Generating-Malta03

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Gliwice July 25-29, 2005

Static and dynamic surfaces representation

• Matilde P. Legua
• Jose A. Moraño
• Luis M. Sánchez-Ruiz

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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

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Surfaces representation

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

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Usual software

• DPGRAPH
• DERIVE
• MATLAB
• MATHEMATICA

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Implicit expressions Transparencies

Visualize Intersections

• DERIVE
• - Explicit expressions
• Transparencies See_through()

• DPGRAPH
• - Implicit expressions
• Gradual Transparencies

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Surfaces defined by inequalities

Defined regions

• DERIVE
• Several surfaces but without inequalities
• Other possibilities Revolution, lines, ...

DPGRAPH - ? is between inequalities
RevY_Iplot(xsin(y/2) ? z3, y, -PI, PI)
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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-x2/k2)(1/2) ? y0, 0
? x ? k), k,1, 5)
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Graphs in movement

Meaning of parameters by dynamic representations
DPGRAPH Including another dimension (time)
DERIVE time also exists
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Parametric surfaces

Several coordinate systems Rectangular,
cylindrical, spherical, ...
DPGRAPH Trace a graph only over a rectangle
RECTANGULAR (u,v,2COS(1.75u)SIN(v))
It can plot surfaces whose domain is not a
rectangle
RECTANGULAR (3/4(v-u),1/4(u3v),3-v/2u/6)
DERIVE It has more types of coordinate system
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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)

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3D-Curves

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