[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][right][i]If the mountain won't come to Muhammad, then Muhammad must go to the mountain.[/i][/right]The main objective of this GeoGebra book is to show various ways of representing curves and surfaces, including a local perspective, as if we were navigating (surfing) along the path. [br][br]Using GeoGebra's 3D view to display what a local observer, a camera, would see while traveling through space entails, at the time of writing this, a fundamental difficulty. The 3D view always displays a parallelepiped with edges parallel to the Cartesian axes. It is not possible to define an arbitrary orthoedric space; instead, an interval between minimum and maximum values must be established for each coordinate. This means that out of the infinite number of rectangles sharing two given opposite vertices, GeoGebra only accepts the one with edges parallel to the axes as the base of the 3D scene.[br][br]For example, it is not possible to set the green parallelepiped, with opposite corners A(0, 0, 0) and B(3, 4, 1), and another vertex at C(4, 2, 0), as the scene. Attempting to do so, as shown in the following construction, results in creating the red orthoedron, with corners A and B, but with edges parallel to the axes.[br][br]To overcome this difficulty, we will follow the aforementioned proverb: keep the camera fixed and dynamically move the figure (curve or surface) in front of the camera. Mathematically (and, I would add, physically, as Galileo would [url=https://en.wikipedia.org/wiki/Galilean_invariance][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url]), this is equivalent to changing the reference system. Additionally, we will make use of four concepts: projection, homotopy, the Frenet frame, and the normal vector to a surface at a given point.

[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]