Visualisations of Gussian and Mean Curvatures by Using Mathematica and webmathematica

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1 Visalisations of Gssian and Mean Cratres by Using Mathematica and webmathematica Vladimir Benić, B. sc., Sonja Gorjanc, Ph. D., Faclty of Ciil Engineering, Kačićea 6, Zagreb, Croatia Abstract. In this paper we hae gien a short oeriew on calclation of the Gassian and mean cratres of a reglar srface and in six examples we hae shown isalisations of the properties of that fnctions by sing Mathematica and color fnction He. We hae described the program webmathematica and presented one web page which is powered by this program.. Introdction Specific conditions concerning the installation of Mathematica in Croatia (the program is aailable on all niersity compters) stimlated some teachers of geometry and mathematics at the Faclties of Ciil Engineering and Geodesy to try to improe teaching and learning process by means of Mathematica and webmathematica. Within the IT project Selected Chapters of Geometry and Mathematics Treated by Means of Mathematica for Ftre Strctral Engineers we designed edcational material which enhances isally standard lectres and stimlates interactie and ttorial way of learning on the Internet. The parts of the edcational material that we hae created so far can be fond, mostly in Croatian langage, at the following address: math/ In this paper we present the part of that edcational material related to the Gassian and mean cratres of a reglar srface, which has been translated into English.. Mathematica isalisations of Gassian and mean cratres For ftre strctral engineers it is important to hae the knowledge of the Gassian and mean cratres. For example: Tensile fabric strctre (e.g. membrane roof) in a niform state of tensile prestress behaes like a soap film stretched oer a wire which is bent in a shape of a closed space cre. Soap film assmes a form which has the minimal area relatie to all other srfaces stretched oer the same wire; this srface is therefore called minimal srface. It can be shown that mean cratre anishes at each point of that srface. The project has been spported by the Ministry of Science and Technology of the Repblic of Croatia since /3.

2 .. Gassian and mean cratre of a reglar srface A reglar srface Φ R 3 is the set of points whose position ectors are the ales of one-to-one continos ector-aled fnction r : U R 3 r(, ) = (x(, ), y(, ), z(, )) () where U R is open and connected, x, y, z : U R are differentiable fnctions and (, ) U, r (, ) r (, ). () At each point of a reglar srface a niqe tangent plane and a normal ector exist. The nit normal ector n at the point with a position ector r(, ) is gien by the following formla: n = r r r r. (3) The qadratic form Ed + Fdd + Gd, where E = r r, F = r r and G = r r, is called the first fndamental form of Φ. The qadratic form Ld + Mdd + Nd, where L = n r, M = n r and N = n r, is called the second fndamental form of Φ. The Gassian cratre K and the mean cratre H of a srface Φ are fnctions K, H : U R gien by the following formlas: K = LN M EG F, EN FM + GL H = (EG F. (4) ).. Visalisations by sing Mathematica The geometrical interpretations of normal, Gassian and mean cratres of a reglar srface can be fond at the following address: math/links/sonja/gasseng/gasseng.html According to Eq. 4 we can define (in the program Mathematica) the fnctions gcratre and mcratre [3, p.394] which calclate the Gassian and mean cratres at each point of a reglar srface. These fnctions enable s to plot the graphs of the Gassian and mean cratres of reglar srfaces and to color srfaces with colors which depend of that cratres. For the following isalizations we sed the periodical Mathematica color fnction He (period ). In black-white print this color-fnction is defined in the following way : He is a color-fnction with ales on a color-spectrm. It is clear that in black-white print in this paper, which for example does not differentiate orange from ble, the distinction of isal data gien by the fnction He is decreased. Een in sch conditions the fnction He can be applied to show some properties of Gassian and mean cratres.

3 Example The parametric eqations of an ellipsoid with a center at the origin O(,, ) are: x(, ) = a cossin, y(, ) = b sin sin, z(, ) = c cos, (, ) [, ] (, ) where a, b, c R +. In Figre we show the nit sphere with radii a = b = c = (Fig. a), the oblate ellipsoid with axes lengths a = b = 4, c = (Fig. b) and the ellipsoid with axes lengths a=.5, b=4, c= (Fig. c) colored by the fnction He[3gcrate] and the graphs of their Gassian cratres (Fig. a, Fig. b, Fig. c ). Figre a Figre b Figre c K(,) K(,) K(,) Figre a Figre b Figre c 3 Example The parametric eqations of the circlar helicoid are: x(, ) = a cos, y(, ) = a cos, z(, ) = b, where a, b R \ {}. In Figre we show the circlar helicoid (a =, b=.5) oer the domain [ 4 7 3, ] [, ] colored by the fnctions He[4gcratre] (Fig. a) and He[mcratre] (Fig. c), as well as the graphs of its Gassian (Fig. b) and mean (Fig. d) cratres. It is clear from Fig. c and Fig. d that it is a minimal srface. K(,) - H(,) Figre a Figre b Figre c Figre d -

