- Difficulty
- Advanced
- Build time
- 150-240 min
- Estimated cost
- $0-$28
- Age range
- 14-18
- Workspace
- A clear table about 90 cm wide
The finish line
What you will build
The tracer moves through a clear near-straight segment while the linkage completes its intended range without crossing or binding.
Learning goals
- Identify how guided arc motion at one linkage point produces straight-line tracer motion.
- Construct and explain a oscillating angular-to-linear path system.
- Measure how the radius-link pivot spacing changes performance.
- Diagnose losses caused by eight pivot joints and rhombus distortion.
Before you build
Materials, tools, and safety
Reuse-material cost: Usually under $5 with an existing kit. Supervision: Adult help recommended for sharp or heated tools.
Tools
- Ruler
- Removable tape for motion marks
Low-cost swaps
- Use equivalent brick-compatible parts from any kit.
- Use cardboard beams and straw bearings for a larger demonstration model.
- Use carefully measured cardboard strips with wide paper-fastener pivots for a large geometry demonstration.
Project-specific safety
- Keep fingers, hair, and loose sleeves clear of moving parts.
- Turn the mechanism by hand; do not attach a high-speed motor.
- Move the linkage slowly and keep fingers outside the folding rhombus; never force it through a singular position.
Orient the build
Place the build so guided arc motion at one linkage point is on your left and straight-line tracer motion is on your right. Call the side facing you the front, the far side the back, the tabletop the bottom, and the opposite face the top.
Build it
Step-by-step instructions
Step 1
Prepare matched links
Overlay the four rhombus links and two longer links to verify hole alignment.
Label equal-length groups before assembly.
Step 2
Build the fixed base
Reinforce the central pivot and circular-guide pivot at the chosen spacing.
Keep both axes square to the work plane.
Step 3
Assemble the rhombus
Join four equal links into a free diamond without crossing.
Check all sides match center to center.
Builder checkpoint: After assemble the rhombus, the first subassembly should stay aligned when handled gently.
Step 4
Attach the long links
Connect two equal longer links from the fixed center to opposite rhombus nodes.
Separate layers with equal spacers.
Watch for: If this stage binds or drifts, inspect base flex before adding more parts.
Step 5
Add the radius constraint
Join the input rhombus node to the second fixed pivot with the radius link.
Choose the intended assembly branch.
Step 6
Place the tracer
Mount a blunt point at the output node opposite the constrained input.
Support paper in the same plane.
Builder checkpoint: After place the tracer, operate the build slowly and confirm that straight-line tracer motion begins without binding.
Step 7
Sweep the safe range
Move in small increments and inspect every pivot and layer.
Mark limits before any link nears a straight-line lock.
Step 8
Trace and verify
Draw the output path through the safe range and compare with a ruler.
Repeat after tightening only loose, not free, joints.
Builder checkpoint: At the final checkpoint, The tracer moves through a clear near-straight segment while the linkage completes its intended range without crossing or binding.
See the engineering
Why it works
- Input
- guided arc motion at one linkage point
- Output
- straight-line tracer motion
- Motion
- oscillating angular-to-linear path
- Energy losses
- eight pivot joints, rhombus distortion, link-plane collisions, base flex
Why this works
Geometric inversion
Equal-length diamond links and two longer equal links preserve a constant product of distances from a fixed center. Constraining one point on a circle forces the opposite point onto a line.
Look for: Trace the output on paper and compare it with a ruler while inspecting whether the rhombus remains equal-sided.
Where the energy goes
Efficiency and losses
The ideal model leaves out eight pivot joints, rhombus distortion, link-plane collisions, base flex. These effects turn some input energy into heat, sound, vibration, or unwanted motion, so measured performance will be lower than an ideal calculation.
Look for: Run the build slowly and locate the first place where eight pivot joints becomes visible or audible.
Math bite
Check the inversion constant
Formula: OA × OC = OB² - AB²
- Long link OB = 100 mm
- Rhombus side AB = 60 mm
Substitute: constant = 100² - 60² = 6400 mm²
Result: Distances OA and OC should multiply to about 6400 mm².
That constant relationship underlies the straight-line constraint.
Joint clearance and flexible links disturb exact geometry.
Make it behave
Test, troubleshoot, and tune
Controlled test
Start here: Sweep the unloaded linkage through its marked safe range.
Success looks like: The tracer follows one continuous line-like path and the rhombus stays uncrossed.
Measure: Maximum deviation from a ruler line and inversion-product variation.
Change: the radius-link pivot spacing
Keep constant: all link lengths, fixed center, frame, assembly branch, and tracer
- reference spacing
- slightly shorter spacing
- slightly longer spacing after clearance check
| Symptom | Likely cause | Confirm it | Fix |
|---|---|---|---|
| The tracer draws an arc | A length equality or ground constraint is wrong | Overlay equal link groups and inspect radius pivot | Correct lengths and fixed-pivot placement |
| The rhombus collapses or crosses | The linkage entered the wrong branch | Return slowly to the start shape | Add safe range stops and follow the assembly map |
| Pivots bind | Layer spacing is unequal or pins are tight | Disconnect one joint at a time | Add equal spacers and running clearance |
| The path changes each pass | Ground pivots or tracer paper move | Mark base position and repeat | Clamp the base and tape paper firmly |
Choose your tradeoff
Accuracy depends on matched center distances and rigid ground pivots. Reduce play only after the mechanism moves freely; tight joints can fake a straight segment by binding.
Keep experimenting
Try another version
Paper geometry study
Assemble the linkage flat and move it without tracing.
Deviation measurement
Record output coordinates every 10 mm of travel.
Inversion check
Measure OA and OC at several positions and compare their product.
Build together
Classroom and access options
Classroom version
Teams can compare the radius-link pivot spacing while keeping all link lengths, fixed center, frame, assembly branch, and tracer. Assign builder, tester, recorder, and explainer roles; have each team predict the result before collecting three trials.
Access adaptations
- Use high-contrast tape to distinguish input and output parts.
- Replace a small crank with a wider handle for an easier grip.
- Number every link and provide a full-size assembly layout with tactile pivot dots.
Reflect on the design
- How did the radius-link pivot spacing change the measured result?
- Where did eight pivot joints affect the build most strongly?
- What evidence shows that geometric inversion explains the motion?
- Which change would improve straight-line tracer motion without creating a new problem?
Glossary
- Geometric inversion
- Equal-length diamond links and two longer equal links preserve a constant product of distances from a fixed center.
- Input
- The action or energy supplied to a system; here it is guided arc motion at one linkage point.
- Output
- The useful response produced by a system; here it is straight-line tracer motion.
- Efficiency
- The fraction of input energy that becomes useful output instead of friction, sound, heat, or unwanted motion.
Build your dreams
One build can start the next.
Share what you learned, change one variable, and help another builder understand what worked.
Explore more guidesSources and build notes
An original BrickLabClips interpretation of a standard mechanical mechanism.
- Mechanism verification: Standard kinematics were checked for motion direction, constraint, clearance, and likely friction points.
Written and edited by BrickLabClips. Published 2026-07-22; updated 2026-07-22.