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# math animation online software 数学动画 4

## 4. Quadratic hypersurfaces in 4D

Content

1. plane curve in 2D
2. space curve in 3D
3. surface in 3D
4. surface in 4D

#### the 4-dimensional object (x,y,z,t) in 3D space

Have you ever tried to see objects in dimension 4? This is possible! Our space is in 3 dimensions. Adding the dimension of time, we get a space-time of dimension 4. The objects in this space-time are therefore 3D objects which deform with the time. following this principle, this tool proposes to you the visualization of some of these objects as deforming 3D objects, with the aim of helping you to establish this vision in dimension 4.

• Input your equation, click the implicit3D button, or Input implicit3D(x*y*z-t=0), hit the ENTER key in your keybroad, manually change the t value by a slider.

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

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). Click the blue equation to load, manually change the t value by a slider in following table.

.gifNo. #equationdescription
1x2+y2+z2+t2=1hypersphere S4
2x2+y2+z2-t2=1spherical cone with principal axis on the axis of t
3x2+y2-z2+t2=0spherical cone with principal axis on the axis of z
4x2-y2+z2+t2=0spherical cone with principal axis on the axis of y
5x2+y2-z*t=0spherical cone whose principal axis is the line x=y=z+t=0
6x2+z2-y*t=0spherical cone whose principal axis is the line x=z=y+t=0
7x2+y2+z2-t2=1spherical hyperboloid whose principal axis is the axis of t
8x2+y2-z2+t2=1spherical hyperboloid whose principal axis is the axis of z
9x2-y2+z2+t2=1spherical hyperboloid whose principal axis is the axis of y
10x2+y2-z*t=1spherical hyperboloid whose principal axis is the line x=y=z+t=0
11x2+z2-y*t=1spherical hyperboloid whose principal axis is the line x=z=y+t=0
12x2+y2-z2-t2=0vertical hyperboloidal cone
13x2-y2+z2-t2=0horizontal hyperboloidal cone
14x2-y2-z*t=0hyperboloidal cone
15x2-z2-y*t=0hyperboloidal cone
16x2+y2-z2-t2=1hyperboloidal hyperboloid
17x2-y2+z2-t2=1hyperboloidal hyperboloid
18x2+y2-z2-t2= -1hyperboloidal hyperboloid
19x2-y2+z2-t2= -1hyperboloidal hyperboloid
20x2-y2-z*t=1hyperboloidal hyperboloid
21x2-z2-y*t=1hyperboloidal hyperboloid
22x2+y2+z2-t=0spherical paraboloid oriented towards the axis of t
23x2+y2-z+t2=0spherical paraboloid oriented towards the axis of z
24x2-y+z2+t2=0spherical paraboloid oriented towards the axis of y
25x2+y2-z2-t=0hyperboloidal paraboloid oriented towards the axis of t, vertical
26x2-y2+z2-t=0hyperboloidal paraboloid oriented towards the axis of t, horizontal
27x2+y2-z-t2=0hyperboloidal paraboloid oriented towards the axis of z
28x2-y2-z+t2=0hyperboloidal paraboloid oriented towards the axis of z
29x2-y+z2-t2=0hyperboloidal paraboloid oriented towards the axis of y
30x2-y-z2+t2=0hyperboloidal paraboloid oriented towards the axis of y
31x2+y2+t2=1vertical spherical cylinder
32x2+z2+t2=1horizontal spherical cylinder
33x2+y2-t2=0conic cylinder with a singular line on the axis of z
34x2+z2-t2=0conic cylinder with a singular line on the axis of y
35x2+y2-t2=1hyperboloidal cylinder with one sheet, vertical
36x2+z2-t2=1hyperboloidal cylinder with one sheet, horizontal
37x2+y2-t2= 1hyperboloidal cylinder with two sheets, vertical
38y2+z2-t2= 1hyperboloidal cylinder with two sheets, horizontal
? 39x*y*z-t=0 Dr hypersurface
? 40 z1.im*cos(a) + z2.im*sin(a) Fermat hypersurface

## Reference 参阅

• example 例题
• function graph 制图
• graphics 制图法
• animation 动画 ﻿