Title: Geometric Constructions with the Compasses Alone
Speaker: Jen-chung Chuan
Affiliation: Deparment of Mathematics, Tsing Hua University, Hsinchu,
Taiwan
e-mail: jcchuan@math.nthu.edu.tw

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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 this talk 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.
  4. the Bowditch curve.

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. [E. H. Lockwood, A Book of Curves; p. 116]

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. [E. H. Lockwood, A Book of Curves; p. 180]
 

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.

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).
 

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.
 

Nephroidoid and its Osculating Circle

Total number of intermediate circles: 10
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.

Lemniscate

Total number of intermediate circles: 5
Location of the CabriJava file:
http://poncelet.math.nthu.edu.tw/disk3/cabrijava/lemniscate-with-compass.html
Principle: the curve can be constructed with the aid of a linkage.

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: reflection and the mid-point construction.
 

References:

  1. E. H. Lockwood, A Book of Curves
  2. Heinrich Dorrie, 100 Great Problems of Elementary Mathematics
  3. A.B. Kempe, How to draw a straight line; a lecture on linkage, reprinted by Chelsea in the collection ¡§Squaring the Circle¡¨
  4. Cabri World 2001, 2nd Cabri Geometry International Conference