## Cubinder's net

Discussion of shapes with curves and holes in various dimensions.

### Re: Toratope's nets and cross-sections

Thanks very much for sharing this service! I will try to use it. I like your responsiveness.
icebreaker
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### Re: Toratope's nets and cross-sections

Finally I found a time to visualize cross-sections of so hard form - coninder.
Attachments
Coninder_c-s_90dg.gif (34.6 KiB) Viewed 2563 times
icebreaker
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### Re: Toratope's nets and cross-sections

I still have a problem to realize the rest particular cross-section of dicone. Here is the face-first one.
Attachments
Dicone_c-s_face-first.gif (16.83 KiB) Viewed 2557 times
icebreaker
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### Re: Toratope's nets and cross-sections

I assume that the slices of dicone could be this:
Attachments
It's really not a true cross-section of a dicone
Dicone_c-s_90gr_MAY_BE_POSSIBLE.gif (25.18 KiB) Viewed 2554 times
Last edited by icebreaker on Sat Dec 23, 2017 10:04 am, edited 1 time in total.
icebreaker
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### Re: Toratope's nets and cross-sections

I decided to create more complex variant of slices of the objects. Is it more felicitous than previous ones? Then I would make this for other toratopes.
Attachments
Cubinder_c-sv2_round-first.gif (78.81 KiB) Viewed 2533 times
Cubinder_c-sv2_cylinder-first.gif (9.73 KiB) Viewed 2533 times
Cubinder_c-sv2_disk-first.gif (41.23 KiB) Viewed 2533 times
Cubinder_c-sv2_round_face-first.gif (55.88 KiB) Viewed 2533 times
Cubinder_c-sv2_edge-first.gif (43.64 KiB) Viewed 2533 times
icebreaker
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### Re: Toratope's nets and cross-sections

And net...
Attachments
Cubinder_net_v2.png (32 KiB) Viewed 2533 times
The third variant
Cubinder_net_v2_.png (28.12 KiB) Viewed 2069 times
Last edited by icebreaker on Wed Jan 24, 2018 1:15 pm, edited 2 times in total.
icebreaker
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### Re: Toratope's nets and cross-sections

The same thing with duocylinder?
Attachments
Duocylinder_net_1_v2.png (27.93 KiB) Viewed 2528 times
Duocylinder_net_2_v2.png (27.89 KiB) Viewed 2528 times
Duocylinder_c-sv2_blue-first.gif (56.47 KiB) Viewed 2528 times
Duocylinder_c-sv2_red-first.gif (55.36 KiB) Viewed 2528 times
P.S. Forgot the face-first one
Duocylinder_c-sv2_face-first.gif (49.9 KiB) Viewed 2519 times
icebreaker
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### Re: Toratope's nets and cross-sections

There is an analogy between the cross-sections of a duocylinder, a cubinder and a tesseract.

Cubinder - cylinder-first
Tesseract - cube-first

Cubinder - round cell-first and disk-first
Duocylinder - cell-first
Tesseract - square-first

Cubinder - round face-first
Tesseract - edge-first

Cubinder - edge-first
Duocylinder - face-first
Tesseract - corner-first
icebreaker
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### Re: Cubinder's net

These are great, man. Nice to see someone else out there making hypershape gifs.
in search of combinatorial objects of finite extent
ICN5D
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### Re: Toratope's nets and cross-sections

Creating this animation seemed to me very long but it's not true.
Attachments
Tesseract_c-sv2_corner-first.gif (32.01 KiB) Viewed 2477 times
Tesseract_c-sv2_edge-first.gif (24.17 KiB) Viewed 2469 times
Tesseract_c-sv2_face-first.gif (21.61 KiB) Viewed 2469 times
Tesseract_c-sv2_cell-first.gif (8.52 KiB) Viewed 2469 times
Tesseract_net_v3.png (20.24 KiB) Viewed 2469 times
icebreaker
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### Re: Toratope's nets and cross-sections

