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Gunter Weiss 5 “e-learning Geometry” The needs of teaching classical 3D-geometry on one hand and the lack of teachers on the other could force the development of e-learning material in Geometry. Such e-courses are generously supported by European Community programs, as they aim at a unification of educational systems in Europe. What can be found of e-courses up to now supports the respective Geometry courses of each author of those e-learning modules, such that they are not really of general use. By the way, not everything is correct in these e-learning materials! For example, in an Analysis e-teaching project supported by the EC the left figure 2 represents the graph of a cubic surface. This seemingly complicated surface is depicted also by its top view, using the same rough discretisation based on default values of MATHEMATICA.. Here mathematicians neglect essential mathematical fundamentals: the influence of discretisation, parameterisation, regularity. (The right figure 2 shows a ‘correct’ image of the left surface, which is the well known PLÜCKER conoid.) Cartesian graph Graph in cylinder coordinates Figure 2: PLÜCKER conoid plots with MATHEMATICA ® New media require other methodical approaches than classical ruler and compass constructions involve. For example, it can no longer be the central task of Descriptive Geometry to dissolve a 3D-problems into incidence and measure problems in the drawing plane. The classical set of geometric objects should be enriched by freeform objects, a listing of the single steps of the solution of an arbitrary problem should be based on argumentation in space. A one-to-one translation of former strategies to 3D-Cad-software surely is the wrong conception! As long as e-learning units have authors, whose education in Constructive Geometry only is based on engineering standard knowledge and who have not fathomed the subject scientifically, I have great reservations about the use of such teaching software. 28
GEOMETRY BETWEEN PISA AND BOLOGNA 6 Geometry: old knowledge to solve modern problems Resent research in reverse engineering, computer vision, robotics seems to be most successful, if based on classical knowledge in Geometry. Here H. POTTMAN, H. STACHEL, B. JÜTTLER, O. RÖSCHEL, H. HAVLICEK, M. HUSTY have to be named as special representatives. They use Projective Geometry, (non Euclidean geometry, linear mappings), classical Differential Geometry (line geometry, kinematics), Circle Geometries (of MÖBIUS, LAGUERRE, LIE) and classical Algebraic Geometry, often in a generalised form, to treat a technical problem in a very elegant and successful manner. When we recognise that all these subjects are standard courses for teachers of Descriptive Geometry in Austria, it is no longer strange that the mentioned persons all come from Austria. It seems to be like in sports: For a nation the chance of gaining medals in a certain branch of sports is higher, if that special sport is commonly practiced. Reference points at „known“ places and main sections 3D Model with centre of projection O, optical ref. point R, focal distance ck Sketch close to a perspective projection Figure 3: 3D-Reconstruction from Sketches (F. HENSCHEL) 29
- Page 1 and 2: Katedra matematiky Fakulty stavebn
- Page 3 and 4: Programový výbor konference: Doc.
- Page 5: OBSAH TABLE OF CONTENTS
- Page 8 and 9: Table of Contents Petr Kahánek, Al
- Page 10 and 11: Table of Contents Daniela Velichov
- Page 13: PLENÁRNÍ PŘEDNÁŠKY PLENARY LEC
- Page 16 and 17: František Kuřina sítě čtyřdim
- Page 18 and 19: František Kuřina Kombinace těcht
- Page 20 and 21: František Kuřina 3 Matematika a v
- Page 22 and 23: František Kuřina řešení je alt
- Page 24 and 25: Gunter Weiss should show an own sci
- Page 26 and 27: Gunter Weiss “industrial reality
- Page 30 and 31: Gunter Weiss In the following some
- Page 32 and 33: Gunter Weiss screen. This made it f
- Page 35: REFERÁTY CONFERENCE PAPERS
- Page 38 and 39: Eva Baranová, Kamil Maleček S a a
- Page 40 and 41: Eva Baranová, Kamil Maleček extr
- Page 42 and 43: Eva Baranová, Kamil Maleček Obrá
- Page 44 and 45: ÊÓÞÔ×ÑÓ×ÓÙÖÒÔÞ×Ñ Å
- Page 46 and 47: ÞÓÓÑÓÒÒ×ÓÙÖÒ×ÓÙÊ´
- Page 48 and 49: ÔÖØÓÑØÓÔÓÝÙ×ÓÙÖÓÚ
- Page 50 and 51: Bohumír Bastl The second reason fo
- Page 52 and 53: Bohumír Bastl 4 3 2 1 0 2 4 0 2 4
- Page 54 and 55: Bohumír Bastl the package that bot
- Page 56 and 57: Zuzana Benáková x B y z B B = u c
- Page 58 and 59: Zuzana Benáková Obrázek 4: třet
- Page 60 and 61: Michal Benes Since des t 2 = ds 2 +
- Page 62 and 63: Michal Benes The latter condition i
- Page 64 and 65: Michal Benes 4 Innitesimal deformat
- Page 66 and 67: Michal Benes where , 1, 2, ¢1, ¢2
- Page 68 and 69: Milan Bořík, Vojtěch Honzík sys
- Page 70 and 71: Milan Bořík, Vojtěch Honzík Dá
- Page 72 and 73: Milan Bořík, Vojtěch Honzík Obr
- Page 74 and 75: Jaromír Dobrý Example 2 Let’s c
- Page 76 and 77: Jaromír Dobrý M ϕ ϕ(M) ϑ V {o}
GEOMETRY BETWEEN PISA AND BOLOGNA<br />
6 Geometry: old knowledge to solve modern<br />
problems<br />
Resent research in reverse engineering, computer vision, robotics seems to<br />
be most successful, if based on classical knowledge in Geometry. Here H.<br />
POTTMAN, H. STACHEL, B. JÜTTLER, O. RÖSCHEL, H. HAVLICEK, M. HUSTY<br />
have to be named as special representatives. They use Projective Geometry,<br />
(non Euclidean geometry, linear mappings), classical Differential Geometry<br />
(line geometry, kinematics), Circle Geometries (of MÖBIUS, LAGUERRE,<br />
LIE) and classical Algebraic Geometry, often in a generalised form, to treat<br />
a technical problem in a very elegant and successful manner. When we<br />
recognise that all these subjects are standard courses for teachers of<br />
Descriptive Geometry in Austria, it is no longer strange that the mentioned<br />
persons all come from Austria. It seems to be like in sports: For a nation the<br />
chance of gaining medals in a certain branch of sports is higher, if that<br />
special sport is commonly practiced.<br />
Reference points at „known“<br />
places and main sections<br />
3D Model with centre of projection O,<br />
optical ref. point R, focal distance ck<br />
Sketch close to a<br />
perspective projection<br />
Figure 3: 3D-Reconstruction from Sketches (F. HENSCHEL)<br />
29
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