I made my first projection animation of a Klein Bottle. Instead of accumulating screenshots and making a gif, I just ran calcplot in realtime and used a screencapture recorder.

https://www.youtube.com/watch?v=aobRi4KQcKA&feature=youtu.be

This animation was made with the parametric equation:

x(u,v) = (3 + sin(v))cos(u)

y(u,v) = (3 + sin(v))sin(u)

z(u,v) = cos(v)cos(u/2)

w(u,v) = cos(v)sin(u/2)

0 < u,v < 2π

rotate on plane xw:

x = (x)*cos(t) + (w)*sin(t)

y = y

z = z

w = (x)*sin(t) - (w)*cos(t)

becomes,

x = ((3 + sin(v))cos(u))*cos(t) + (cos(v)sin(u/2))*sin(t)

y = (3 + sin(v))sin(u)

z = cos(v)cos(u/2)

w = ((3 + sin(v))cos(u))*sin(t) - (cos(v)sin(u/2))*cos(t)

Project on plane xyz:

x(u,v) = ((3 + sin(v))cos(u))*cos(t) + (cos(v)sin(u/2))*sin(t)

y(u,v) = (3 + sin(v))sin(u)

z(u,v) = cos(v)cos(u/2)

w(u,v) = 0

Vary parameter t between 0 and 2π to rotate on plane xw

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For exploring the cross sections of a mobius tube (klein bottle) , use the implicitized polynomial:

(x^2 +y^2 +z^2 +w^2 -a^2 -b^2)^2 -4a^2(b^2 -z^2 -w^2) = 0

which can also be represented as

(sqrt(x^2 +y^2) -a)^2 +z^2 +w^2 = b^2

rotating on plane xw, set w=d :

x = (x*cos(t) + d*sin(t))

w = (x*sin(t) - d*cos(t))

becomes a movable 3D cross section:

((x*cos(t)+d*sin(t))^2 +y^2 +z^2 +(x*sin(t)-d*cos(t))^2 -a^2 -b^2)^2 -4a^2(b^2 -z^2 -(x*sin(t)-d*cos(t))^2) = 0

set a = 3 ; b = 1

Exploring this function using ‘d’ to translate along w , ’t’ to rotate on xw , you will see all of the slices of a spheritorus:

Rotating at origin : http://i.imgur.com/9AZPAua.mp4

Passing through 3D at various angles : http://i.imgur.com/cYzvzEl.mp4