Re: Slur with left and/or right arrow head

[David Kastrup]

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