[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]The Frenet frame [url=https://en.wikipedia.org/wiki/Differentiable_curve#Frenet_frame][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] (or [b][color=#cc0000]T[/color][color=#6aa84f]N[/color][color=#0000ff]B [/color][/b]frame) associated with a point C on a curve c(t) is a local reference system formed by three orthonormal vectors [b][color=#cc0000]T[/color][/b], [b][color=#6aa84f]N[/color][/b] and [b][color=#0000ff]B[/color][/b], which are, respectively, the [color=#cc0000]Tangent[/color], [color=#6aa84f]Normal[/color], and [color=#0000ff]Binormal [/color]to the curve at that point.[br][br]If C represents a physical particle moving in space, [b][color=#cc0000]T[/color][/b] would be the unit vector corresponding to velocity, and [b][color=#6aa84f]N[/color][/b] would be the unit vector corresponding to normal acceleration. [br][br]In our "move towards the mountain" (as Muhammad would say), this frame will be of great help. In fact, the colors red, green, and blue with which we have colored each letter are a hint of what we are going to do. Since we cannot change the 3D scene to show what is seen from point C, we will move C to the coordinate center (0, 0, 0) and rotate the Frenet frame of C until [b][color=#cc0000]T[/color][/b] coincides with the [color=#cc0000]xAxis[/color], [b][color=#6aa84f]N[/color][/b] with the [color=#6aa84f]yAxis[/color] and [b][color=#0000ff]B[/color][/b] with the [color=#0000ff]zAxis[/color]. In summary, we will change [url=https://www.geogebra.org/m/z5d7n5n4][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url] the global reference system to a local reference system. [br][br]If [b][color=#ff7700]p[/color][/b] is the (path) parameter of C on the curve c(t), that is, if C = c([b][color=#ff7700]p[/color][/b]), then the [b][color=#cc0000]T[/color][color=#6aa84f]N[/color][color=#0000ff]B[/color][/b] vectors are usually calculated as: [br][center][math]\Big\begin{array}{rc}\vec{T}=\frac{c'(p)}{||c'(p)||}\;\;\;\;\;\;\;\;\;\vec{B}=\frac{c'(p)\otimes c''(p)}{||c'(p)\otimes c''(p)||}\;\;\;\;\;\;\;\;\;\vec{N}=\vec{B}(p)\otimes \vec{T}(p)\end{array}[/math][/center]However, we have not followed these formulas, but have taken advantage of GeoGebra commands to avoid the use of differentiation, thus streamlining the representation. We have defined the vectors of the frame, named [b][color=#cc0000]vT[/color][/b], [color=#6aa84f][b]vN[/b][/color], and [b][color=#0000ff]vB[/color][/b], as follows:[br][br] [b][color=#cc0000]vT[/color][/b] = UnitVector(Tangent(C, c))[br] [color=#6aa84f][b]vN[/b][/color] = [code]UnitVector(CurvatureVector[/code](C, c))[br] [b][color=#0000ff]vB[/color][/b] = [code]UnitVector[/code](vT ⊗ vN)

[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]