How to calculate visible vertical beam-height

[Thomas Morley]

Hi,

here a problem which should be solveable, but the math escapes me,
although all values are known, afaict.

Please look at the attached image, the black polygons are created by
the beam-markup-command (slightly changed to make blot-diameter
settable with an override).
The red lines represent the vertical visible height of the beam.
Below the code for the attached image.

I'm interested in a formula how the length of the red line changes
with different blot-diameters.

Any hint is highly appreciated.

Thanks,
  Harm

\version "2.19.82"

#(define-markup-command (beam layout props width slope thickness)
  (number? number? number?)
  #:properties ((blot #f))
  (let* ((y (* slope width))
         (yext (cons (min 0 y) (max 0 y)))
         (half (/ thickness 2)))

    (ly:make-stencil
     `(polygon ',(list
                  0 (/ thickness -2)
                  width (+ (* width slope)  (/ thickness -2))
                  width (+ (* width slope)  (/ thickness 2))
                  0 (/ thickness 2))
               ,(or blot (ly:output-def-lookup layout 'blot-diameter))
               #t)
     (cons 0 width)
     (cons (+ (- half) (car yext))
           (+ half (cdr yext))))))

\markup {
  \overlay {
    \override #'(blot . 0) \beam #5 #1 #2
    \translate #'(1 . 0) \with-color #red \draw-line #'(0 . 2)
  }

  \overlay {
    \override #'(blot . 2) \beam #5 #1 #2
    \translate #'(2.35 . 0) \with-color #red \draw-line #'(0 . 4.7)
  }
}

visible-vertical-beam-thick.png
Description: PNG image