Projections
This activity belongs to the GeoGebra Road Runner (beep, beep) book.
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 
on version 4, in these polyhedra , or in these surfaces , 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 , or for simultaneously viewing the 3D view with other perspectives, as is the case here.
A three-dimensional point P can be projected in the graphical view as:
on version 4, in these polyhedra , or in these surfaces , 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 , or for simultaneously viewing the 3D view with other perspectives, as is the case here.
A three-dimensional point P can be projected in the graphical view as:
(x(P) sen(β) + y(P) cos(β), -x(P) cos(β) sen(α) + y(P) sen(β) sen(α) + z(P) cos(α))
where α and β are the inclination and rotation angles of the projection. If we call the list "base":base = {(sen(α), -cos(α) sen(β)), (cos(α), sen(α) sen(β)), (0, cos(β))}
then, a parametric curve c(t) = {fx(t), fy(t), fz(z)} can be projected as:proy = fx(t) base(1) + fy(t) base(2) + fz(t) base(3)
and a point C = Point(c, p) = c(p) on the curve c(t) can be projected as:ProyC = x(C) base(1) + y(C) base(2) + z(C) base(3)
The following construction projects in this way the spatial curve c(t) = (cos(t), sin(t), cos(2t)) and a point C=c(p) on it (below, with a white background), in the 2D graphical view (above, with a black background). Note that p is the parameter of movement for C on the curve c(t), meaning p always varies between 0 and 1. Move the sliders to observe their effect.Author of the construction of GeoGebra: Rafael Losada