Quadriken im Raum

Geometrische Algebra in der Computergrafik, Semesterarbeit

Transcript

1. QIR Einleitung Problem Grundlagen Quadriken Kollineation Stereo Konzept DMD Umsetzung GUI Repo FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC Quadriken im Raum und ihre Schnittbilder an ebenen Fl¨ achen Geometrische Algebra in der Computergrafik Studiengang: Informatik, Modul BZG1310 Objektorientiere Geometrie Autor: Roland Bruggmann, brugr9@bfh.ch Dozent: Marx Stampfli, marx.stampfli@bfh.ch Datum: 12. Januar 2015 Berner Fachhochschule | Haute ´ ecole sp´ ecialis´ ee bernoise | Bern University of Applied Sciences
2. QIR Einleitung Problem Grundlagen Quadriken Kollineation Stereo Konzept DMD Umsetzung GUI Repo ¨ Ubersicht 1 Einleitung Problemstellung 2 Grundlagen Quadriken und Schnittbilder Kollineation Stereobildwiedergabe 3 Konzept Dom¨ anenmodell-Diagramm 4 Umsetzung Grafische Benutzerschnittstelle (Demo) Repository Berner Fachhochschule | Haute ´ ecole sp´ ecialis´ ee bernoise | Bern University of Applied Sciences
3. QIR Einleitung Problem Grundlagen Quadriken Kollineation Stereo Konzept DMD Umsetzung GUI Repo Einleitung Quadriken im Raum und ihre Schnittbilder an ebenen Fl¨ achen Problemstellung Applikation in C/C++: Quadrik im Raum soll . . . mit Computergrafik (OpenGL) dargestellt werden. mit ebener Fl¨ ache geschnitten, das Schnittbild akzentuiert dargestellt werden. durch geometrische Transformation erkundet werden k¨ onnen. durch Kollineation ver¨ andert werden k¨ onnen. FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the Berner Fachhochschule | Haute ´ ecole sp´ ecialis´ ee bernoise | Bern University of Applied Sciences
4. QIR Einleitung Problem Grundlagen Quadriken Kollineation Stereo Konzept DMD Umsetzung GUI Repo Grundlagen Quadriken im Raum und ihre Schnittbilder an ebenen Fl¨ achen Quadriken und Schnittbilder Quadrik (engl. quadric)1: gekr¨ ummte Fl¨ ache in R3 Als gemischt-quadratische Koordinatengleichung: ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy + 2px + 2qy + 2rz + d = 0 (1) Als Matrizenmultiplikation im projektiven Raum (w = 1): vT · Q · v = 0 (2) mit v =     x y z 1     und symmetrischer Koeffizientenmatrize Q =     a h g p h b f q g f c r p q r d     1Zwillinger, Daniel: Standard Mathematical Tables and Formulae, Boca Raton, FL: Chapman & Hall/CRC, 2003, page 578. Berner Fachhochschule | Haute ´ ecole sp´ ecialis´ ee bernoise | Bern University of Applied Sciences
5. QIR Einleitung Problem Grundlagen Quadriken Kollineation Stereo Konzept DMD Umsetzung GUI Repo Grundlagen Quadriken im Raum und ihre Schnittbilder an ebenen Fl¨ achen Quadriken und Schnittbilder Ellipsoid QEllipsoid =   +a 0 0 0 0 +b 0 0 0 0 +c 0 0 0 0 −d   (Kugel: a = b = c) FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Schnittbild: Ellipse. Hyperboloid QHyperboloid =   +a 0 0 0 0 −b 0 0 0 0 +c 0 0 0 0 ±d   (einschalig: d < 0, zweischalig: d > 0) FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC Schnittbild: Hyperbel. Paraboloid QParaboloid =    +a 0 0 0 0 ±b 0 0 0 0 0 ±r 0 0 ±r d    (elliptisch: b > 0, hyperbolisch: b < 0) FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid o sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom m hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4. defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the rad Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the p have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ © 2003 by CRC Press LLC FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC Schnittbild: Parabel. Berner Fachhochschule | Haute ´ ecole sp´ ecialis´ ee bernoise | Bern University of Applied Sciences
