolate the zero crossing and compare with the exact circle. This separates implicit representation from visual drawing.

Applications include flames, image segmentation, free surfaces, crystal growth and shape optimisation. Conservation-sensitive flows often add volume corrections or couple another interface method.

Grid convergence should track interface position, normals, curvature and conserved volume separately. One small positional error does not guarantee accurate curvature.

For parents and students, the transferable idea is to describe a complicated boundary through a simple rule whose sign tells where a point lies and whose zero set carries the edge.

A responsible report states the level-set sign, initial phi, speed law, evolution equation, spatial scheme, time integrator, reinitialisation, narrow band, curvature treatment, convergence and validation.

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Frequently asked questions

What is a level set?

A scalar grid function whose chosen contour, usually zero, represents an interface.

Why use a signed distance function?

Its magnitude measures distance near the interface and its gradient supplies a normal.

How does the interface move?

A Hamilton–Jacobi equation evolves the level-set function from a speed law.

What is reinitialisation?

A process that restores a distance-like profile while aiming to keep the zero contour fixed.

Does automatic topology change guarantee conservation?

No. Volume or mass can drift and needs measurement or correction.

What must be reported?

Sign convention, initial function, speed, scheme, reinitialisation, curvature, convergence, conservation and validation.

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Primary source and fact-check note

Stanley Osher and James A. Sethian, “Fronts Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton–Jacobi Formulations,” Journal of Computational Physics 79, 1988. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• Finite difference method
• Adaptive mesh refinement
• Computational mathematics hub
• How Mathematics Works
• Study and learning methods

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Method audit: representation, transfer and evidence

Sign audit

Declare inside, outside, normal and curvature conventions.

Contour audit

Measure zero-set motion during advection and reinitialisation.

Derivative audit

Use eligible upwinding and curvature stencils.

Conservation audit

Track enclosed volume as well as interface position.

Mechanism workshop: calculate, challenge and transfer

Sign map

Evaluate phi inside, outside and on a circle.

Normal map

Calculate gradients at four compass points.

Curvature map

Compare exact and discrete circle curvature.

Advection

Advance constant-speed radius analytically and numerically.

Reinitialise

Measure zero-contour displacement.

Topology test

Merge two circles through evolving zero sets.

Volume test

Track area before and after motion.

Narrow band

Count updated nodes and verify stencil support.

Student explanation

Explain an invisible function carrying a visible edge.

Parent explanation

Connect sign rules with moving boundaries.

Evidence notebook: reconstruct every step

Evidence notebook 1: Function

Step 1 isolates function. Set phi=sqrt(x^2+y^2)-1. The checked result is: Zero contour is unit circle. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Declare sign. Record phi. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 2: Inside

Step 2 isolates inside. Evaluate at origin conceptually. The checked result is: phi=-1. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Gradient singular there but interface is away. Record sign. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 3: Outside

Step 3 isolates outside. Evaluate at (1.05,0). The checked result is: phi=.05. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Point is outside initially. Record value. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 4: Normal

Step 4 isolates normal. At (1,0), compute grad phi. The checked result is: Normal=(1,0). Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Magnitude one. Record normal. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 5: Curvature

Step 5 isolates curvature. Compute divergence of normal on circle. The checked result is: Curvature=1. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Sign convention matters. Record value. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 6: Speed

Step 6 isolates speed. Set outward F=.2. The checked result is: Radius rate=.2. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Constant normal speed. Record law. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 7: Time

Step 7 isolates time. Set t=.5. The checked result is: Radius becomes 1.1. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Exact benchmark. Record radius. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 8: Updated sign

Step 8 isolates updated sign. Re-evaluate (1.05,0). The checked result is: phi=-.05. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Point is now inside. Record value. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 9: Grid contour

Step 9 isolates grid contour. Sample phi on nodes. The checked result is: Sign changes bracket interface. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Interpolate crossings. Record contour. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 10: Upwinding

