DrGeo Smalltalk Sketches for Theory and Problems of Projective Geometry
# ayres
DrGeo Smalltalk Sketches for :
Frank Ayres Jr.
Theory and Problems of Projective Geometry (Schaum Outline Series)
Schaum Publishing Co. New York, 1967
DrGeo is a Smalltalk system for geometry (running on Cuis Smalltalk).
See http://cuis.st and http://gnu.org/s/dr-geo.
The book by Frank Ayres Jr. provides in Chapter1 - 12 the basic propositions of plane projective geometry developed entirely by synthetic methods, and discusses some space (3D) configurations in Chapter 3.
The Frank Ayres book is Copyright (c) 1967 Schaum Publishing Company. "All rights reserved. This book or any part thereof may not be reproduced in any form without written permission from the publishers."
REFERENCES
* Abraham Bosse with preface by Desargues, 1643 (early modern French)
"La pratique du trait a preuves de Mr. Desargues [lyonnois] pour la coupe des pierres en l'architecture", Paris, Pierre Des-Hayes
text in early modern French, with a preface by Desargues, with 114 technical drawings at the end of the book (after the text)
book focuses on technical (rigourous) drawings and techniques for cutting stones
* Abraham Bosse with preface by Desargues, 1664 (early modern Dutch by J.Bara)
"Algemeene Manier van de Hr. Desargues, Tot de practijck der Perspectiven, gelijck tot die der Meetkunde, met de kleine voetmaat"
"Mitsgaders der plaatsen, en proportien van de stercke en flaauwe Rakingen, of Kleuren"
"By-een-gevoeght door Abraham Bosse, en uyt het Frans vertaalt van J.Bara"
Book in Dutch published in 1664 by Dancker Dancertsz (or Dancker Dancertsen) in Amsterdam in 1664.
Contains about 155 plates (drawings) about perspective drawing, shading (high / low contract grayshades) with mathematical proofs on perspectivities.
drawings and text side by side (text and drawings on different pages)
with a preface by Girard Desargues (signed DESARGUES and later once as SGDL)
The preface by Desargues in the 1664 book differs from the one in the 1643 book (perhaps it was written before 1661 when Desargues died ?)
The book focuses on applications, illustrations for architecture (perspective)
* Philippe de la Hire, Traité des epicycloïdes &de leurs usages dans les mechaniques, Paris, Imprimerie Royale, 1694
Book with theory on cycloids and epicycloids in the style of Euclid, but with applications to machines; de la Hire mentions Gerard Desargues as 'an excellent geometer" and explains that de la Hire had ordered to rebuild a machine originally built by Desargues as engineer; de la Hire mentions that he suspects that Desargues built this machine, but did not publish anything about it; the old machine was in ruins at the time of de la Hire.
The book of de la Hire also contains a discussion on von Tchirnhaus visit to Paris and a proof of a Tchirnhaus conjecture on epicycloides.
The painter Laurent de la Hyre, father of the mathematician Philippe de la Hire, was a friend of Desargues; he painted Allégorie de la Géométrie.
* Monge, Géometrie descriptive, 1795, Paris
Originally "Géométrie Descriptive, Leçons données aux Ecoles normales" an III
Published "an VII de la république"
4th edition in 1820 "Géométrie Descriptive, Théorie des ombres et de la perspective" Gaspard Monge, Barnabé Brisson
Republished in 1922 : Les mâitres de la pensée scientifique, publiée par les soins de Maurice Solovine, Paris Gauthiers-Villars, 1922
* Michel Chasles, Aperçu historique sur l'origine et le développement des méthodes en géométrie, 1837, Hayez, Bruxelles
* Michel Chasles, Traité de géométrie supérieure, 1880, Gauthier-Villars, Paris
* L.N.M. Carnot, Traité de la corrélation des figures, Paris, IX (1801)
* L.N.M. Carnot, Géométrie de Position, XI (1803), Imprimerie de Crapelet, Paris
* Karl Doehlemann, Projektive Geometrie in synthetischer Behandlung, 1905, Leipzig
* Luigi Cremona, Elementi di geometria projettiva, Torino, 1873 (Italian, later translated into German and French)
* Luigi Cremona, Elements of projective geometry, Clarendon Press 1885 (English translation with foreword from Cremona)
* Luigi Cremona, Elements of projective geometry, Oxford University Press, 1913
* Luigi Cremona, Elements of projective geometry, 1960, Dover Pubs, New York
* André Delachet, La Géometrie Projective, 1964, Presses Univ. de France
Chapter 1 Introduction
8a. pencil of points (POP)
8b. pencil of lines (POL)
9a. elementary perspectivity
9b. perspective POPs with center of perspectivity, invariant point
9b2. perspective POPs defined by two points and their correspondents
9c. perspective POLs with axis (line) of perspectivity, invariant line
9c2. perspective POLs defined by two lines and their correspondents
10. projectivity (projective correspondence) between POP : 2 perspectivities
11. projectivity (projective correspondence) between POP : 3 perspectivities
12. augmented lines: project point at infinity to intersection parallel line
13. two projectively related superposed POLs
14a. projectivity between three collinear points A1,B1,C1 to A2,B2,C2 (A1=A2)
14b. projectivity between three collinear points on different lines (A1!=A2)
14c. projectivity between pair of three collinear points on same line
15. projectivity of points p(A,B,C,D) -^ p(C,D,A,B)
16. projectivity of lines P(a,b,c,d) -^ P(b,a,d,c)
Chapter 2 Cross Ratio
1. ratio of division in Euclidean plane
1b. cross ratio (=double ratio) in Euclidean plane
1c. cross ratio interchange any two of the four points simultaneously
1d. cross ratio interchange of one pair
1e. cross ratio interchange of inner two points or outer two points
2. invariance of cross ratio
3. cross ratio of four lines
4. cross ratio of of two points and two lines
7a. cross ratio lambda >0 for (A,B;C,X) three distince points A,B,C
7b. cross ratio lambda <0 for (A,B;C,X) three distince points A,B,C
7c. cross ratio zero for (A,B;C,X) three distince points A,B,C
7d. cross ratio one for (A,B;C,X) three distinct points A,B,C
7e. cross ratio = ratio of division of three distinct points A,B,C
8. projectivity of points for which Q is self-corresponding is perspectivity / construction of point of perspectivity
8b.projectivity of lines for which PQ is self-corresponding is perspectivity / construction of line (axis) of perspectivity (??)