6. QIR Einleitung Problem Grundlagen Quadriken Kollineation Stereo Konzept DMD Umsetzung GUI Repo Grundlagen Quadriken im Raum und ihre Schnittbilder an ebenen Fl¨ achen Kollineation Gegebene Normalform in ¨ aquivalente Quadriken abbilden: Typ Normalform ¨ Aquivalente Mittelpunktsquadrik Kugel FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC Kegeliger Typ Zylinder FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC Parabolischer Typ Scheibe FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ · ¾ Ü · ¾ Ý · ¾ Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ and the radius is Ô ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü½ ݽ Þ½µ, ´Ü¾ ݾ Þ¾µ, ´Ü¿ Ý¿ Þ¿µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC Berner Fachhochschule | Haute ´ ecole sp´ ecialis´ ee bernoise | Bern University of Applied Sciences
7. QIR Einleitung Problem Grundlagen Quadriken Kollineation Stereo Konzept DMD Umsetzung GUI Repo Grundlagen Quadriken im Raum und ihre Schnittbilder an ebenen Fl¨ achen Kollineation Abbildung durch projektive Transformation H: vn = H · vn (3) mit vn =     xn yn zn 1     und H =     h11 h12 h13 0 h21 h22 h23 0 h31 h32 h33 0 h41 h42 h43 h44     Koeffizientenmatrize der Abbildung: Q = HT · Q · H−1 (4) Berner Fachhochschule | Haute ´ ecole sp´ ecialis´ ee bernoise | Bern University of Applied Sciences
8. QIR Einleitung Problem Grundlagen Quadriken Kollineation Stereo Konzept DMD Umsetzung GUI Repo Grundlagen Quadriken im Raum und ihre Schnittbilder an ebenen Fl¨ achen Stereobildwiedergabe Spektrales Multiplexing mit Rot-Gr¨ un-Anaglyphen Perspektivische Projektion zweier asymmetrischer Sichtvolumen in dasselbe Bild: Pleft =      2n r−l+2d 0 r+l r−l+2d 0 0 2n t−b t+b t−b 0 0 0 − f +n f −n − 2fn f −n 0 0 −1 0      Pright =      2n r−l−2d 0 r+l r−l−2d 0 0 2n t−b t+b t−b 0 0 0 − f +n f −n − 2fn f −n 0 0 −1 0      mit d = 1 2 × eyeSep × n focalDist eyeSep (eye separation): Abstand der Augen des menschlichen Binokulars focalDist (focal distance): Distanz des Binokulars zur ’near clipping plane’ n Berner Fachhochschule | Haute ´ ecole sp´ ecialis´ ee bernoise | Bern University of Applied Sciences
9. QIR Einleitung Problem Grundlagen Quadriken Kollineation Stereo Konzept DMD Umsetzung GUI Repo Konzept Quadriken im Raum und ihre Schnittbilder an ebenen Fl¨ achen Dom¨ anenmodell-Diagramm Auswahl Quadrik Liste mit Normalformen Liste reeller Quadriken Quadrik NF: v Koeffizienten NF: Q 1 1 auswählen Benutzer- schnittstelle 1 1 erzeugen Kollineation H Abb. Quadrik v'=Hv (Objekt-Koodinaten) Normalengleichung ax^2+...+d=0 Abb. Koeffizienten Q'=H^TQH^-1 1 1 editieren 1 1 auswählen 1 1 parametrisieren 1 1 abbilden 1 1 visualisieren 1 1 parametrisieren 1 1 visualisieren Auswahl Projektion Orthografische P. Perspektivische P. Stereoskopische P. Auswahl Affine Transf. Zoom Rotation Animierte Transf. 1 1 auswählen 1 1 abbilden 1 1 transformieren 1 1 projzieren Visualisierung Quadrik (Welt-Koordinaten) Berner Fachhochschule | Haute ´ ecole sp´ ecialis´ ee bernoise | Bern University of Applied Sciences
10. QIR Einleitung Problem Grundlagen Quadriken Kollineation Stereo Konzept DMD Umsetzung GUI Repo Umsetzung Quadriken im Raum und ihre Schnittbilder an ebenen Fl¨ achen Grafische Benutzerschnittstelle (Demo) QIR Berner Fachhochschule | Haute ´ ecole sp´ ecialis´ ee bernoise | Bern University of Applied Sciences
11. QIR Einleitung Problem Grundlagen Quadriken Kollineation Stereo Konzept DMD Umsetzung GUI Repo Umsetzung Quadriken im Raum und ihre Schnittbilder an ebenen Fl¨ achen Repository https://github.com/brugr9/qir Bildnachweis: Figure 4.39: The five non-degenerated real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. (Die f¨ unf nicht-degenerierten reellen Quadriken. Oben links: Ellipsoid. Oben rechts: zweischaliges Hyperboloid (eine Schale nach oben und eine nach unten gerichtet). Unten links: elliptisches Paraboloid. Unten Mitte: einschalges Hyperboloid. Unten rechts: hyperolisches Paraboloid.) In: Daniel Zwillinger: Standard Mathematical Tables and Formulae. 31. Aufl. Boca Raton, FL: Chapman & Hall/CRC, 2003. S. 580. Berner Fachhochschule | Haute ´ ecole sp´ ecialis´ ee bernoise | Bern University of Applied Sciences