Coninder's cross-sections.
Attachments
Coninder_c-sv2_cylinder-first.gif (25.67 KiB) Viewed 2450 times
Coninder_c-sv2_cone-first.gif (10.71 KiB) Viewed 2450 times
Coninder_c-sv2_hyperbola.gif (57.65 KiB) Viewed 2447 times
Coninder_c-sv2_parabola.gif (21.1 KiB) Viewed 2447 times
icebreaker
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### Re: Toratope's nets and cross-sections

Cross-sections of dicone.
Attachments
Dicone_c-sv2_disk-first.gif (19.37 KiB) Viewed 2421 times
Dicone_c-sv2_edge-first.gif (35.96 KiB) Viewed 2421 times
Dicone_c-sv2_cone-first.gif (19.5 KiB) Viewed 2348 times
Last edited by icebreaker on Sun Dec 24, 2017 9:02 am, edited 1 time in total.
icebreaker
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### Re: Toratope's nets and cross-sections

I need to know what is an implicit equation of dicone in order to check out (by using CalcPlot) whether there is more cross-sections than I think.
icebreaker
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### Re: Toratope's nets and cross-sections

icebreaker wrote:I need to know what is an implicit equation of dicone in order to check out (by using CalcPlot) whether there is more cross-sections than I think.

I think that the unit dicone should be:

sqrt(x^2 + y^2) + sqrt(z^2 + t^2) = 1
Marek14
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### Re: Toratope's nets and cross-sections

Thank you very much. But where can I find the same things for any another object?

UPD: How to determine whether are my previous cross-sections of a coninder and a dicone true?
icebreaker
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### Re: Cubinder's net

Not sure where to find them...

Coninder is a bit harder. I guess you can start from cone as sqrt(x^2 + y^2) + abs(z) = 1, but that gives double cone. To get single cone, you'd have to limit it to z >=0, but then you won't get base. You might have to compose the full implicit function using several logical tricks:

f(x) = 0 & g(x) = 0 -> f(x)^2 + g(x)^2 = 0
f(x) = 0 or g(x) = 0 -> f(x)g(x) = 0
f(x) >= 0 -> f(x) - abs(f(x)) = 0
Marek14
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### Re: Cubinder's net

Here you go. These equations are ready made, so you can copy-paste into calcplot3d (implicit function input window)

Explore Functions for 4D Shapes

Listed for each object is an equation for a moveable 3D hyperplane, that can slide and rotate in 4D.

- Use ‘a’ to slide up/down along the 4th axis.

- Use b, c, and d to rotate the sliding direction around in 4D.

- Adjusting the rotate sliders full left/right will make a 90 degree turn. You can set 0 < b,c,d < 2π for a full 360° turn

Parameter Ranges to Set:

XYZbox = ±10

-10 < a < 10

0 < b,c,d < π/2

Syntax:
notation - name : translate/rotate1,rotate2,rotate3

• ((II)(II)) - TIGER : a/b
(sqrt(x^2 + (y*sin(b) + a*cos(b))^2) - 6)^2 + (sqrt(z^2 + (y*cos(b) - a*sin(b))^2) - 6)^2 = 2

• (((II)I)I) - DITORUS : a/b,c
(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + y^2) -6)^2 + (z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -3)^2 + (z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2 = 1

• ((III)I) - TORISPHERE : a/b
(sqrt((x*sin(b) + a*cos(b))^2 + y^2 + z^2) -6)^2 + (x*cos(b) - a*sin(b))^2 = 2

• ((II)II) - SPHERITORUS : a/b
(sqrt((x*sin(b) + a*cos(b))^2 + y^2) -6)^2 + z^2 + (x*cos(b) - a*sin(b))^2 = 2

• IO(O)I - TORINDER : a/b,c
abs(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + y^2) -4)^2 + (z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) -(z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) + abs(sqrt((sqrt((x*sin(b) + a*cos(b))^2 + y^2) -4)^2 + (z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))^2) +(z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) = 3

