Re: Slur with left and/or right arrow head

[Thomas Morley]

Am Mi., 17. Apr. 2019 um 22:53 Uhr schrieb David Kastrup <address@hidden>:
>
> Thomas Morley <address@hidden> writes:
>
> > Am Mi., 17. Apr. 2019 um 21:52 Uhr schrieb David Kastrup <address@hidden>:
> >>
> >> >> "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

Well, I'd like to understand beziers better, continuing my afford here
also means I shouldn't ignore such things ;)

Thanks,
  Harm