Am 19.09.2016 um 00:32 schrieb David Kastrup:
> Urs Liska <> writes:
>> Am 18.09.2016 um 20:54 schrieb David Kastrup:
>>> Do you know how to split a bezier at a given ratio into equivalent
>>> beziers?  It's a comparatively simple operation and I think it's already
>>> somewhere in the C++ code though without access from Scheme.
>> No, but I should be able to figure it out (if noone sends a pointer
>> before I manage to do so).
> Well, METAFONT uses the notation
> a[z1, z2]
> for z1 + a*[z2-z1], mapping a range of 0..1 for a linearly between z1
> and z2.
> If we have points z1, z2, z3, z4 defining a Bezier, then the two split
> beziers are defined with the points
> z1, a[z1, z2], a[a[z1, z2], a[z2, z3]], a[a[a[z1, z2], a[z2, z3]],
>                                           a[a[z2, z3], a[z3, z4]]
> and
> a[a[a[z1, z2], a[z2, z3]], a[a[z2, z3], a[z3, z4]]],
> a[a[z2, z3], a[z3, z4]], a[z3, z4], z4
> Basically, calculation of a point a on an n-grade Bezier is done using a
> recursive formula to depth n, and keeping the intermediate results will
> give you the control points for the Bezier curves split at that point.

I think before diving into that I share what I currently have, so we may
discuss which approach should actually be continued.

The attached solution does the following:

  * Apply offsets for the start/end points and to the second and
    second-to-last control-points, based on the original points of the
    non-compound slur
  * Add an inflection point, which is specified as a point between the
    (actual) end points of the slur, given X and Y ratios (as a pair of
    numbers between 0 and 1
  * Determine the length and slope of the line going through the
    inflection point.
      o Currently this is done through specifying one point relative to
        the inflection point and mirroring it symmetrically
      o Instead I'd like to specify an angle and a length.
      o It would be nice to have the angle relative to the slope of the
        slur as a whole, but that may not be a good idea, as we have
        actually two separate lines with different slopes
      o Length should be given as a ratio, presumable relative to the
        length of the line between the inflection point and the
        respective end point.
      o There should be one optional argument to enforce symmetrical
        points here.

BTW I've spiced up the control points display a bit. I hope it's

I would like to integrate this with Janek's \shapeII functions
as I think there'll be quite some code (and interface?) that can be shared.



\version "2.19.47"
\language "deutsch"
#(define (make-cross-stencil coord col)
   "Draw a cross-stencil at coord."
   (let ((thick 0.1)
         (sz 0.2))
        (- (car coord) sz)
        (- (cdr coord) sz)
        (+ (car coord) sz)
        (+ (cdr coord) sz))
        (- (car coord) sz)
        (+ (cdr coord) sz)
        (+ (car coord) sz)
        (- (cdr coord) sz)))


#(define (connect-dots pt1 pt2 col)
      (car pt1)
      (cdr pt1)
      (car pt2)
      (cdr pt2)))

#(define (mirror-point pt1 pt2)
   "Mirror pt2 against pt1"
    (- (* 2 (car pt1)) (car pt2))
    (- (* 2 (cdr pt1)) (cdr pt2))))

#(define (add-points pt1 pt2)
   "Add two points"
   (cons (+ (car pt1) (car pt2))
     (+ (cdr pt1) (cdr pt2))))

#(define (divide pt1 pt2 ratio)
    ((x1 (car pt1))
     (x2 (car pt2))
     (y1 (cdr pt1))
     (y2 (cdr pt2))
     (xratio (car ratio))
     (yratio (cdr ratio))
     (newcenter    (cons
                    (+ (* (- 1 xratio) x1) (* xratio x2))
                    (+ (* (- 1 yratio) y1) (* yratio y2)))))
    (display newcenter)