9. dual pappus (pappos) theorem lines are concurrent (point of Pappos)
9b. pappus (pappos) theorem points are collinear (line of Pappos)
Chapter 3 Desargues two triangle theorem
0a. complete 2-point
0b. complete 2-line
0c. complete 3-point (triangle)
0d. complete 3-line (triangle)
1a. complete (4-point) quadrangle
1b. complete (4-line) quadrilateral
1c. complete (4-point) quadrangle (different, better layout)
1d. complete (4-line) quadrilateral (different, better layout)
1e. complete 4-point in space (tetrahedron)
2. complete 5-point in space
3. Desargues configuration (center perspectivity,two triangles,axis 3pt)
3b. Desargues two triangle theorem coplanar-noncoplanar
3c. Desargues two triangle theorem affine parallel lines (Euclides 6,2)
4. obtain intersection X of AB and line p by using Desargues theorem
5. perspective quadrangles
6. perspective quadrangles with diagonal side in common
7. Desargues two triangle theorem
8. dual of 4 : obtain join of intersection lines a and b and point P
9. special case of Pappus: A1,A2,A3 and B1,B2,B3 perspective from O
10. reduce 2 perspectivities between points on 3 concurrent lines
four triangles in perspective (corollary Desargues)
11. reduce 3 perspectivities between points on 2 pairs of lines
corollary of reduction of 2 perspectivities (10)
[NOTE: stack including temps is too deep]
12. reduce 3 perspectivities between points on 3 concurrent lines
and a not-concurrent line
Chapter 4 Harmonic Sets
1a. harmonic set (harmonic tetrad) of points
1b. harmonic set (harmonic tetrad) of lines
1c. harmonic conjugate point E for points A,B,D
1d. harmonic conjugate line e for lines a,b,d
1e. harmonic set of lines (complete quadrilateral)
2. harmonic set of points by section of harmonic set of lines
3. harmonic points of a complete quadrangle (diagonal triangle)
3b. quadrilateral of harmonic points of a complete quadrangle
4. harmonic sequence generated by points A,B,C on line o
5a. harmonic conjugate wrt. point at infinity/unit pt on augmented line
5b. harmonic conjugate wrt. point at infinity/unit pt on augmented line
5c. harmonic sequence generated by point at infinity/unit pt on line
5d. harmonic conjugate wrt. point at infinity/origin on augmented line
5e. harmonic conjugate wrt. point at infinity/origin on augmented line
5f. harmonic sequence generated by point at infinity/origin on line
5g. harmonic net of integers
5h. 1/2 as harmonic conjugate of 2 wrt. -1 and 1
5i. 1/n as harmonic conjugate of n wrt. -1 and 1 (or wrt. 1 and -1)
5j. harmonic net of rationals 1,2,3,4 ... 1/2,1/3,1/4 ...
5k. harmonic set -1,1;0,inf (or 1,-1;0,inf)
6. triangle ABC three collinear points A1,B1,C1 1 on each side of ABC
lines that join the harmonic conjugates wrt. 2 vertices on side
with the third vertex, are concurrent
7. quadrangle PQRS diagonal triangle ABC harmonic points DEFGIJ
K intersection of DI and PQ,L intersection of DJ and RS
M intersection of EI and PQ,N intersection of EJ and RS
BKM and BNL are collinear
Chapter 5 Projectivities
1. Projectivities and the Pappos configuration
projectivity as product of two perspectivities (simple case)
this is the case of chap3-10, this projectivity is a perspectivity
axis of projectivity (Pappos line)
if projectivity of points r(A,B,C,D) -^ s(A',B',C',D')
then line A A' double element line : perspectivity between POL's
theorem chap2-9 : any projectivity on A and A',
where the line A A' is self-corresponding is a perspectivity
2. Projectivities and the Pappos configuration: general case
Construction projectivity using axis of projectivity (Pappos)
3. Projectivities and the Pappos config