Speaker: Jen-chung Chuan

Affiliation: Deparment of Mathematics, Tsing Hua University, Hsinchu,

Taiwan

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 this talk we
are to

concentrate the constructions of

- the conics: hyperbola, parabola and ellipse.
- the epicycloids (the cardioid and the nephroid), hypocycloids (the deltoid and the astroid) and their osculating circles.
- the Lemniscate.
- the Bowditch curve.

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]

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]

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.

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).

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.

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.

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.

Location of the CabriJava file:

http://poncelet.math.nthu.edu.tw/disk3/cabrijava/bowditch-with-compass.html

Principle: reflection and the mid-point construction.

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