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Geometric Modeling and Applications Set
coordinated by
Marc Daniel
Volume 5
Geometric Modeling of Fractal Forms for CAD
Christian Gentil
Gilles Gouaty
Dmitry Sokolov
First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd
27-37 St George’s Road
London SW19 4EU
UK
John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA
© ISTE Ltd 2021
The rights of Christian Gentil, Gilles Gouaty and Dmitry Sokolov to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2021932086
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-040-9
Preface
This work introduces a model of geometric representation for describing and manipulating complex non-standard shapes such as rough surfaces or porous volumes. It is aimed at students in scientific education (mathematicians, computer scientists, physicists, etc.), engineers, researchers or anyone familiar with the mathematical concepts addressed at early stages of the graduate level. However, many parts are accessible to all, in particular, all introductory sections that present ideas with examples. People with no prior background, whether they are artists, designers or simply curious, will be able to understand the philosophy of our approach, and discover a new universe of unsuspected and exciting forms.
Geometric representation models are mathematical tools integrated into computer-aided geometric design (CAGD) software. They make the production of numerical representations of forms possible. By means of graphical interfaces or programming tools, users can draw and/or manipulate these shapes. They can also test or evaluate their physical properties (mechanical, electro-magnetic, acoustic, etc.) by communicating geometric descriptions to further specific numerical simulation software.
The geometric representation model we present here is based on the fractal geometry paradigm. The principle behind this consists of studying the properties (signal, geometry, phenomena, etc.) at different scales and identifying the invariants from there. The objects are described as self-referential between two scales: each of the object features (namely, the lower scale level) is described as a reference to the object itself (namely, the higher scale). This approach is not conventional and often confusing at first. We come to perceive its richness and power very quickly, however. The universe of forms that can possibly be created is infinite and has still only partially been explored.
In this book, we present the mathematical foundations,
