This page now separates the exact point-inversion identity from the much richer Sylvester-Kempe linkage that can embody it. The connected diagram shows the declared circle, collinear inverse points and straight output locus; it is a geometry worksheet, not a four-bar linkage synthesis or a dimensioned machine.
Ideal Circle-Inversion Geometry
Enter a circle radius, inversion constant and angular position. The worksheet calculates the exact inverse of a point on a circle through O. It does not supply the links, triangles, joints or dimensions of a physical Sylvester-Kempe inversor.
Equation Used
- O is the origin and the declared input circle has centre (r,0) and radius r, so it passes through O.
- theta is measured at the circle centre from the positive x direction used in P=(r(1+cos theta),r sin theta).
- k is a positive declared inversion constant in squared length units.
- P and Q lie on the same ray from O and satisfy OP times OQ equals k.
- For this circle, Q lies on the exact vertical line x=k/(2r).
- The animation sweeps a separate demonstration point while the highlighted point and numeric tiles use the declared theta.
- A physical Sylvester-Kempe inversor uses a contraparallelogram, a four-bar and similar rigid triangles; this worksheet does not synthesize them.
- Link dimensions, branches, interference, singularities, clearances, stiffness, accuracy, loads, materials, fabrication, maintenance and safe operation are excluded.
Retains the exact circle-inversion arithmetic, replaces the arbitrary pseudo-linkage drawing with a faithful point-map diagram, and removes the false four-bar synthesis plus unsupported construction and performance claims.
The page calculates a point inversion, not a four-bar synthesis
Dijksman states the defining relationship OP times OQ equals a constant, with O, P and Q collinear. For the declared circle through O, substituting its parametric point into Q=kP/|P| squared places every inverse point on one vertical line.
The historical Sylvester-Kempe mechanism is not merely four bars. The cited mechanism study describes a contraparallelogram, a four-bar and similar rigid triangles, while the Smithsonian model uses linked triangular plates. This page therefore draws the exact map and does not invent missing linkage dimensions.
Use this only to inspect ideal circle-inversion geometry
The worksheet is useful for checking the algebraic relationship between an input circle through the inversion centre and its straight inverse locus.
It cannot design, size, build, rate or troubleshoot a physical inversor or linear-motion stage.
Invert the circle point about O
The constant x coordinate proves that the ideal inverse locus is a straight line for 0<theta<180 degrees.
A transparent ideal-map example
For r=50 mm, k=10000 mm squared and theta=60 degrees, P=(75.0,43.3) mm, OP=86.6 mm, OQ=115.5 mm and Q=(100.0,57.7) mm. The product OP times OQ is exactly 10000 mm squared.
An exact identity is not a physical design
The algebra specifies two collinear points and their distance product. A mechanism must impose that relationship through a complete set of rigid-link constraints and a chosen assembly branch.
The former four-link description, invented dimensions, optical-stage claims, tolerances, stroke limits, force behavior, materials, construction, maintenance and performance claims are removed.
Ideal inversion-map questions
Does this provide Sylvester-Kempe link lengths?
No.
Why is Qx constant?
Because |P| squared equals 2r times Px for a circle of radius r through O.
Is the vertical line an approximation?
No; it is exact for this ideal point map.
Does exact algebra guarantee an exact built mechanism?
No.
Quadruplanar inversor and linkage references
- Dijksman, A strong relationship between new and old inversion mechanisms: the Sylvester-Kempe quadruplanar structure and its contraparallelogram, four-bar and similar rigid triangles.
- Dijksman, A Way to Generalize Peaucellier's Inversor: geometric inversion relation OP times OQ equals a constant and the need for a two-mobility inversion cell.
- Smithsonian, Sylvester-Kempe Inversor model 351: museum description of the linked triangular plates and circle-to-line tracing action.
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