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Static and dynamic surfaces representation

Learn to visualize lines, surfaces, and 3-D curves using MATLAB, Mathematica, and DPGRAPH. Understand implicit expressions, transparencies, and real-time scrolling for effective mathematical representation.

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Static and dynamic surfaces representation

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  1. Gliwice July 25-29, 2005 Static and dynamic surfaces representation • Matilde P. Legua • Jose A. Moraño • Luis M. Sánchez-Ruiz

  2. Introduction • Visualization of lines and surfaces • Geometrical application problems • To teach mathematical concepts when we are working with scalar fields • To help to locate the regions which we should pay special attention • The aim is to take advantages of capabilities of commonly used mathematical software to get the proper representation

  3. Surfaces representation • Implicit expressions & Transparencies • Surfaces defined by inequalities • Real Time scrolling • Graphs in movement • Parametric surfaces • More about surfaces representation • 3-D curves

  4. Usual software • DPGRAPH • DERIVE • MATLAB • MATHEMATICA

  5. Implicit expressions & Transparencies Visualize Intersections: • DERIVE • - Explicit expressions • Transparencies: See_through() • DPGRAPH • - Implicit expressions • Gradual Transparencies

  6. Surfaces defined by inequalities Defined regions: • DERIVE • Several surfaces but without inequalities • Other possibilities: Revolution, lines, ... DPGRAPH -  is & between inequalities RevY_Iplot(x=sin(y/2) z=3, y, -PI, PI)

  7. Real time scrolling Meaning of parameters: DPGRAPH: Scroll bar modifies the parameter in real time DERIVE: Can use Vector( ) command. VECTOR(RevZ_IPlot(z = 5(1-x^2/k^2)^(1/2)  y=0, 0  x  k), k,1, 5)

  8. Graphs in movement Meaning of parameters by dynamic representations: DPGRAPH: Including another dimension (time) DERIVE:time also exists

  9. Parametric surfaces Several coordinate systems: Rectangular, cylindrical, spherical, ... DPGRAPH: Trace a graph only over a rectangle RECTANGULAR (u,v,–2*COS(1.75*u)+SIN(v)) It can plot surfaces whose domain is not a rectangle RECTANGULAR (3/4*(v-u),1/4*(u+3*v),3-v/2+u/6) DERIVE: It has more types of coordinate system

  10. More about surfaces representation Matlab: Several interesting commands • mesh, surf • meshc and surfc (with level lines) • Subplot (distributing graphs) • Using OpenGL renderer (transparencies) • MOVIE (dynamical graphs) • As a drawback (Surfaces in implicit form are not admitted) Mathematica is a good tool also • Plot3D, ParametricPlot3D • Hidden Surface (transparencies) • Animate (Variation on a parameter) or ShowAnimation (sequence of graphs)

  11. 3D-Curves DPGRAPH: Parametrical coordinates allow visualize 3D curves RECTANGULAR (2*SIN(4*u),2*COS(4*u),u) DERIVE, MATLAB and MATHEMATICA provide similar graphs

  12. Conclusion • Sofisticated mathematical software should get better results however DPGRAPH seems to be more practical and easier when we are • working with implicit forms • building dynamic surfaces • getting transparencies • needing memory requirements • We must have very clear that it is not a CAS

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