• IO(O)> - TORICONE : a/b,c
equation unknown, though one possible way is to define as an oblique cone spindle-torus, with the vertex as point of self-intersection

• IOOO / (IIII) - GLOME : a
x^2 + y^2 + z^2 + a^2 = 7

• IOO> - SPHONE : a/b
abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2 + z^2) + 2(x*cos(b) - a*sin(b))) + sqrt((x*sin(b) + a*cos(b))^2 + y^2 + z^2) = 7

• IOOI - SPHERINDER : a/b
abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2 + z^2) -(x*cos(b) - a*sin(b))) + abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2 + z^2) +(x*cos(b) - a*sin(b))) = 7

• IO>> - DICONE : a/b,c
abs(abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2) +2(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + sqrt((x*sin(b) + a*cos(b))^2 + y^2) +4(z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) + abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2) +2(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + sqrt((x*sin(b) + a*cos(b))^2 + y^2) = 10

• IO>I - CONINDER : a/b,c
abs(abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2) +2(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + sqrt((x*sin(b) + a*cos(b))^2 + y^2) -2(z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) + abs(abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2) +2(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + sqrt((x*sin(b) + a*cos(b))^2 + y^2) +2(z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) = 15

• IOIO - DUOCYLINDER : a/b
abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2) - sqrt(z^2 + (x*cos(b) - a*sin(b))^2)) + abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2) + sqrt(z^2 + (x*cos(b) - a*sin(b))^2)) = 10

• IIO> - CYLINDRONE : a/b,c
abs(abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2) -(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2) +(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + 4(z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) + abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2) -(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs(sqrt((x*sin(b) + a*cos(b))^2 + y^2) +(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) = 15

• I>>> - PENTACHORON : a/b,c,d
abs(abs(abs(abs((x*sin(b) + a*cos(b))) + 2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))) +2(z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))) + abs(abs((x*sin(b) + a*cos(b))) + 2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))) + 4(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))) + abs(abs(abs((x*sin(b) + a*cos(b))) + 2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))) +2(z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))) + abs(abs((x*sin(b) + a*cos(b))) + 2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))) = 15

• I>>I - TETRAHEDRINDER : a/b,c
abs(abs(abs(abs((x*sin(b) + a*cos(b))) + 2y) + abs((x*sin(b) + a*cos(b))) + 2(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs(abs((x*sin(b) + a*cos(b))) + 2y) + abs((x*sin(b) + a*cos(b))) - 4(z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) + abs(abs(abs(abs((x*sin(b) + a*cos(b))) + 2y) + abs((x*sin(b) + a*cos(b))) + 2(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs(abs((x*sin(b) + a*cos(b))) + 2y) + abs((x*sin(b) + a*cos(b))) + 4(z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) = 20

• I>IO - CYLTRIANGLINDER : a/b,c
abs(abs(abs((x*sin(b) + a*cos(b))) + 2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))) - 2sqrt(z^2 + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2)) + abs(abs(abs((x*sin(b) + a*cos(b))) + 2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))) + 2sqrt(z^2 + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))^2)) = 15

• I>I> - TRIANGLE PRISM PYRAMID : a/b,c,d
abs(abs(abs(abs((x*sin(b) + a*cos(b))) + 2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))) - 2(z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))) + abs(abs(abs((x*sin(b) + a*cos(b))) + 2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))) + 2(z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))) + 4(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d))) + abs(abs(abs((x*sin(b) + a*cos(b))) + 2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))) - 2(z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))) + abs(abs(abs((x*sin(b) + a*cos(b))) + 2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))) + 2(z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))) = 15

• I>II - TRIANGLE DIPRISM : a/b,c
abs(abs(abs((x*sin(b) + a*cos(b))) + 2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))) - abs(z-(y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) - abs(z+(y*cos(c) - (x*cos(b) - a*sin(b))*sin(c)))) + abs(abs(abs((x*sin(b) + a*cos(b))) + 2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))) + abs(z-(y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) + abs(z+(y*cos(c) - (x*cos(b) - a*sin(b))*sin(c)))) = 15

