Geometric Constructions with the Compasses Alone

 
Jen-chung Chuan
Deparment of Mathematics
National Tsing Hua University
Hsinchu, Taiwan 300
e-mail: jcchuan@math.nthu.edu.tw

Mascheroni dedicated one of his books Geometria del compasso (1797) to Napoleon in verse in which he proved that all Euclidean constructions can be made with compasses alone, so a straight edge in not needed. This theorem was (unknown to Mascheroni) proved in 1672 by a little known Danish mathematician Georg Mohr. In the setting of dynamic geometry, the Mohr-Mascheroni constructions ask for specific procedures in which the figures are constructed using the compasses alone. In what follows we are to concentrate the constructions of

  1. the conics: hyperbola, parabola and ellipse.
  2. the epicycloids (the cardioid and the nephroid), hypocycloids (the deltoid and the astroid) and their osculating circles.
  3. the Lemniscate of Bernoulli.
  4. the Bowditch curve.

The dynamic geometry environment provided by CabriJava is essential in our exploration.

Hyperbola
Total number of intermediate circles: 8
Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/hyperbola-with-compass.html
Principle: hyperbolas are the inversions of the lemniscates. [Lockwood; p. 116]

1.gif (16516 bytes)

Parabola
Total number of intermediate circles: 8
Location of the CabriJava file:  http://poncelet.math.nthu.edu.tw/disk3/cabrijava/parabola-with-compass.html
Principle: parabolas are the inversions of the cardioids. [Lockwood; p. 180]

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Ellipse
Total number of intermediate circles: 8
Location of the CabriJava file:  http://poncelet.math.nthu.edu.tw/disk3/cabrijava/ellipse-with-8circles.html
Principle:

1. Center of the reference circle, the inverse and the point itself are collinear.
2. x = a cos t, y = b sin t.

3.gif (12939 bytes)

Cardioid
Total number of intermediate circles: 4
Location of the CabriJava file:  http://poncelet.math.nthu.edu.tw/disk3/cabrijava/cardioid-from-circle.html
Principle: x = 2 cos t - cos(2t), y = 2 sin t - sin(2t).

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Cardioid and its Osculating Circle
Total number of intermediate circles: 10
Location of the CabriJava file:   http://poncelet.math.nthu.edu.tw/disk3/cabrijava/osc-cardioid-compass.html
Principle: the points [cos t,sin t], [cos(2t), sin(2t)] separate the point [2 cos t - cos(2t), 2 sin t - sin(2t)] and the center of curvature harmonically.

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Nephroid and its Osculating Circle
Total number of intermediate circles: 11
Location of the CabriJava file:   http://poncelet.math.nthu.edu.tw/disk3/cabrijava/osc-nephroid-compass.html
Principle: the points [cos t,sin t], [cos(3t), sin(3t)] separate the point [3 cos t - cos(3t), 3 sin t - sin(3t)] and the center of curvature harmonically.

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Deltoid and its Osculating Circle
Total number of intermediate circles: 11
Location of the CabriJava file:   http://poncelet.math.nthu.edu.tw/disk3/cabrijava/osc-deltoid-compass.html
Principle: the points [cos t,sin t], [cos(-2t), sin(-2t)] separate the point [cos (2t) - 2 cos(t), sin (2t) +2 sin(t)] and the center of curvature harmonically.

10.gif (21995 bytes)

Recovering the Center of a Circle
Total number of intermediate circles: 6
Location of the CabriJava file:   http://poncelet.math.nthu.edu.tw/disk3/cabrijava/center.html
Principle: Inversion.

11.gif (15396 bytes)

Fermat Point
Total number of intermediate circles: 16
Location of the CabriJava file:   http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass-fermat.html
Principle: Simititude.

12.gif (11774 bytes)

Peaucellier's Linkage
Total number of intermediate circles: 5
Location of the CabriJava file:   http://poncelet.math.nthu.edu.tw/disk3/cabrijava/peau.html
Principle: Inversion.

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Intersection of a Line and a Circle
Total number of intermediate circles: 4
Location of the CabriJava file:   http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass-line-circle-intersections.html
Principle: Symmetry.

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Envelope Formaing Deltoid and 3-Cusped-Epicycloid
Total number of intermediate circles: 11
Location of the CabriJava file:   http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass-epi-hypo.html
Principle: Symmetry.

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Regular Pentagon
Total number of intermediate circles: 12
Location of the CabriJava file:   http://poncelet.math.nthu.edu.tw/disk3/cabrijava/5-gon-compass.html
Principle: Inversion.

16.gif (17828 bytes)

Linkage Drawing the Ellipse
Total number of intermediate circles: 11
Location of the CabriJava file:  http://poncelet.math.nthu.edu.tw/disk3/cabrijava/ellipse-linkage-with-compass.html
Principle: Inversion.

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Square Constructed from One Diagonal
Total number of intermediate circles: 6
Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass-sq-diag.html
Principle: Symmetry.

18.gif (8983 bytes)

Square Constructed from One Side
Total number of intermediate circles: 6
Location of the CabriJava file:  http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass-sq-side.html
Principle: Symmetry and translation.

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Dividing a Cirlce into Four Equal Parts
Total number of intermediate circles: 6
Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/compass-sq.html
Principle: Symmetry and translation.

20.gif (13180 bytes)

Arc Bisection
Total number of intermediate circles: 7
Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/bisect-arc.html
Principle: Symmetry.

21.gif (9880 bytes)

Bowditch Curve
Total number of intermediate circles: 13
Location of the CabriJava file: http://poncelet.math.nthu.edu.tw/disk3/cabrijava/bowditch-with-compass.html
Principle: Coordinates.

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References

  1. H. M. Cundy and A. P. Rollett, Mathematical Models, Oxford University Press,1961.
  2. Heinrich Dorrie, 100 great problems of elementary mathematics; their history and solution, New York, Dover, 1965.
  3. Howard Eves, A Survey of Geometry, Boston, Allyn and Bacon, 1963-65.
  4. Hilda P. Hudson, Ruler and Compasses, reprinted by Chelsea in the collection "Squaring the Circle"
  5. A.B. Kempe, How to draw a straight line; a lecture on linkage, reprinted by Chelsea in the collection "Squaring the Circle"
  6. A.N. Kostovskii, Geometrical Constructions Using Compasses Only, Popular lectures in mathematics series,v. 4, Translated from the Russian by Halina Moss. Translation editor: Ian N. Sneddon, New York,Blaisdell, 1961.
  7. E. H. Lockwood, A Book of Curves, Cambridge University Press, 1961.
  8. Robert C. Yates, A Handbook on Curves and Their Properties, Ann Arbor, J. W. Edwards, 1947.