Web Textbook for Descriptive Geometry

In this paper we present educational material which is a basis for a web textbook designed for the lectures on geometric subjects, [1]. The authors of the textbook are geometry teachers from the University of Zagreb, three of them, Sonja Gorjanc, Helena Koncul and Iva Kodrnja, coming from the Faculty of Civil Engineering, and one, Ema Jurkin, from the Faculty of Mining, Geology and Petroleum Engineering. The contents of the textbook cover the topics that are taught within the courses Descriptive Geometry and Perspective at the Faculty of Civil Engineering, and courses Descriptive Geometry and Descriptive Geometry with Computer Graphics at the Faculty of Mining, Geology and Petroleum Engineering, however, some of its content is also applicable to other faculties in Croatia. The textbook is available online for free and licensed under CC BY-NC-SA 4.0 licence. One can include the full textbook, or only the parts of the textbook, into various e-learning platforms. Materials can be used free under the same conditions, i.e. must not be used commercially. The importance of clear and careful spatial reaso– ning and its graphic representation in the education of future engineers makes descriptive geometry a tradit– ional mathematical course at technical faculties, parti– cularly at those related to civil engineering. A series of high-quality descriptive geometry textbooks were published in Croatian during last fifty years, and this textbook presents their sequel. However, it really differs from previous textbooks in the application of infor– mation technologies in presenting classic contents and introducing 3D modelling in descriptive geometry teaching. Therefore, a different approach to some course units has been required. Due to specific characteristics of descriptive geometry as a discipline of mathematics, the textbook is graphically rich and it contains 38 interactive files, 493 original drawings and figures, 83 original animations, 82 original presentations and 93 video clips (up to five minutes), a total of more than six hours of audio and video material. It took quite a long time to prepare this extensive teaching material, which required interdisciplinary knowledge and skills, in parallel with lesson prepa– ration. The starting point of the textbook was repository of educational materials created within the project Introducing 3D Modeling into Geometry Education at Technical Colleges that was supported by Fund for Development of the University of Zagreb, [2]. During the year 2012 twelve teachers worked on the project, and four faculties were involved. The head of the pro– ject was Sonja Gorjanc. The main goals of the project were strengthening the professional and scientific colla– boration among teachers, developing teaching metho– dology for 3D computer modeling to enhance geometry courses, and harmonization of educational material sta– ndards and their implementation in the e-learning sys– tems of the technical faculties that were involved. The focus of the project was creating a basic repository of material related to common teaching topics and those customized to profiles of each faculty, [3]. The expe– rience of teachers using created educational materials were described in [4]. During the last few years we enriched the existing repository with new material and increased the number of topics it used to cover, giving it a form of textbook. We tried to connect the benefits of classical and modern approaches to teaching geometry guided by the thoughts presented in [5], as well as in [6] and [7].


INTRODUCTION
In this paper we present educational material which is a basis for a web textbook designed for the lectures on geometric subjects, [1].The authors of the textbook are geometry teachers from the University of Zagreb, three of them, Sonja Gorjanc, Helena Koncul and Iva Kodrnja, coming from the Faculty of Civil Engineering, and one, Ema Jurkin, from the Faculty of Mining, Geology and Petroleum Engineering.The contents of the textbook cover the topics that are taught within the courses Descriptive Geometry and Perspective at the Faculty of Civil Engineering, and courses Descriptive Geometry and Descriptive Geometry with Computer Graphics at the Faculty of Mining, Geology and Petroleum Engineering, however, some of its content is also applicable to other faculties in Croatia.
The textbook is available online for free and licensed under CC BY-NC-SA 4.0 licence.One can include the full textbook, or only the parts of the textbook, into various e-learning platforms.Materials can be used free under the same conditions, i.e. must not be used commercially.
The importance of clear and careful spatial reasoning and its graphic representation in the education of future engineers makes descriptive geometry a traditional mathematical course at technical faculties, particularly at those related to civil engineering.A series of high-quality descriptive geometry textbooks were published in Croatian during last fifty years, and this textbook presents their sequel.However, it really differs from previous textbooks in the application of information technologies in presenting classic contents and introducing 3D modelling in descriptive geometry teaching.Therefore, a different approach to some course units has been required.Due to specific characteristics of descriptive geometry as a discipline of mathematics, the textbook is graphically rich and it contains 38 interactive files, 493 original drawings and figures, 83 original animations, 82 original presentations and 93 video clips (up to five minutes), a total of more than six hours of audio and video material.
It took quite a long time to prepare this extensive teaching material, which required interdisciplinary knowledge and skills, in parallel with lesson preparation.The starting point of the textbook was repository of educational materials created within the project Introducing 3D Modeling into Geometry Education at Technical Colleges that was supported by Fund for Development of the University of Zagreb, [2].During the year 2012 twelve teachers worked on the project, and four faculties were involved.The head of the project was Sonja Gorjanc.The main goals of the project were strengthening the professional and scientific collaboration among teachers, developing teaching methodology for 3D computer modeling to enhance geometry courses, and harmonization of educational material standards and their implementation in the e-learning systems of the technical faculties that were involved.The focus of the project was creating a basic repository of material related to common teaching topics and those customized to profiles of each faculty, [3].The experience of teachers using created educational materials were described in [4].
During the last few years we enriched the existing repository with new material and increased the number of topics it used to cover, giving it a form of textbook.We tried to connect the benefits of classical and modern approaches to teaching geometry guided by the thoughts presented in [5], as well as in [6] and [7].The first chapter deals with plane geometry in more detail than a descriptive geometry courses.The chapter contains more than 30 interactive figures designed by GeoGebra, and by allowing the user to move certain elements one can study the plane constructions and their properties in detail.The links in the textbook enable the user to connect to certain contents when dealing with other contents in the textbook.

CONTENTS OF TEXTBOOK
The second chapter deals with space geometry and methods of projection, i.e. basic classic contents of descriptive geometry.Modern technology is used for visualization.The chapter mostly deals with Monge's method, with solved exercises presented as a slideshow with video clips of spatial representations.
A large number of figures and animations are supported by Mathematica, while slideshows are created using Adobe Illustrator.
The third chapter presents space curves and surfaces in great detail.The chapter contains approximately 70 animations created by Mathematica that visualize the generation of surfaces (rotational, translation, ruled sur-faces, helicoids) and space curves (intersection curves of surfaces, helixes).
In the final chapter the contents are presented through the application of computer technologies.For teaching 3D modeling we use Rhinoceros.It was chosen for the following reasons: a good experience of the colleagues from the University of Innsbruck, a relatively low cost of the educational lab licence and the free trial version available for students' downloads.Therefore, the textbook also contains 88 five-minute video clips helping students to learn basics of Rhinoceros 3D modeling.
They were produced by the screencasting program Jing.By using video clips, students are introduced to the basics of Rhinoceros on their own.Every time when we want to teach them something about Rhinoceros, we use a geometry example.So, at the same time students master the computer program and learn geometrical contents chosen by geometry teachers.

EXAMPLES
In this section we present the textbook material by showing the examples of using aforementioned programs.5 demonstrates the principle and an example of an intersection curve construction of a cone and a cylinder while Fig. 6 exhibits the intersections of a hyperbolic paraboloid and a plane.Finally, the application of helicoid is shown in Fig. 7.

CONCLUSION
This paper discloses the nowel educational material for delivering the lectures in descriptive geometry and perspective on the Faculty of Civil Engineering and the Faculty of Mining, Geology and Petroleum Engineering, University of Zagreb.This approach provides the set of highly efficient educational activities and exercises by which students -the future engineers can develop clear and careful spatial reasoning and its graphic representation.