4 Example 3 The parametric eqations of the presented hyperbolic paraboloid are: x(, ) =, y(, ) =, z(, ) =, (, ) [, ] [, ]. In Figre 3 we show this paraboloid colored by the fnctions He[gcratre] (Fig. 3a) and He[mcratre] (Fig. 3c), as well as the graphs of its Gassian (Fig. 3b) and mean (Fig. 3d) cratres. K(,) H(,) Figre 3a Figre 3b Figre 3c Figre 3d Example 4 The parametric eqations of the presented 3rd degree parabolic conoid are: x(, ) =, y(, ) =, z(, ) =.5( 3 + 3), (, ) [, 5] [, ]. In Figre 4 we show this conoid colored by the fnctions He[gcratre] (Fig. 4a) and He[mcratre] (Fig. 4c), as well as the graphs of its Gassian (Fig. 4b) and mean (Fig. 4d) cratres. K(,) H(,) Figre 4b Figre 4a Figre 4c Figre 4d Example 5 The parametric eqations of the presented monkey saddle are: x(, ) =, y(, ) =, z(, ) = 3, (, ) [.8,.8] [.8,.8]. In Figre 5 we show this monkey saddle colored by the fnctions He[gcratre] (Fig. 5a) and He[mcratre] (Fig. 5c), as well as the graphs of its Gassian (Fig. 5b) and mean (Fig. 5d) cratres.

5 K(,) H(,) Figre 5a Figre 5b Figre 5c Figre 5d Example 6 The parametric eqations of the presented srface are: 5 3 x(, ) =, y(, ) =, z(, ) = sin sin, (, ) [, ] [, ]. In Figre 6 we show this srface colored by the fnctions He[gcratre] (Fig. 6a) and He[mcratre] (Fig. 6c), as well as the graphs of its Gassian (Fig. 6b) and mean (Fig. 6d) cratres. K(,) H(,) - - Figre 6a 3. Figre 6b Figre 6c Figre 6d webmathematica As it is well known web is based on client/serer architectre. When a ser wants to see some web page on his browser (Internet explorer, Opera, Mozilla etc), the browser (client) sends reqirement to the respectie serer to display that page. Then the serer sends the content of the page to the client which shows the page to the ser. The langage for designing web pages is Hyper Text Markp Langage (HTML). This langage does not spport interactie soltions of mathematically formlated problems and interactie isalisations of reslts. To be able to sole mathematical problems it is necessary to se some of specialised programs, sch as Mathematica. Bt the program Mathematica can not be directly actiated by HTML. Therefore, HTML serer has to be connected with Mathematica and it is done with the program webmathematica. In other words, webmathematica bridges web serer and the program Mathematica which enables interactie calclations and isalisations on web pages. The procedre of interactie commnication is the following: - A ser feeds data for a certain mathematical problem into his client compter. - The display of the page with reslts (nmerical, symbolic or graphical) is reqired from web serer.

6 - Web serer throgh webmathematica actiates the program Mathematica which prodces reslts and forwards them to webmathematica. Then webmathematica sends the reslts to web serer. - Web serer sends the page with the reslts to the client which displays web page to the ser. In order to be able to se webmathematica, it is necessary to install webmathematica and Mathematica on the compter with web serer. The web pages on serer hae to be written in extended HTML which is defined by the rles of webmathematica. In Fig. 7 we show the print-screen of web page powered by webmathematica which we designed within IT project mentioned in the introdction. A ser can write his inpts in white rectangles. Visalize is the command btton to start interactie commnication. The reslts (LieGraphics3D on compter) are shown in Fig. 8. Figre 7

7 Figre 8: The print-screen of web page with reslts.

8 4. Conclsion In teaching geometry Mathematica and can be sed for designing diersified and highqality edcational material. Moreoer, it is an ideal program for connecting the content of geometrical and mathematical sbjects. New technology webmathemaica opens the door to interactie compting and isalization of data directly from the ser s web proider. We hope that the presented edcation material, created for the first year stdents, wold improe stdents nderstanding of the terms related to the normal, Gassian and mean cratres of a reglar srface. References [] Benić V., Gorjanc S., 3, Compting in Geometrical Edcation at the Faclty of Ciil Engineering in Zagreb, pp.77-83, Proceeding of st Symposim on Compting in Engineering, Zagreb [] Goetz A., 97, Introdction to Differential Geometry. Addison Wesley, Reading, Massachsetts [3] Gray A., 998, Modern Differential Geometry of Cres and Srfaces with Mathematica. CRC Press, Boca Raton [4] Wickham-Jones T.,, webmathematica: User Gide. Wolfram Research Inc., pdf format [5] Wolfram S., 993, Mathematica. Addison Wesley