N. M. Beskin
Mir Publishers, Moscow | 1985 | 78 páginas | djvu | 663 kb
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Drawing a plane figure is not geometrically difficult because the image drawn is either an exact copy of the original or a similar figure, e.g. the drawing of a circle looks like the original circle. Drawing geometric solids is quite a different matter. Unfortunately, there are no “spatial pencils” which can trace an object in the air. Such a pencil would “draw” a cube by tracing along its edges. Hence, we have to sketch a cube on paper with an ordinary pencil. A plane image will never be an exact copy of a solid and, therefore, a certain routine ought to be followed in drawing a solid that would create an image of the original in the best way.
What is the book about. Descriptive geometry embraces so
many methods that even a brief account would make up a rather thick volume. Therefore, we shall discuss just one of these methods, so as to enable the reader to make stereometric drawings and solve the respective problems…
This book presents a geometric theory of constructing
stereometric drawings. Having mastered this theory, a reader will be able to make the drawings himself rather than have to stick to the few sample ones.
The first chapter presents the theory, the second one is devoted
to its applications (drawing of a cube, a cone, a cylinder, etc.),
and the third one describes a method of plotting the points of an
image if their coordinates are known.
It is these strategies and routine that this book discusses. Though many things are possible with modern computer programs, but the logic may not be known to people who are using them.
The book was translated from the Russian by Valery Barvashov and was first published by Mir in 1985.
Contents
Chapter 1. Theory
1. The subject matter of the Theory of images .7
2. Requirements of an image. 7
3. What is the book about .8
4. The method of parallel projection …8
5. A comment on notation .10
6. Properties of parallel projections . 11
7. Free images .13
8. Constructing the images of plane figures ..14
9. Some examples of representing polygons .15
10. The image of a circle .16
11. Another viewpoint of constructing the images of plane figures .17
12. Pohlke-Schwartz theorem.20
13. Representing geometric solids…26
14. Reversibility of an image .28
15. Specified images .30
Chapter 2. Practical Exercises
16. Cross sections of polyhedrons .32
17. Metric problems ..35
18. Solids of revolution…37
19. The image of a plane..42
20. Inscribed and circumscribed solids.…43
21. Some drawing conventions..46
22. Drawing obvious images…47
Chapter 3. A computation method
23. Theory… 50
24. Application of the computation method …53
Appendix 1. Expression of the Coordinates of the Image Points Using the Coordinates of the Original Points
25. A characteristic property of a linear homogeneous function .62
26. Formulas for the coordinates of the points of an image .64
Appendix 2. The Ellipse
27. Uniform compression .67
28. The definition of an ellipse ..70
29. Some properties of ellipse …70
30. The ellipse as the projection of a circle .73
31. The cross sections of a circular cylinder …75
3;2. Some constructions connected with the ellipse .76
Representación de figuras espaciales
88 páginas | PDF | 15,7 Mb
Al representar una figura plana no aparece ninguna dificultad geométrica. El dibujo resulta una copia exacta del original o bien representa una figura semejante al mismo. Cuando consideramos el dibujo de un círculo, nuestra percepción visual es la misma que al considerar el círculo original. La situación es otra por completo cuando se trata de la representación de figuras espaciales. Desgraciadamente, no existe un "lápiz espacial" cuya punta deje huella en el aire. Tal lápiz permitiría "dibujar" un cubo auténtico trazando sus aristas. Pero como no existe, nos vemos obligados a dibujar el cubo desplazando sobre el papel la punta de un lápiz corriente. La imagen plana no puede ser copia exacta de una figura espacial. Esta discordancia plantea el problema: ¿a qué normas atenerse para construir la representación a fin de que ésta dé la imagen más adecuada del original?.
