1. 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

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

6. 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)

7. 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)

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

9. 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

10. 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)

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