[color=#999999]This activity belongs to the GeoGebra [i][url=https://www.geogebra.org/m/r2cexbgp]Road Runner (beep, beep)[/url][/i] book. [/color][br][br]Before GeoGebra had a 3D view, it was already possible to visualize polyhedra, curves, and surfaces using projections in the 2D view. Some examples can be seen in this (spanish) course [url=https://geogebra.es/cvg/12/2.html][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] on version 4, in these polyhedra [url=https://www.geogebra.org/m/jt5r98a8#chapter/916689][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], or in these surfaces [url=https://www.geogebra.org/m/jt5r98a8#chapter/916692][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], all created before 2009. Even now that we have the 3D view, such projections can be useful for visualizing objects of higher dimensions, like the hypercube [url=https://www.geogebra.org/m/cSu6jFh9][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url], or for simultaneously viewing the 3D view with other perspectives, as is the case here.[br][br]A three-dimensional point P can be projected in the graphical view as:[br][center](x(P) sen(β) + y(P) cos(β), -x(P) cos(β) sen(α) + y(P) sen(β) sen(α) + z(P) cos(α))[/center]where α and β are the inclination and rotation angles of the projection.[br][br]If we call the list "base":[br][center]base = {(sen(α), -cos(α) sen(β)), (cos(α), sen(α) sen(β)), (0, cos(β))}[/center]then, a parametric curve c(t) = {fx(t), fy(t), fz(z)} can be projected as:[br][center]proy = fx(t) base(1) + fy(t) base(2) + fz(t) base(3)[/center]and a point C = Point(c, p) = c(p) on the curve c(t) can be projected as:[br][center]ProyC = x(C) base(1) + y(C) base(2) + z(C) base(3)[/center]The following construction projects in this way the spatial curve c(t) = (cos(t), sin(t), cos(2t)) and a point C=c([b][color=#ff7700]p[/color][/b]) on it (below, with a white background), in the 2D graphical view (above, with a black background). Note that [b][color=#ff7700]p[/color][/b] is the parameter of movement for C on the curve c(t), meaning [b][color=#ff7700]p[/color][/b] always varies between 0 and 1.[br]Move the sliders to observe their effect.

[color=#999999][color=#999999]Author of the construction of GeoGebra: [color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url][/color][/color][/color]