[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 representation of the vectors of the Frenet frame suggests using them to easily turn a curve into a tube that follows the same path. Unfortunately, GeoGebra does not dynamically execute (i.e., it does not update the parameter value) the Tangent and CurvatureVector commands when used in the definition of a surface. This prevents their use, so to create the tube, we have no choice but to use differentiation formulas, which, in addition to complicating algebraic expressions, consume many resources during execution, slowing it down and losing fluidity.[br][br]An alternative for visualizing the tube is to dispense with the Surface command and represent the tube as a sequence of circles. The result is visually quite acceptable and, above all, does not lose agility in representation.[br][br]For example, to create a tube with a radius of 0.4 around the curve c(t), with t varying between t1 and t2, as shown in the construction (Tube1), the instruction is simply:[br][br] Sequence(Circle(c(t), 0.4, Tangent(c(t), c)), t, t1, t2, 0.01)[br][br]Alternatively (Tube2), we can replace the circles with separate cylinders (to avoid overlaps):[br][br] Sequence(Cylinder(c(t), c(t + 0.05), 0.4), t, t1, t2, 0.1)

[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]