Geometrische Algebra in der Computergrafik

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 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
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 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 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

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 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 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

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
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 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