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
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]
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]
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
