[color=#999999][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][/color][br]We are now ready to change the point of view. That is, to replace the reference system of the Cartesian axes, with canonical vectors [b][color=#cc0000]i[/color][/b][b][color=#6aa84f] j[/color][/b][b][color=#0000ff] k[/color][/b], by the local reference system [b][color=#cc0000]T[/color][color=#6aa84f]N[/color][color=#0000ff]B[/color][/b] of the Frenet frame. You can find everything related to a change of 3D reference system in this (spanish) GeoGebra book [url=https://www.geogebra.org/m/z5d7n5n4][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url].[br][br]At the beginning, in the construction, the [b]Point view[/b] checkbox is activated. Now, the point that moves along the curve remains fixed in the 3D view, exactly at the coordinate center (0, 0, 0), while the entire curve moves in its place. Note that the same happens in its projection in the 2D view (in this view, we have preserved, on the left, the global projection to appreciate the difference).[br][br]To achieve this transformation, we have performed the following steps:[br][list=1][*]We define the rotation matrix [url=https://www.geogebra.org/m/z5d7n5n4#material/xh9zencf][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] [b]M = Mz My Mx[/b], where:[br][b]Mz[/b] is the rotation matrix of angle -arg([b][color=#cc0000]T[/color][/b]) around the [color=#0000ff]zAxis[/color].[br][b]My[/b] is the rotation matrix of angle alt([b][color=#cc0000]T[/color][/b]) around the [color=#6aa84f]yAxis[/color].[br][b]Mx[/b] is the rotation matrix of angle If(arg([b]aux[/b]) < 0, π + alt([b]aux[/b]), -alt([b]aux[/b])) around the [color=#cc0000]xAxis[/color], where [b]aux[/b] is the auxiliary vector defined as: [br][center]Rotate(Rotate([b][color=#6aa84f]N[/color][/b], -arg([b][color=#cc0000]T[/color][/b]), [color=#0000ff]zAxis[/color]), alt([b][color=#cc0000]T[/color][/b]), [color=#6aa84f]yAxis[/color])[/center]In case we travel along the surface, we replace [b][color=#6aa84f]N[/color][/b] with [color=#00ffff][b]normal [/b][/color]⊗[b][color=#6aa84f] [b][color=#cc0000]T[/color][/b][/color][/b].[br][br][/*][*]We subject all points [b]P[/b] that we want to display to the isometry, composed of rotation and translation, which transforms P into [b]P' = (P - C) M[/b]. For example, the shown curve, parameterized by f(t), becomes:[br][center]c' = Curve((f(t) - C) M, t, t1, t2)[/center]In the case of a surface sup, parameterized by F(u, v), we would obtain:[br][center]sup' = Surface((F(u, v) - C) M, u, u1, u2, v, v1, v2)[/center]Note that this transformation brings C to the origin of coordinates, since C - C = (0, 0, 0). Likewise, it aligns the [b][color=#cc0000]T[/color][color=#6aa84f]N[/color][color=#0000ff]B[/color][/b] vectors with the canonical base [b][color=#cc0000]i[/color][/b][b][color=#6aa84f] j[/color][/b][b][color=#0000ff] k[/color][/b] (in case of choosing to travel along the surface, it will be the [color=#00ffff][b]normal [/b][/color]vector that coincides with [b][color=#0000ff]k[/color][/b]).[/*][/list][br]Now, we just need to adjust the scenario to the local point of view. First, observe in the construction that the curve does not appear as usual in the 3D view. This happens because we have changed the representation mode of this scenario, switching from orthographic projection (by default) to [b]perspective projection[/b] (at a distance of 250), which will help represent the local vision.[br][br]To align the observer's gaze direction with the movement direction (remembering that vector [b][color=#cc0000]T[/color][/b] now replaces vector [color=#cc0000][b]i[/b][/color]) when the [b]Surf [/b]and [b]Surf (front)[/b] checkboxes are activated, the following instruction is executed: SetViewDirection(Vector((1,0,0)), false).[br][br]Finally, we reduce the scenario until the coordinate origin is just in front, when the [b]Surf [/b]checkbox is activated:[br][center]ZoomIn(-0.45, -2, -0.7, 2, 2, 1)[/center]or even until it is out of sight, as if we were right above it, looking straight ahead, when the [b]Surf (front)[/b] checkbox is activated:[br][center]ZoomIn(-0.3, -2, -0.7, 2, 2, 1)[/center]

[color=#999999][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][/color]