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

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

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

Usual software DPGRAPH DERIVE MATLAB MATHEMATICA

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

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

Meaning of parameters: 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)

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

Several coordinate systems: Rectangular, cylindrical, spherical, ... 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

Matlab: Several interesting commands Mathematica is a good tool also 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)

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

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