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.

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