his app can be opened directly in the geogebra site using this link: https://www.geogebra.org/classic/m9jt3swa
The left side presents an Argand (complex) plane, with three sets of points, each with a different color.-In black, 4 points can be placed freely. If you want, you can select coordinates without many decimals, sticking on the corners of the grid. These points are joined by three segments between the first one (zo1) and the remaining three points.-In blue, a set of 4 points represents the initial black set after experiencing some transformations which will be explained below.-In red, a set of 6 points represents the values for the cross-ratios that can be established rotating the ordering of the 4 initial points. The value of the usual cross-rate is also presented: cr=(z1-z3)(z2-z4)/{(z1-z4)(z2-z3). These six points are joined by several red segments in a way that highlights their geometric relationships.There are two additional points, one (moeb) in green color, which applies a certain Möbius transformation to the initial seto of points, and the other one (Proj) in blue color, which defines an additional projective transformation of the resulting set.his app can be opened directly in the geogebra site using this link: https://www.geogebra.org/classic/edk7kauf
The right side presents a projective arrangement where a set of four points are drawn on an inclined plane (in violet color). The plane can be tilted by dragging the violet point linked to the origin O by a segment.Two checkboxes allow the selection of two different sets of points-Line: a linear arrangement of four equally spaced points, which can be varied moving the red point.-Sqare: a square formed with four points, which can be also varied moving the red point.his app can be opened directly in the geogebra site using this link: https://www.geogebra.org/classic/dywd2qt3
iT allows to explore the effect of a change in the cross-ratio on a givenj arrangement of points.
You can freely move three points (z1, z2, z3) in blue, joined by segments, as well as a fourth point z4 (in greencolor) which establishes an initial cross-ratio r0. An alternative cross-ratio can be chosen by moving the red point r. The app renders the corresponding fourth point zx (in green, joined by a grren segment to z4.
Selecting a checkbox presents an arrangement of four points joined by a line (Line, Circle or logrithmic spiral) where they are evenly spaced following the first one, which yoiu can move freely. If you move the blue points (z1, z2, z3) as well as the green point z4 so that they coincide with the black ladder of points it is possible to relate the changes with respect to r0 with the position of the point zx in relation to the common curve.