visual metaphor of mathematical abstractions and their visualization through newly uprised...

Visual Metaphor of Mathematical Abstractions and their Visualizationthrough Newly Uprised PDF-document Facielities

PhD Alexander L. Krochin

Ural Federal University, Yekaterinburg, Russia

15.03.2017

PhD Alexander L. Krochin (Ural Federal University, Yekaterinburg, Russia)Visual Metaphor of Mathematical Abstractions 15.03.2017 1 / 20

"Tell me and I forget. Show me and I remember. Involve me and Iunderstand."— Chinese proverb

Key words for open education: visualization + dynamic + interactivity


Visualization

There are many different meanings of the word visualization. Outsidethe domain of computer graphics, accepted definitions includemaking visible, especially to one’s mind (thing not visible to the eye),forming a mental image of something (thing not present to the sight, anabstraction, etc.) [Webster70].


Visualization and modern teaching

The visualization has been used in many areas of knowledge: architec-ture, medicine, science, software, information visualization, “big data”visualization, and so on.Here, the visualization of mathematical abstract entities and concepts isused to facilitate teaching/learning process and communication betweenteacher and students.


PDF document standard

Portable Document Format (PDF) (developed, improved and maintainedby Adobe) is one of the wide-spreading document standards for platform-independent exchange.The main features of the file formats in the use of learning are:

presence of the built-in JavaScript programming language;controlled multi-layered structure of the slide, which supports anima-tion and quasi-animation;possibility of introducing (embedding) multimedia objects;a built-in mechanism for interactive viewing of 3-dimensional objectsfrom the embedded U3D and PRC formats source.


PDF features in action

PDF format documents with new interactive and dynamic facilities essen-tially improve the effectiveness of teaching/ learning process, especiallyin mathematics education.The main problem in preparation of PDF documents is developing anadequate visual metaphor and a suitable and inexpensive program in-strument for metaphor implementation.


LATEX tool

The LATEX (pfdLATEX) is the most popular text processor in academicsociety, that produces an exellent mathematical papers.Graphs and pictures can be produced by many graphical programs (Meta-Post, TikZ, Asymptote, etc.) and systems like the Maple with variousoutput formats.


Didactical capabilities

Modern graphical tools allow one display dynamical images. One cancontrol the level of inputs and outputs for logical transformations,formulae exploration;cover and recover the text and image fragments;possibility of introducing (embedding) multimedia objects;onscreen modelling (Buffon pin);


Didactical capabilities

Modern graphical tools allow one display dynamical images. One cancontrol the level of inputs and outputs for logical transformations,formulae exploration;cover and recover the text and image fragments;possibility of introducing (embedding) multimedia objects;onscreen modelling (Buffon pin);

The instructor want to explain to students how to solve some difficultproplem.Every steps of his explanation connect with certain picture associatedwith a particular image via appearing arrows.


Dynamical explanation


Nice and effective tool is animation — a consecutive display of the pictorialsequence (animate.sty).The more faster animation can illustrate the Law of Large Numbersgraphically. Convolution of a probability density functions of uniformdistribution. Number of terms grows: two, three, four, to ten. As aresult one can see: a triangle, and other curves, shown


Frames of didactical animation


Instrumental program tools for PDF format

Graphical parts of documents are prepared by means of numerous freetools, occasionally, commercial programs.Popular assembly software based on LATEX contains almost all free graphic-aimed tools (MetaPost, TikZ, and Ps-tricks). So, the LATEX source scripts“collecting” text, graphics, and multimedia components and transform allin whole standard PDF document.The choice of specific programs for creating graphical objects is deter-mined by the specific features of visualization of mathematical mentalimages.


3D — images

Let us distinguish an ordinary static drawing of 3D object named as quasi3D and real or dynamical 3D images.Somebody can rotate letter, vary size, and change angle and direction ofvision.Beside that, real 3D picture is much more informative and expressive,and that is important for educational purposes.


There is free tool for demonstrate PDF documents in their entirety —Adobe Reader. But a convenient and suitable tool for making teachingmaterials is absent.

Adobe Reader screen shots 3D model with drop-down control menu.PhD Alexander L. Krochin (Ural Federal University, Yekaterinburg, Russia)Visual Metaphor of Mathematical Abstractions 15.03.2017 14 / 20

Sources for 3D images

The 3D file convertionMaple and Mathematica — many output formats (real 3D): X3D,Collada(dae) .Useful for PDF — U3D, PRC — not available..dae (Collada) ⇒ (via something — Blender) U3D ⇒ PDF (via movie15)..x3d ⇒ (maplex3d2prc) PRC ⇒ PDF (media9).


