1. A Software for Visualization and Animation in Mathematics Eberhard Malkowsky Vesna Veličković Seminar o nastavi matematike i računarstva Niš, 14, 15 February, 2003

2. Introduction Visualization and animation play an important role in the understanding of mathematics. Therefore they should extensively be used in teaching. Here we present our own software for visualization and animation in mathematics.

3. The Software We use our own software [1]. The software is aimed at visualizing mathematics and to support teaching and research. It has applications to the natural and engineering sciences. It is open, that is the source files are available to the users. Thus it can be extended. It uses OOP. The programming language is PASCAL.

4. References [1] E. Malkowsky, W. Nickel, Computergrafik in der Differentialgeometrie, Vieweg-Verlag Braunschweig/Wiesbaden,1993

5. The Main Principles of the Software 1.Strict separation of geometry and the technique of drawing 2.Line graphics 3.Central projection 4.Independent visibility check

6. Lines of Intersection

7. Level Lines

8. Central Projection We use central projection with a free choice of parameters.

9. Central projection of a cylinder

10. Independent Visibility Check Independent procedure Check Immediate analytical test Interpolation to close occasional gaps Dotting or omitting invisible parts

11. Dandelin’s Spheres

13. Distance The usual distance of O and P is (1). It is called Euclidean distance.

14. Circles The points of a circle have a constant distance from its centre. If we take this as a definition of a circle then its shape depends on the choice of the distance function.

15. A Square Another way of measuring the distance of O and P is shown in the figure. The formula is given in (2).

16. Squares Circles with respect to the distance function d 1.

17. Generalizations More generally we consider distance functions d q with q>0 given by

18. Astroids Astroids are circles with respect to the distance function d q for q=2/3.

19. Circles Circles with respect to different distance functions.

20. 2D Stars

21. 3D Stars

22. Introducing a Metric If X is a set, Y a metric space with d and That is is a one-to-one map Then a metric can be defined on X by

23. A Map From a Rectangle Onto the Plane maps The map Let

24. Concentric circles in R and their images under f

25. Images of circles under f

32. Chain It is a well known fact that, if we hold a chain at its ends and let it hang freely then it hangs in a shape which looks very much like a quadratic parabola.

33. Catenary The curve the chain describes is called a catenary.

34. Catenoid If we rotate a catenary around an axis as shown in the figure we will get a surface which is called a catenoid.

35. Catenoid

36. Minimal Surfaces Let two curves be given. A minimal surface is a surface which has minimal surface area of all the surfaces bounded by the given curves. A Catenoid is a minimal surface. Its boundary are two circles.A Catenoid is a minimal surface

37. Helikoid If we connect each point of helix with the point of z-axis at the same height, we will get a helikoid (spiral stair, strip). A helikoid is a minimal surface. Its boundary curves are two helices or a helix and a straight line.

38. From a Catenary to a Helikoid There is a continuous transformation from a catenoid to a helikoid. All surfaces in between are minimal surfaces.from a catenoid to a helikoid

39. Geometric Construction of a Double Egg Line

40. Geometric Construction of a Rosette

41. Geometric Construction of an Epicycloid

42. Deformations of Torus to Double Egg (Torus to Double Egg)(Torus to Double Egg) Torus (Torus)(Torus) Torus to Lemniscate (Torus to Lemniscate)(Torus to Lemniscate)

43. Rotation Rotation of a potential surface (Rotation of a potential surface)(Rotation of a potential surface)