Thomas Morley <thomasmorle...@gmail.com> writes: > Am Mi., 17. Apr. 2019 um 21:52 Uhr schrieb David Kastrup <d...@gnu.org>: >> >> >> "time". >> > >> > Well, actually I read that in some papers trying to explain beziers, >> > already. >> > But what means "time"? >> > I'm arranging pixels on a screen, or tell a printer what to print >> > where or draw points and lines with a pencil on a sheet of paper. >> > This may be "time"-consuming lol >> >> Lol to you: it's the drawing time of drawing the curve, so yes, this is >> exactly the meaning assigned to t. It is normalised from 0 to 1 instead >> of measuring it in pencil-seconds. >> >> > But what does "time" means here in the mathematical sense, this part I >> > didn't get yet. >> >> How far you have progressed with drawing the curve. > > Well, as already said, making things visible may help. > If I look at the dotted bezier of my recently posted pdf (the dots are > made by splitting t from 0 to 1 in sixty pieces) then it seems drawing > more or less straight lines takes less effort, i.e. is less > time-consuming than to draw curves, more steeper curves means more > work, i.e. more time. > > Though, I'm used to think of mathematical functions assigning one > x-value to one (or more) y-values, at least if we keep thinking in two > dimensions.
It's not a function of y from x or vice versa: x and y are separate functions of t. In that manner, you can rotate (or otherwise lineary transform) a curve by rotating the control points. A function of y from x is not something you can rotate by 90 degrees, in contrast. Making both x and y separate functions from an artifical parameter t allows not having x or y be different in character. > Thus I still have problems to accept this thinking. It's not a manner of thinking. It's just trying to give an artifical construct in the form of an independent arbitrary parameter that has been arbitrarily normalized from 0 to 1 (and indeed, in LilyPond a normalization from -1 to 1, namely #LEFT to #RIGHT might be better justifiable but diverging from most formulas in literature) some more tangible image/meaning. If you don't find it helpful, you can just forget it. -- David Kastrup _______________________________________________ lilypond-user mailing list lilypond-user@gnu.org https://lists.gnu.org/mailman/listinfo/lilypond-user
