This application proposes for you to visualize the following list of quadratic hypersurfaces in the space-time of dimension 4. In the equations, x,y,z are the coordinates of the space, and we use the variable t to designate the coordinate time. A same hypersurface is often presented several times, under different viewing angles (in particular with respect to t).

equation | hypersurface |
---|---|

x*y*z-n=0 | corner hypersurface |

z1.im*cos(a) + z2.im*sin(a) | Fermat hypersurface |

[cosα cosγ,sinα cosγ,cosβ sinγ,sinβ sinγ] | flat tori |

A quadratic hypersurface is the set of points verifying an algebraic equation of degree 2.

Click the blue equation to load, manually change the t value by a slider.

.gif | No. # | equation | description |
---|---|---|---|

1 | x^{2}+y^{2}+z^{2}+t^{2}=1 | sphere S^{3}
| |

2 | x^{2}+y^{2}+z^{2}-t^{2}=0 | spherical cone with principal axis on the axis of t | |

3 | x^{2}+y^{2}-z^{2}+t^{2}=0 | spherical cone with principal axis on the axis of z | |

4 | x^{2}-y^{2}+z^{2}+t^{2}=0 | spherical cone with principal axis on the axis of y | |

5 | x^{2}+y^{2}-z*t=0 | spherical cone whose principal axis is the line x=y=z+t=0 | |

6 | x^{2}+z^{2}-y*t=0 | spherical cone whose principal axis is the line x=z=y+t=0 | |

7 | x^{2}+y^{2}+z^{2}-t^{2}=1 | spherical hyperboloid whose principal axis is the axis of t | |

8 | x^{2}+y^{2}-z^{2}+t^{2}=1 | spherical hyperboloid whose principal axis is the axis of z | |

9 | x^{2}-y^{2}+z^{2}+t^{2}=1 | spherical hyperboloid whose principal axis is the axis of y | |

10 | x^{2}+y^{2}-z*t=1 | spherical hyperboloid whose principal axis is the line x=y=z+t=0 | |

11 | x^{2}+z^{2}-y*t=1 | spherical hyperboloid whose principal axis is the line x=z=y+t=0 | |

12 | x^{2}+y^{2}-z^{2}-t^{2}=0 | vertical hyperboloidal cone | |

13 | x^{2}-y^{2}+z^{2}-t^{2}=0 | horizontal hyperboloidal cone | |

14 | x^{2}-y^{2}-z*t=0 | hyperboloidal cone | |

15 | x^{2}-z^{2}-y*t=0 | hyperboloidal cone | |

16 | x^{2}+y^{2}-z^{2}-t^{2}=1 | hyperboloidal hyperboloid | |

17 | x^{2}-y^{2}+z^{2}-t^{2}=1 | hyperboloidal hyperboloid | |

18 | x^{2}+y^{2}-z^{2}-t^{2}= -1 | hyperboloidal hyperboloid | |

19 | x^{2}-y^{2}+z^{2}-t^{2}= -1 | hyperboloidal hyperboloid | |

20 | x^{2}-y^{2}-z*t=1 | hyperboloidal hyperboloid | |

21 | x^{2}-z^{2}-y*t=1 | hyperboloidal hyperboloid | |

22 | x^{2}+y^{2}+z^{2}-t=0 | spherical paraboloid oriented towards the axis of t | |

23 | x^{2}+y^{2}-z+t^{2}=0 | spherical paraboloid oriented towards the axis of z | |

24 | x^{2}-y+z^{2}+t^{2}=0 | spherical paraboloid oriented towards the axis of y | |

25 | x^{2}+y^{2}-z^{2}-t=0 | hyperboloidal paraboloid oriented towards the axis of t, vertical | |

26 | x^{2}-y^{2}+z^{2}-t=0 | hyperboloidal paraboloid oriented towards the axis of t, horizontal | |

27 | x^{2}+y^{2}-z-t^{2}=0 | hyperboloidal paraboloid oriented towards the axis of z | |

28 | x^{2}-y^{2}-z+t^{2}=0 | hyperboloidal paraboloid oriented towards the axis of z | |

29 | x^{2}-y+z^{2}-t^{2}=0 | hyperboloidal paraboloid oriented towards the axis of y | |

30 | x^{2}-y-z^{2}+t^{2}=0 | hyperboloidal paraboloid oriented towards the axis of y | |

31 | x^{2}+y^{2}+t^{2}=1 | vertical spherical cylinder | |

32 | x^{2}+z^{2}+t^{2}=1 | horizontal spherical cylinder | |

33 | x^{2}+y^{2}-t^{2}=0 | conic cylinder with a singular line on the axis of z | |

34 | x^{2}+z^{2}-t^{2}=0 | conic cylinder with a singular line on the axis of y | |

35 | x^{2}+y^{2}-t^{2}=1 | hyperboloidal cylinder with one sheet, vertical | |

36 | x^{2}+z^{2}-t^{2}=1 | hyperboloidal cylinder with one sheet, horizontal | |

37 | x^{2}+y^{2}-t^{2}=1 | hyperboloidal cylinder with two sheets, vertical | |

38 | x^{2}+z^{2}-t^{2}=1 | hyperboloidal cylinder with two sheets, horizontal | |

? | 39 | x*y*z-t=0 | corner hypersurface |

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