• I>[I>] - DUOTRIANGLINDER : a/b,c,d
abs(abs(abs((x*sin(b) + a*cos(b)))+2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c)))+abs((x*sin(b) + a*cos(b))) - abs(abs((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))+2(z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d)))-abs((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))) + abs(abs(abs((x*sin(b) + a*cos(b)))+2(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c)))+abs((x*sin(b) + a*cos(b))) + abs(abs((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))+2(z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d)))+abs((z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))) = 17

• II>> - DIPYRAMID : a/b,c
abs(abs(abs((x*sin(b) + a*cos(b))-y) + abs((x*sin(b) + a*cos(b))+y) + 3(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))-y) + abs((x*sin(b) + a*cos(b))+y) + 4(z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) + abs(abs((x*sin(b) + a*cos(b))-y) + abs((x*sin(b) + a*cos(b))+y) + 3(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))-y) + abs((x*sin(b) + a*cos(b))+y) = 17

• II>I - PYRAMID PRISM : a/b,c
abs(abs(abs((x*sin(b) + a*cos(b))-y) + abs((x*sin(b) + a*cos(b))+y) + 3(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))-y) + abs((x*sin(b) + a*cos(b))+y) - 4(z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) + abs(abs(abs((x*sin(b) + a*cos(b))-y) + abs((x*sin(b) + a*cos(b))+y) + 3(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((x*sin(b) + a*cos(b))-y) + abs((x*sin(b) + a*cos(b))+y) + 4(z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) = 17

• IIIO - CUBINDER : a/b
abs(2sqrt((x*sin(b) + a*cos(b))^2 + y^2) - abs(z-(x*cos(b) - a*sin(b))) - abs(z+(x*cos(b) - a*sin(b)))) + abs(2sqrt((x*sin(b) + a*cos(b))^2 + y^2) + abs(z-(x*cos(b) - a*sin(b))) + abs(z+(x*cos(b) - a*sin(b)))) = 15

• III> - CUBE PYRAMID : a/b,c
abs(abs(abs((x*sin(b) + a*cos(b))-y) + abs((x*sin(b) + a*cos(b))+y) -2(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs(abs((x*sin(b) + a*cos(b))-y) + abs((x*sin(b) + a*cos(b))+y) +2(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + 3(z*cos(c) - (x*cos(b) - a*sin(b))*sin(c))) + abs(abs((x*sin(b) + a*cos(b))-y) + abs((x*sin(b) + a*cos(b))+y) -2(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs(abs((x*sin(b) + a*cos(b))-y) + abs((x*sin(b) + a*cos(b))+y) +2(z*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) = 21

• IIII - TESSERACT : a/b,c,d
abs(abs((x*sin(b) + a*cos(b))-(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c)))+abs((x*sin(b) + a*cos(b))+(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) - abs((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))-(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))-abs((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))+(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))) + abs(abs((x*sin(b) + a*cos(b))-(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c)))+abs((x*sin(b) + a*cos(b))+(y*sin(c) + (x*cos(b) - a*sin(b))*cos(c))) + abs((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))-(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))+abs((z*sin(d) + (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*cos(d))+(z*cos(d) - (y*cos(c) - (x*cos(b) - a*sin(b))*sin(c))*sin(d)))) = 10

****** Special Values to use with Tesseract *******

Corner First: b=pi/3 , c=0.9554 , d=pi/4

Line First: b=pi/2 , c=0.9554 , d=pi/4

Square First: b=pi/2 , c=pi/2 , d=pi/4

Cube First: b=pi/2 , c=pi/2 , d=pi/2
in search of combinatorial objects of finite extent
ICN5D
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### Re: Toratope's nets and cross-sections

It's a very helpful material! That's exactly what I needed. Thank you.
icebreaker
Dionian

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### Re: Cubinder's net

Yeah, man. I explored these shapes some time ago. For the dicone, you could add the slice where a large cone appears, and gradually shrinks to a point. If you're interested, I use a simple algorithm to derive these equations. They are the functions of STEMP notation (using I, >, O symbols).
in search of combinatorial objects of finite extent
ICN5D
Pentonian

Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

### Re: Toratope's nets and cross-sections

ICN5D wrote:Yeah, man. I explored these shapes some time ago. For the dicone, you could add the slice where a large cone appears, and gradually shrinks to a point. If you're interested, I use a simple algorithm to derive these equations. They are the functions of STEMP notation (using I, >, O symbols).

