In this Archimedes-trammel arrangement, two points on a rigid beam slide on perpendicular axes and the tracing point lies between them. Its distances to the two shoe pivots become the two ellipse semi-axis magnitudes.
Trammel of Archimedes Interactive Calculator
Enter the tracing point's collinear distances to the vertical-track and horizontal-track shoe pivots. Results describe the ideal ellipse locus only.
Equation Used
- The tracks are straight, intersecting and perpendicular.
- Both shoe pivots and the tracing point are collinear on a rigid beam.
- The tracing point lies between the two shoes, making their separation dV+dH.
- The inputs are positive over the supplied range and are sorted before major/minor naming.
- Clearance, friction, deformation, tool offset, force, speed, tolerance and safe fabrication are omitted.
Tracing Point Between Perpendicular-Slot Shoes
Trammel patents describe two slider blocks moving in crossed perpendicular tracks and pivoted to points on one rigid beam. With a working point fixed to that beam, the constraints generate an elliptical path.
For the arrangement used here, the tracing point divides the shoe-to-shoe beam segment into distances dV and dH. The horizontal-track shoe is at ((dV+dH)cos theta,0), the vertical-track shoe is at (0,(dV+dH)sin theta), and the intermediate tracing point is P=(dV cos theta,dH sin theta).
What This Calculator Does
The tool sorts the two entered point-to-shoe distances to report semi-major and semi-minor axes, then calculates ideal ellipse area and dimensionless eccentricity.
It does not size tracks, shoes, beam, pivots, cutters or a drive.
Ideal Ellipse Locus
P = (dV cos theta,dH sin theta)
a = max(dV,dH)
b = min(dV,dH)
Area = pi ab
e = sqrt(1-b2/a2)
When dV=dH, the ideal locus is a circle and eccentricity is zero; the shoes remain separated by 2dV in this between-shoes arrangement.
Worked Ideal Example
For dV=100 mm and dH=50 mm, the semi-major axis is 100 mm, the semi-minor axis is 50 mm, the ideal area is about 15,708 mm² and eccentricity is about 0.866. The two shoe pivots are 150 mm apart along the beam.
Model Boundary
- Mechanical arrangement: two beam pivots carry sliders in perpendicular crossed tracks and constrain a beam-mounted point to an ellipse.
- Calculated: ideal semi-axis sorting, ellipse area and eccentricity.
- Not calculated: physical track length, slider size, end clearance, retention, backlash, perpendicularity error, beam or pin deformation, tool offset, cutter diameter, friction, drive force, velocity, acceleration, vibration, wear, tolerance, accuracy, material, strength, workpiece restraint or safe operation.
Frequently Asked Questions
Why are the two input distances the semi-axis magnitudes?
The ideal tracing coordinates are dV cos theta and dH sin theta.
Why sort the values?
Semi-major names the larger coordinate amplitude and semi-minor the smaller, regardless of which track is assigned each one.
Can equal distances draw a circle?
Yes in this between-shoes arrangement: the ideal locus is a circle while the two shoes remain separated.
Is ellipse area the material area of a cut part?
It is only the area enclosed by the ideal centreline locus.
Does eccentricity measure manufacturing error?
No. It describes ideal ellipse shape.
Can this define a powered cutting fixture?
No. Hardware geometry, loads, speed, restraint, guarding and applicable safety requirements are outside the model.
References
- U.S. Patent 4,306,598, Ellipse cutting machine: perpendicular crossed tracks, slider blocks, adjustable trammel points and a beam-mounted working tool.
- U.S. Patent 517,522, Ellipsograph: intersecting slots, sliding blocks and beam adjustment distances corresponding to ellipse semi-axes.
- U.S. Patent 2,870,761, Radius and ellipse dresser for grinding wheels: trammel definition using right-angle grooves, two sliding pins and a describing point.
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