[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]In the construction of the previous activity, the point C=c(p) is the point (path) parameter [b][color=#ff7700]p[/color][/b] of the curve c(t). Now, while the parameter [b][color=#ff7700]p[/color][/b] always varies between 0 and 1, the parameter t of the curve c(t) can vary between two arbitrary values t1 and t2 (for example, between -π and 3π).[br][br]In this GeoGebra book, we will need later on (to calculate the normal vector to a surface at a given point) to find the value of t that corresponds to the value of [b][b][color=#ff7700]p[/color][/b][/b]. That is, we will need a [i]linear transformation[/i] from the interval [0, 1] to the interval [t1, t2]. Let [b]h(x) = a x + b[/b] be this transformation. Then it must hold that [b]t1 = a 0 + b[/b] and [b]2 = a 1 + b[/b]. Therefore, the sought transformation is [b]h(x) = (t2 - t1) x + t1[/b]. This transformation maps the value [b][b][color=#ff7700]p[/color][/b][/b] to the value [b](t2 - t1) [b][color=#ff7700]p[/color][/b] + t1[/b]. [br][br]Let's see a faster and more natural way to reach the same result. When [b][color=#ff7700]p[/color][/b] is 0, (1 - [b][color=#ff7700]p[/color][/b]) is 1, and vice versa. So the transformation:[br][center][b](1 - [b][color=#ff7700]p[/color][/b]) [/b]t1 [b]+ [b][color=#ff7700]p[/color][/b] [/b]t2[/center]will continuously transition from the value t1 (when [b][b][color=#ff7700]p[/color][/b][/b]=0) to the value t2 (when [b][b][color=#ff7700]p[/color][/b][/b]=1). This reasoning not only applies to values in one dimension but is equally valid for any dimension. If A and B are points (of any dimension), the following transformation will continuously transition from point A to point B:[br][center][b](1 - [b][color=#ff7700]p[/color][/b]) [/b]A + [b][b][color=#ff7700]p[/color][/b][/b] B[/center]We can replace points A and B with curves or surfaces. This transformation is fundamental for continuously transitioning from one object to another. It's so important that it has its own name: [i]linear homotopy [/i][url=https://en.wikipedia.org/wiki/Homotopy][img]https://www.geogebra.org/resource/scjbyz2p/0tuzuVw455vxurEw/material-scjbyz2p.png[/img][/url]. [br][br]In the following construction, you can see some examples (we've also used homotopy to transition from one color to another in them). You can find other examples of homotopy in these GeoGebra books by [url=https://www.geogebra.org/m/MSNNQCmE#chapter/284391]José Manuel Arranz[/url] and [url=https://www.geogebra.org/m/DCRzbTbQ#material/KFAPeA7E]Juan Carlos Ponce Campuzano[/url]. Homotopy is a widely used resource in GeoGebra to naturally connect two curves, creating a surface that joins them (like the glass in the construction, with an elliptical base and circular mouth). In this book, we will only use it, as we've already mentioned, to map the interval [0, 1] to the interval [t1, t2].

[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]