Yes, I am interested in all this.
icebreaker
Dionian

Posts: 44
Joined: Thu Sep 14, 2017 8:07 am
Location: Omsk, Russian Federation (UTC+6)

### Re: Cubinder's net

Okay, fortunately, I detailed most of it in this post. It's not very formal, just explained in my own words that can still use improvement.

These functions are what I use to generate this list of shape equations. Also posted here.

And the rest of the equation is a hardcoded rotate function to move the 3D slice around. That thread in the first and last link is worth a read.

I can provide more examples if needed.
in search of combinatorial objects of finite extent
ICN5D
Pentonian

Posts: 1047
Joined: Mon Jul 28, 2008 4:25 am
Location: Orlando, FL

### Re: Toratope's nets and cross-sections

I found a very interesting addon for Blender which allows to build a mathematical surface. I can't understand how to do it because this addon has another way of defining surfaces (by parametric equations). How to transform an implicit equation to a parametric one?
Attachments
Blender-XYZ-surface.png
Blender-XYZ-surface-example.png
Blender-XYZ-surface-example-.png
icebreaker
Dionian

Posts: 44
Joined: Thu Sep 14, 2017 8:07 am
Location: Omsk, Russian Federation (UTC+6)

### Re: Cubinder's net

Oh, well look at that! Parametric functions aren't that bad. But, converting them to implicit can be a really tricky process. I nailed down a method to convert curved surfaces only: i.e. spheres and tori. I have not yet gone looking for this (I did actually, just not very hard) regarding flat sided shapes with sharp edges. I feel that I don't need to, since I have a process for deriving both implicit and parametric, for the same shape.

A few examples:

