[b]Visualizing Functions of Two Variables z = f (x, y) onto its xy-projection[/b] [i][b] Scanning the extrema lines in Cartesian und Polar coordinate systems.[/b][/i] Points of intersection of these lines are critical points. Using test point PTest you can determine the types of this points. Coordinates are given in Tables on the right side of the screen. (x,y) - independent scanning extrema coordinate in Cartesian coordinate system und (r;φ) - independent scanning extrema coordinate in Polar coordinate system. The position of the pole: (r;φ)_0 is a movable. Moving the pole can be adjusted position of the critical points. Intersection of lines near (r;φ)_0- don't take into account. Here is used 2 options to generate these lines: -first (methode ExtremaByLocus) ploted the extreme colored lines max and min. Maximum points are at the intersection of 'red'/'red' lines und minimum points- at the intersection of 'blue'/'blue' lines. Saddle points-at the intersections 'red'-'blue' lines. -second (GeoGebra Command –Extremum) ploted the monochrome lines. [i][b]Clarify coordinates:[/b][/i] Switch checkbox ''precise setting of extrema''. Another points appears (in the intervals of δ-radius) around the initial points. Using GeoGebra Command -Max and Min are computed more accurate minimum and maximum. Revised values are given in Tables. [i] [b]Contour picture.[/b][/i] In the section 'plotting a two dimensional contour graph' you can construct contour lines sequentially from the minimum to the maximum level. Their values are specified with z-values. It is possible to draw a more detailed contour picture. Density of these contour lines (lines with the same z-coordinate) can be set with regler ω_c. Type of critical points can be determine using the curvature of the contour lines. Sequence of closed contours determines the point of local extreme. Region bounded by the four lines (looks very similar to a hyperbola) defines a saddle point. Regions with a different arrangement of lines may be qualified as inflection points in the two-dimensional case. Increasing the frequency ω_c you could to examine in more details the functions region with a small changes of coordinates z. All parameters can be changed in the settings section.