[color=#0000ff][i][color=#0000ff][i][color=#999999]This activity belongs to the GeoGebra book [url=https://www.geogebra.org/m/mes4bgft]The Domain of the Time[/url].[/color][/i][/color][/i][/color][br][br]This animation simulates the motion of a [i]cycloidal pendulum[/i] (or Huygens' pendulum) in [b]real time[/b], neglecting friction. The animation [b]does not use formulas[/b] (neither trigonometry, equations, nor differential calculus), but only makes the necessary variations in the vectors that guide the movement.[br][quote][color=#0000ff][i]The simple pendulum cannot be considered a reliable and uniform time measurement because wide oscillations take longer than smaller ones; with the help of geometry, I found a method, unknown until now, of suspending the pendulum. I have investigated the curvature of a particular curve that lends itself admirably to achieving the desired uniformity. Once I applied this form of suspension to clocks, their operation became so steady and reliable that, after numerous experiments on land and water, it is beyond doubt that these clocks offer the greatest security for astronomy and navigation. The mentioned line is the same as that traced in the air by a nail attached to a wheel when it rotates forward; [b]mathematicians call it the cycloid[/b], and it has been carefully studied for its many other properties. But I studied it for its application to the mentioned time measurement, which I discovered while studying it with purely scientific interest, without suspecting the outcome.[/i][br] [br][/color][right][color=#0000ff] Christian Huygens, [i]Horologium oscillatorium[/i] (1673)[/color][/right][br][/quote]Observe the figure that appears when the construction starts. The masses at points [color=#0000ff]M[/color] and [color=#ff7700]A[/color] are released to fall by their own weight, both on the cycloid (green), generated by a circle of radius [i]r[/i]. As we have seen (and as Huygens discovered), this curve is [i]tautochrone[/i], so both masses have the same period.[br][br]Press the [img]https://www.geogebra.org/resource/hwdawgnn/MmhoDfF5M6lNH9D4/material-hwdawgnn.png[/img] button. You can reposition points [color=#0000ff]M[/color] and [color=#ff7700]A[/color] anywhere on the arc of the cycloid. You'll see that[color=#ff7700] A[/color] crosses the lowest point of the cycloid at the same time as [color=#0000ff]M[/color].[br][br]Also observe that the string holding the masses, with a length of 4[i]r[/i], in this cycloidal pendulum curves along the yellow cycloid (generated by a circle with radius [i]r[/i]), winding and unwinding, so that its end traces the green cycloid (or an arc of it).[br][list][*]Note: The green curve, traced by point [color=#0000ff]M[/color] [i][/i][i]as it unwinds or winds on the yellow curve[/i], is called the [i][color=#6aa84f][b]involute [/b][/color][/i](or [i]evolvent[/i][i][/i]) of the yellow one. In the construction, you can observe that the [color=#6aa84f][b]involute [/b][/color]of the cycloid is the same cycloid (yellow) from which it originates, just translated. [br][br]Another way to express this is that the curve gathering the [i]centers of curvature[/i] (the [color=#f1c232][b][i]evolute[/i][/b][/color]) of the green cycloid is, when translated, the same cycloid (yellow). Activate the Osculating Circle checkbox (whose center is, at each instant, the center of curvature of point [color=#0000ff]M[/color]) to verify this. This happens because, for any curve, "the [color=#f1c232][b]evolute [/b][/color]of the [color=#6aa84f][b]involute[/b][/color]"[color=#6aa84f][b] [/b][/color] is the original curve (yellow). [/*][/list]

[b]SCRIPT FOR SLIDER anima[/b][br][br][color=#cc0000][color=#cc0000]# Calculate the elapsed seconds dt; add one second if t1(1) < tt[/color][/color][br][color=#999999]SetValue(tt, t1(1))[br]SetValue(t1, First(GetTime(), 3))[br]SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)[/color][br][br][color=#cc0000]# Move M and A[/color][br][color=#999999][color=#999999]SetValue[/color](v, vt + dt gt)[/color][br][color=#999999][color=#999999]SetValue[/color](vA, vtA + dt gtA)[br][color=#999999]SetValue[/color](M, M + dt v)[/color][br][color=#999999][color=#999999]SetValue[/color](A, A + dt vA)[/color][br][br][br][br][color=#999999][color=#999999][color=#0000ff][color=#0000ff][color=#999999][color=#999999]Author of the activity and GeoGebra construction: [/color][/color][/color][color=#0000ff][color=#999999][color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url].[/color][/color][/color][/color][/color][/color]