Code: Select all
generating tiger parametric equation, and converting to implicit polynomialx = rcos(u)y = rsin(u)x = rcos(u)+Ry = rsin(u)x = (rcos(u)+R)cos(v)y = (rcos(u)+R)sin(v)z = rsin(u)x = (rcos(u)+R)cos(v)y = (rcos(u)+R)sin(v)z = rsin(u) + Px = (rcos(u)+R)cos(v)y = (rcos(u)+R)sin(v)z = (rsin(u)+P)cos(t)w = (rsin(u)+P)sin(t)implicitize:x = (rcos(u)+R)cos(v)y = (rcos(u)+R)sin(v)z = (rsin(u)+P)cos(t)w = (rsin(u)+P)sin(t)solve for variable t,x = (rcos(u)+R)cos(v)y = (rcos(u)+R)sin(v)z/(rsin(u)+P) = cos(t)arcsin(w/(rsin(u)+P)) = tx = (rcos(u)+R)cos(v)y = (rcos(u)+R)sin(v)z/(rsin(u)+P) = cos(arcsin(w/(rsin(u)+P)))solve for v,x/(rcos(u)+R) = cos(v)y/(rcos(u)+R) = sin(v)z/(rsin(u)+P) = cos(arcsin(w/(rsin(u)+P)))x/(rcos(u)+R) = cos(v)arcsin(y/(rcos(u)+R)) = vz/(rsin(u)+P) = cos(arcsin(w/(rsin(u)+P)))x/(rcos(u)+R) = cos(arcsin(y/(rcos(u)+R)))z/(rsin(u)+P) = cos(arcsin(w/(rsin(u)+P)))x = (rcos(u)+R)cos(arcsin(y/(rcos(u)+R)))z = (rsin(u)+P)cos(arcsin(w/(rsin(u)+P)))solve for u, from x(u,y):u = cos^(-1)((sqrt(r^2 x^2+r^2 y^2)-r R)/r^2)substitute in z(u,w):z = (rsin(cos^(-1)((sqrt(r^2 x^2+r^2 y^2)-r R)/r^2))+P)cos(arcsin(w/(rsin(cos^(-1)((sqrt(r^2 x^2+r^2 y^2)-r R)/r^2))+P)))z = (P+r sqrt(1-(sqrt(r^2 x^2+r^2 y^2)-r R)^2/r^4)) sqrt(1-w^2/(P+r sqrt(1-(sqrt(r^2 x^2+r^2 y^2)-r R)^2/r^4))^2)solving for r, 8 solutions:r = -sqrt(-2 sqrt(-2 sqrt(P^2R^2w^2x^2 + P^2R^2w^2y^2 + P^2R^2x^2z^2 + P^2R^2y^2z^2) + P^2w^2+P^2z^2 + R^2x^2 + R^2y^2) + P^2 + R^2 + w^2 + x^2 + y^2 + z^2)eliminating square roots,r^2 = (-sqrt(-2 sqrt(-2 sqrt(P^2R^2((x^2 + y^2)(w^2 + z^2))) + P^2(w^2 + z^2) + R^2(x^2 + y^2)) + P^2 + R^2 + w^2 + x^2 + y^2 + z^2))^2-r^2 = 2 sqrt(-2 sqrt(P^2R^2((x^2 + y^2)(w^2 + z^2))) + P^2(w^2 + z^2) + R^2(x^2 + y^2)) - (P^2 + R^2 + w^2 + x^2 + y^2 + z^2)(x^2 +y^2 +z^2 +w^2 +P^2 +R^2 -r^2)^2 = (2 sqrt(-2 sqrt(P^2R^2((x^2 + y^2)(w^2 + z^2))) + P^2(w^2 + z^2) + R^2(x^2 + y^2)))^2 (x^2 +y^2 +z^2 +w^2 +P^2 +R^2 -r^2)^2 = 4(-2 sqrt(P^2R^2((x^2 + y^2)(w^2 + z^2))) + P^2(w^2 + z^2) + R^2(x^2 + y^2))(x^2 +y^2 +z^2 +w^2 +P^2 +R^2 -r^2)^2 -4P^2(w^2 + z^2) -4R^2(x^2 + y^2) = -8sqrt(P^2R^2((x^2 + y^2)(w^2 + z^2)))((x^2 +y^2 +z^2 +w^2 +P^2 +R^2 -r^2)^2 -4P^2(w^2 + z^2) -4R^2(x^2 + y^2))^2 = (-8sqrt(P^2R^2((x^2 + y^2)(w^2 + z^2))))^2((x^2 +y^2 +z^2 +w^2 +P^2 +R^2 -r^2)^2 -4P^2(w^2 + z^2) -4R^2(x^2 + y^2))^2 = 64(P^2R^2((x^2 + y^2)(w^2 + z^2)))((x^2 +y^2 +z^2 +w^2 +P^2 +R^2 -r^2)^2 -4P^2(w^2 + z^2) -4R^2(x^2 + y^2))^2 -64(P^2R^2((x^2 + y^2)(w^2 + z^2))) = 0Degree-8 Polynomial with 4 variables, 3 coefficients((x^2 +y^2 +z^2 +w^2 +a^2 +b^2 -c^2)^2 -4a^2(x^2 + y^2) -4b^2(z^2 + w^2))^2 -64a^2b^2(x^2 + y^2)(z^2 + w^2) Rotate/Translate cross section:x = (x*cos(t) + d*sin(t))w = (x*sin(t) - d*cos(t))(((x*cos(t) + d*sin(t))^2 +y^2 +z^2 +(x*sin(t) - d*cos(t))^2 +a^2 +b^2 -c^2)^2 -4a^2((x*sin(t) - d*cos(t))^2 + z^2) -4b^2((x*cos(t) + d*sin(t))^2 + y^2))^2 -64a^2b^2((x*cos(t) + d*sin(t))^2 + y^2)((x*sin(t) - d*cos(t))^2 + z^2) = 0Graphs a tiger! Rotate/translate morphs, radius sizes all check out.\begin{align*}& \left(\left(x^2 +y^2 +z^2 +w^2 +a^2 +b^2 -c^2\right)^2 -4a^2\left(x^2 + y^2\right) -4b^2\left(z^2 + w^2\right)\right)^2 -64a^2b^2\left(x^2 + y^2\right)\left(z^2 + w^2\right)  \\\end{align*}

