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Quadriken im Raum

The document discusses quadrics in space and their intersection images on planar surfaces. It introduces the problem of developing an application in C/C++ to visualize and transform quadrics in 3D space using computer graphics and OpenGL. Key quadrics like ellipsoids, hyperboloids, and paraboloids are defined through their quadratic coordinate equations and coefficient matrices. Their intersection images with planar surfaces are also described.

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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 isÔ 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 isÔ 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 isÔ 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 isÔ 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 isÔ 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
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
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 isÔ 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 isÔ 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 isÔ 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 isÔ 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
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     1 Zwillinger, 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
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 isÔ 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 isÔ 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 radÔ 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 isÔ 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
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 isÔ 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 FIGURE4.39 Thevenon-degeneraterealquadrics.Topleft:ellipsoid.Topright:hyperboloidoftwo sheets(onefacingupandonefacingdown).Bottomleft:ellipticparaboloid.Bottommiddle: hyperboloidofonesheet.Bottomright:hyperbolicparaboloid. Conversely,anequationoftheform Ü ¾ ·Ý ¾ ·Þ ¾ ·¾Ü·¾Ý·¾Þ·¼(4.18.7) definesasphereif¾·¾·¾;thecenteris´   µandtheradiusisÔ¾·¾·¾ . 1.Fourpointsnotinthesameplanedetermineauniquesphere.Ifthepoints havecoordinates´Ü½Ý½Þ½µ,´Ü¾Ý¾Þ¾µ,´Ü¿Ý¿Þ¿µ,and´ÜÜÞµ,the ©2003byCRCPressLLC FIGURE4.39 Thevenon-degeneraterealquadrics.Topleft:ellipsoid.Topright:hyperboloidoftwo sheets(onefacingupandonefacingdown).Bottomleft:ellipticparaboloid.Bottommiddle: hyperboloidofonesheet.Bottomright:hyperbolicparaboloid. Conversely,anequationoftheform Ü ¾ ·Ý ¾ ·Þ ¾ ·¾Ü·¾Ý·¾Þ·¼(4.18.7) definesasphereif¾·¾·¾;thecenteris´   µandtheradiusisÔ¾·¾·¾ . 1.Fourpointsnotinthesameplanedetermineauniquesphere.Ifthepoints havecoordinates´Ü½Ý½Þ½µ,´Ü¾Ý¾Þ¾µ,´Ü¿Ý¿Þ¿µ,and´ÜÜÞµ,the ©2003byCRCPressLLC 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 isÔ 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 isÔ 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
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
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
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 1projzieren Visualisierung Quadrik (Welt-Koordinaten) Berner Fachhochschule | Haute ´ecole sp´ecialis´ee bernoise | Bern University of Applied Sciences
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
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

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Quadriken im Raum

  • 1.
    QIR Einleitung Problem Grundlagen Quadriken Kollineation Stereo Konzept DMD Umsetzung GUI Repo FIGURE 4.39 The venon-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 isÔ 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 isÔ 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 isÔ 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 isÔ 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 isÔ 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 Quadrikenund 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 Raumund 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 isÔ 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 isÔ 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 isÔ 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 isÔ 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 Raumund 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     1 Zwillinger, 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 Raumund 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 isÔ 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 isÔ 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 radÔ 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 isÔ 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 Raumund 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 isÔ 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 FIGURE4.39 Thevenon-degeneraterealquadrics.Topleft:ellipsoid.Topright:hyperboloidoftwo sheets(onefacingupandonefacingdown).Bottomleft:ellipticparaboloid.Bottommiddle: hyperboloidofonesheet.Bottomright:hyperbolicparaboloid. Conversely,anequationoftheform Ü ¾ ·Ý ¾ ·Þ ¾ ·¾Ü·¾Ý·¾Þ·¼(4.18.7) definesasphereif¾·¾·¾;thecenteris´   µandtheradiusisÔ¾·¾·¾ . 1.Fourpointsnotinthesameplanedetermineauniquesphere.Ifthepoints havecoordinates´Ü½Ý½Þ½µ,´Ü¾Ý¾Þ¾µ,´Ü¿Ý¿Þ¿µ,and´ÜÜÞµ,the ©2003byCRCPressLLC FIGURE4.39 Thevenon-degeneraterealquadrics.Topleft:ellipsoid.Topright:hyperboloidoftwo sheets(onefacingupandonefacingdown).Bottomleft:ellipticparaboloid.Bottommiddle: hyperboloidofonesheet.Bottomright:hyperbolicparaboloid. Conversely,anequationoftheform Ü ¾ ·Ý ¾ ·Þ ¾ ·¾Ü·¾Ý·¾Þ·¼(4.18.7) definesasphereif¾·¾·¾;thecenteris´   µandtheradiusisÔ¾·¾·¾ . 1.Fourpointsnotinthesameplanedetermineauniquesphere.Ifthepoints havecoordinates´Ü½Ý½Þ½µ,´Ü¾Ý¾Þ¾µ,´Ü¿Ý¿Þ¿µ,and´ÜÜÞµ,the ©2003byCRCPressLLC 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 isÔ 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 isÔ 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 Raumund 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 Raumund 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 Raumund 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 1projzieren 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 Raumund 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 Raumund 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