Step 10 isolates upwinding. Choose derivatives by speed direction. The checked result is: Controls Hamilton-Jacobi stability. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Avoid blind centring. Record stencil. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 11: Reinitialise

Step 11 isolates reinitialise. Restore distance-like gradient. The checked result is: Zero contour should barely move. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Measure displacement. Record change. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 12: Volume

Step 12 isolates volume. Calculate enclosed area. The checked result is: Exact pi R squared. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Track numerical loss. Record error. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 13: Refine

Step 13 isolates refine. Halve grid and time step. The checked result is: Compare position and curvature. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Separate errors. Record table. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 14: Conclusion

Step 14 isolates conclusion. State sign,speed,scheme,reinitialisation and checks. The checked result is: Worked radius at .5 is 1.1. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Topology ability is not accuracy proof. Write conclusion. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Failure cases and responsible recovery

Zero contour is whole field

Only one level represents the interface. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Sign unstated

Normals and curvature become ambiguous. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Centred advection

Hamilton-Jacobi stability may fail. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Reinitialisation unmeasured

The interface can move. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Topology equals conservation

Mass can drift. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Curvature from coarse grid

Second derivatives amplify error. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Narrow band too thin

Stencils or moving fronts escape. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Pretty contour proof

Position, normal and volume errors need numbers. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Verified case study: fourteen steps from model to decision

1. Define the model

Write the PDE or kinetic rule, variables, units, domain and boundary or initial data.

2. Identify the representation

Name boundary densities, lattice populations, a level-set field or Lagrangian markers.

3. State the transfer rule

Write the kernel, collision-streaming map, Hamilton-Jacobi speed or delta coupling.

4. Calculate a tiny case

Use the smallest nontrivial example with labelled indices.

5. Check a conserved quantity

Verify source weight, mass, volume, force or work as eligible.

6. Audit signs and geometry

Check normals, directions, level-set sign, marker coordinates and Jacobians.

7. Test a singular or sharp feature

Approach a kernel source, interface, wall or concentrated force.

8. Refine

Improve panels, lattice, interface grid and marker spacing consistently.

9. Separate errors

Distinguish modelling, discretisation, quadrature, iteration and rounding.

10. Compare a baseline

Use an exact, manufactured, direct or independently implemented reference.

11. Count cost

Record unknowns, nonzeros or dense entries, cell updates, memory and elapsed time.

12. Validate

Compare application outputs with independent measurements or accepted benchmarks.

13. Communicate limits

State what the result supports and what remains untested.

14. Preserve evidence

Save equations, data, code, meshes, parameters, logs and corrected arithmetic.

Learner lab: ten checks for transferable understanding

Lab question 1

Create a reproducible mini-investigation of signed distance, normals, curvature, upwinding, reinitialisation, topology and volume. For task 1, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 2

Create a reproducible mini-investigation of signed distance, normals, curvature, upwinding, reinitialisation, topology and volume. For task 2, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 3

Create a reproducible mini-investigation of signed distance, normals, curvature, upwinding, reinitialisation, topology and volume. For task 3, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 4

Create a reproducible mini-investigation of signed distance, normals, curvature, upwinding, reinitialisation, topology and volume. For task 4, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 5

Create a reproducible mini-investigation of signed distance, normals, curvature, upwinding, reinitialisation, topology and volume. For task 5, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 6

Create a reproducible mini-investigation of signed distance, normals, curvature, upwinding, reinitialisation, topology and volume. For task 6, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 7

Create a reproducible mini-investigation of signed distance, normals, curvature, upwinding, reinitialisation, topology and volume. For task 7, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 8

Create a reproducible mini-investigation of signed distance, normals, curvature, upwinding, reinitialisation, topology and volume. For task 8, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 9

Create a reproducible mini-investigation of signed distance, normals, curvature, upwinding, reinitialisation, topology and volume. For task 9, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 10

Create a reproducible mini-investigation of signed distance, normals, curvature, upwinding, reinitialisation, topology and volume. For task 10, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.