Code: Select all
Generate 3-torus parametric equation, and convert to implicitx = rcos(u)y = rsin(u)x = rcos(u)+Ry = rsin(u)x = (rcos(u)+R)cos(v)y = (rcos(u)+R)sin(v)z = rsin(u)x = (rcos(u)+R)cos(v)+Py = (rcos(u)+R)sin(v)z = rsin(u)x = ((rcos(u)+R)cos(v)+P)cos(t)w = ((rcos(u)+R)cos(v)+P)sin(t)y = (rcos(u)+R)sin(v)z = rsin(u)R1 > R2 > rx(t,u,v) = ((rcos(u)+R2)cos(v)+R1)cos(t)y(t,u,v) = ((rcos(u)+R2)cos(v)+R1)sin(t)z(u,v) = (rcos(u)+R2)sin(v)w(u) = rsin(u)Convert to Implicit Equation:x = ((rcos(u)+R)cos(v)+P)cos(t)y = ((rcos(u)+R)cos(v)+P)sin(t)z = (rcos(u)+R)sin(v)w = rsin(u)Using the cos^2(t) + sin^2(t) = 1 identityx = ((rcos(u)+R)cos(v)+P)cos(t)y = ((rcos(u)+R)cos(v)+P)sin(t)z = (rcos(u)+R)sin(v)w = rsin(u)x/((rcos(u)+R)cos(v)+P) = cos(t)y/((rcos(u)+R)cos(v)+P) = sin(t)z = (rcos(u)+R)sin(v)w = rsin(u)(x/((rcos(u)+R)cos(v)+P))^2 = cos^2(t)(y/((rcos(u)+R)cos(v)+P))^2 = sin^2(t)z = (rcos(u)+R)sin(v)w = rsin(u)(x/((rcos(u)+R)cos(v)+P))^2 + (y/((rcos(u)+R)cos(v)+P))^2 = 1z = (rcos(u)+R)sin(v)w = rsin(u)x^2 + y^2 = ((rcos(u)+R)cos(v)+P)^2z = (rcos(u)+R)sin(v)w = rsin(u)sqrt(x^2 + y^2) = (rcos(u)+R)cos(v)+Pz = (rcos(u)+R)sin(v)w = rsin(u)sqrt(x^2 + y^2)-P = (rcos(u)+R)cos(v)z = (rcos(u)+R)sin(v)w = rsin(u)(sqrt(x^2 + y^2)-P)/(rcos(u)+R) = cos(v)z/(rcos(u)+R) = sin(v)w = rsin(u)((sqrt(x^2 + y^2)-P)/(rcos(u)+R))^2 = cos^2(v)(z/(rcos(u)+R))^2 = sin^2(v)w = rsin(u)((sqrt(x^2 + y^2)-P)/(rcos(u)+R))^2 + (z/(rcos(u)+R))^2 = 1w = rsin(u)(sqrt(x^2 + y^2)-P)^2 + z^2 = (rcos(u)+R)^2w = rsin(u)sqrt((sqrt(x^2 + y^2)-P)^2 + z^2) = rcos(u)+Rw = rsin(u)sqrt((sqrt(x^2 + y^2)-P)^2 + z^2)-R = rcos(u)w = rsin(u)(sqrt((sqrt(x^2 + y^2)-P)^2 + z^2)-R)/r = cos(u)w/r = sin(u)((sqrt((sqrt(x^2 + y^2)-P)^2 + z^2)-R)/r)^2 = cos^2(u)(w/r)^2 = sin^2(u)((sqrt((sqrt(x^2 + y^2)-P)^2 + z^2)-R)/r)^2 + (w/r)^2 = 1(sqrt((sqrt(x^2 + y^2)-P)^2 + z^2)-R)^2 + w^2 = r^2