#(define (distance pt1 pt2)
    ((square (lambda (x) (* x x))))
      (square (- (car pt1) (car pt2)))
      (square (- (cdr pt1) (cdr pt2)))))))

compoundSlur =
#(define-event-function (options)(ly:context-mod?)
   ; TODO: Change inner-segment to
   ; - a ratio (to the length of the line between cp1 and cp4)
   ; - an angle (relative to the same line cp1--cp2)
      (lambda (grob)
         ((opts (map (lambda (o)
                       (cons (second o) (third o))) (ly:get-context-mods options)))
          ; TODO: factor out requiring/defaulting options
          ;; let offsets default to taking the control-points of the automatic slur
           (let ((prop (assq 'offsets opts)))
             (if prop (cdr prop) '((0 . 0)(0 . 0)(0 . 0)(0 . 0)))))
          ;; let center-ratio default to 0.5/0.5
           (let ((prop (assq 'center-ratio opts)))
             (if prop (cdr prop) '(.5 . .5))))
           (let ((prop (assq 'center-ratio opts)))
             (if prop (cdr prop) 0.5)))
          (inner-segment (assq-ref opts 'inner-segment))

           (let ((prop (assq 'ann opts)))
             (if prop (cdr prop) #f)))
          ;; automatic control points of the non-compound slur
          (cps (ly:slur::calc-control-points grob))
          ;; add offsets to the four control points
          (cp1 (add-points (first cps) (first offsets)))
          (cp2 (add-points (second cps) (second offsets)))
          (cp6 (add-points (third cps) (third offsets)))
          (cp7 (add-points (fourth cps) (fourth offsets)))

          ;; calculate central connecting point and surrounding handles
          (cp4 (divide cp1 cp7 center-ratio))
          (inner-len (distance cp1 cp7))
          (slope (/ (- (cdr cp7) (cdr cp1)) (- (car cp7) (car cp1))))

          (cp5 (add-points cp4 inner-segment))
          (cp3 (mirror-point cp4 cp5))

          (cps1 (list cp1 cp2 cp3 cp4))
          (cps2 (list cp4 cp5 cp6 cp7))
            (ly:grob-set-property! grob 'control-points cps1)
            (ly:slur::print grob)))
            (ly:grob-set-property! grob 'control-points cps2)
            (ly:slur::print grob)))
          ;; annotates all new control-points
           (if ann?
                 ;; display control points of the original, non-compound slur
                  (lambda (c)
                    (make-cross-stencil c cyan))
                 ;; display actual control points of the compound slur
                  (lambda (c)
                    (make-cross-stencil c red))
                  (append cps1 cps2))
                 ;; display connections between original and offset control points
                  (lambda (c1 c2)
                    (connect-dots c1 c2 cyan))
                  (list cp1 cp2 cp6 cp7))
                 ;; display handles indicating the
                  (lambda (c1 c2)
                    (connect-dots c1 c2 red))
                  (list cp1 cp3 cp4 cp7)
                  (list cp2 cp4 cp5 cp6))

         (ly:message "angle: ~a" slope)
    #{ -\tweak stencil $proc ( #}))

upper = \relative {
  \key d \major
  \clef bass
  s2 r8 d,16 g h d g h
  d8 r s2.
  s4 \voiceTwo h8.(-- c16-- h2--)
lower = \relative {
  \key d \major
  \clef bass
  <d' h g=>~-^
  % Uncomment one out of the following lines to compare the
  % default slur with the compound one
  -\compoundSlur \with {
    offsets =       % offsets against the automatic control points
                    % of the original, non-compound slur
    #'((0 . -1.5)   ; left starting point
        (-2 . -1)   ; second control point
        (-2 . 1)    ; second-to-last control point
        (0 . 0))    % right end point

    center-ratio =
    #'(0.6 . 0.65)  % X/Y ratio of the inflection point
                    % calculated between the actual end points of the slur

    % TODO: change to ratio and angle props (then create defaults)
    inner-segment =
    #'(4 . 6)       % X/Y of the (so far symmetric) line around the
                    % inflection point
    ann = ##t 	    % Display control points
      <d h g>4
      % Uncomment one of the following lines to see the robustness of the slur
      <c g e>4
      %      <c g e>16 q q q
      <h g>4 \voiceOne cis
      \change Staff = upper
      \clef treble \voiceOne d e fis2 )
    \new Voice {
      s2. <g,= e>4 \oneVoice
      <fis h,> <g e> <fis dis>2\fermata
\score {
    \new PianoStaff <<
      \new Staff = upper \upper
      \new Staff = lower \lower

Attachment: compound-slur-relative.preview.pdf
Description: Adobe PDF document

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