Sources for 3D images

The 3D file convertionMaple and Mathematica — many output formats (real 3D): X3D,Collada(dae) .Useful for PDF — U3D, PRC — not available..dae (Collada) ⇒ (via something — Blender) U3D ⇒ PDF (via movie15)..x3d ⇒ (maplex3d2prc) PRC ⇒ PDF (media9).

Asymptote — embedding into LATEX(via pdfLATEX) or PRC and ⇒ PDF(media9)


Simple examples

(dice3D)


(dissabbiatoremeshlab.u3d)

Click here!PhD Alexander L. Krochin (Ural Federal University, Yekaterinburg, Russia)Visual Metaphor of Mathematical Abstractions 15.03.2017 17 / 20

(plot3a.prc)

Click here!PhD Alexander L. Krochin (Ural Federal University, Yekaterinburg, Russia)Visual Metaphor of Mathematical Abstractions 15.03.2017 18 / 20

Example58 TUGboat, Volume 30 (2009), No. 1

Graphics

Asymptote: Lifting TEX to three dimensions

John C. Bowman and Orest Shardt

Abstract

Asymptote, a modern successor to the METAPOST

vector graphics language that features robust floating-point numerics, high-order functions, and deferreddrawing, has recently been enhanced to generatefully interactive three-dimensional output. Thisdata can either be viewed with Asymptote’s nativeOpenGL-based renderer or internally converted toAdobe’s highly compressed PRC format for embed-ding within a PDF file. Asymptote thus providesthe scientific community with a self-contained andpowerful TEX-aware facility for generating portableinteractive three-dimensional PDF files.

1 Introduction

The descriptive vector graphics language Asymptote1

was developed to provide a standard for drawingmathematical figures, just as TEX and LATEX havebecome the standard for typesetting equations in themathematics, physics, and computer science commu-nities [1]. For professional quality and portability,Asymptote natively generates PostScript, PDF, andPRC vector graphics output. The latter is a highlycompressed 3D format that is typically embeddedwithin a PDF file and viewed with Adobe Reader.

In both two and three dimensions, consistentfonts and equations should be used in the graphicsand text portions of a document. This implies thatlabels must be typeset directly by TEX. This articleprovides an overview of the major advances in thecurrent version (1.82) of Asymptote that allow it toextract and lift Bezier font descriptions generated byTEX and Dvips into 3D, using efficient algorithmsfor partitioning planar regions into nondegenerateCoons patches [3]. Together with 3D generalizationsof the METAFONT path operators and a method forcomputing twist-free tubes and arrowheads, thesealgorithms provide the 3D foundation of Asymptote.

2 Bezier surfaces

A major recent advance in Asymptote is the abilityto embed Bezier surfaces as interactive PRC content

1 Andy Hammerlindl, John Bowman, and Tom Prince,available under the GNU Lesser General Public License fromhttp://asymptote.sourceforge.net/.

Figure 1: An interactive 3D PDF of a Bezier surfacerepresentation of the Utah Teapot.

within a PDF file, as illustrated in Fig. 1.2 In contrast,the version of U3D supported by Adobe can onlyrender surfaces described by polygons and hence isnot a suitable vector graphics format.

3 Three-dimensional TEX

TEX produces output in a special device independentformat (DVI). While this output can be easily turnedinto PostScript, one needs a way of extracting Beziercurves that describe properly kerned font characters.Asymptote does this by overloading the PostScript/show operator, as described in Appendix A. Specialcare was required to handle the filled rectangles thatTEX uses to draw square root symbols and fractionbars. The resulting exact 2D vector representationof the original TEX input is treated by Asymptoteas an array of paths to be filled with the PostScriptnonzero winding number fill rule.

The routine bezulate described in Figs. 2 and 3,along with the nondegenerate patch splitting algo-rithms described in [3], is used to convert the result-ing Bezier paths to Bezier surfaces. These surfacesare then output in the PRC format, along with arendered preview image for noninteractive viewingand printing. Using these techniques, Asymptote isthen able to typeset the Gaussian integral in Fig. 4as an interactive 3D diagram.

4 Thick lines in 3D

Figure 5 depicts capped thick lines and Asymptote’sfive (METAPOST-inspired) path connectors [2]:

-- .. & --- ::

for the following path, when lifted to the x–y plane:

2 An interactive PDF version of this article may be foundat http://asymptote.sourceforge.net/articles/.

John C. Bowman and Orest Shardt


Usage of 3D PDF

Big massives of 3D data — the result of mesurement and calculation in many sciencies. Andall these results may be embedded into PDF documents for easier perception, analysis, anddiscussion.Next slide is the complex moleculae image, calculatede from data.

15th March 2017 1 / 1

(plot3a.prc)

Click here!15th March 2017 2 / 1

Thank you for your attention!


visual metaphor of mathematical abstractions and their visualization through newly uprised...

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