In the end, I discovered that the most efficient way to convert a closed toratope parametric equation is with the identity:

cos^2(t) + sin^2(t) = 1

This identity, along with some manipulation, will derive the simplest expression in implicit form.

But, here's the kicker: implicit is best suited for taking slices ; parametric is best for taking projections.

Now, I did also nail down a process, albeit ugly and complex, for getting slice equations in parametric form, for spheres and tori only. I detailed it in this Quora answer to my own question.

However, once you introduce flat sides into the equation, it gets hairy, and I haven't pursued it much. So, if you are looking for a method to slice flat-sided parametric surfaces, then you're into new territory (which probably isn't that bad; one's persistence will pay off like it always does). In the end, I recommend using parametric for when you're ready to make projections. I started a new thread that goes over this process.
in search of combinatorial objects of finite extent
ICN5D
Pentonian

Posts: 1047
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Location: Orlando, FL

### Re: Toratope's nets and cross-sections

I finally made slices of a 3-sphere (the colors corresponds to the article in Wikipedia https://en.wikipedia.org/wiki/3-sphere)
Red - parallels
Green - hypermeridians
Blue - meridians

P.S. I want to create the equator-first cross-section but it's difficult for me.
Attachments
Glome_c-s_pole-first.gif (363.8 KiB) Viewed 2065 times
Better variant
Glome_c-sv2_pole-first.gif (320.66 KiB) Viewed 1261 times
icebreaker
Dionian

Posts: 44
Joined: Thu Sep 14, 2017 8:07 am
Location: Omsk, Russian Federation (UTC+6)

### Re: Toratope's nets and cross-sections

The cross-sections of a cylindrone.
Attachments
Cylindrone_c-s_cylinder-first.gif (32 KiB) Viewed 930 times
Cylindrone_c-s_disk-first.gif (57.53 KiB) Viewed 930 times
Cylindrone_c-s_edge-first.gif (51.34 KiB) Viewed 930 times
icebreaker
Dionian

Posts: 44
Joined: Thu Sep 14, 2017 8:07 am
Location: Omsk, Russian Federation (UTC+6)

### Re: Toratope's nets and cross-sections

There is an analogy between the cross-sections of coninder, dicone and cylindrone.

Coninder - cylinder-first
Dicone - disk-first
Cylindrone - cylinder-first

Coninder - cone-first
Dicone - cone-first
Cylindrone - disk-first

Coninder - edge-first (hyperbola)
Dicone - edge-first
Cylindrone - edge-first
Last edited by icebreaker on Thu Mar 08, 2018 2:40 pm, edited 1 time in total.
icebreaker
Dionian

Posts: 44
Joined: Thu Sep 14, 2017 8:07 am
Location: Omsk, Russian Federation (UTC+6)

### Re: Cubinder's net

I still can't visualize the equator-first slices of 3-sphere so I need some help. It's approximate shadows looks like those http://eusebeia.dyndns.org/4d/600-cell.
icebreaker
Dionian

Posts: 44
Joined: Thu Sep 14, 2017 8:07 am
Location: Omsk, Russian Federation (UTC+6)

### Re: Toratope's nets and cross-sections

In order to help myself I made these animations. These two are the same approximate glome.
Attachments
Glome_c-s_approximate_pole-first.gif (150.62 KiB) Viewed 671 times
Glome_c-s_approximate_equator-first.gif (318.65 KiB) Viewed 671 times
icebreaker
Dionian

Posts: 44
Joined: Thu Sep 14, 2017 8:07 am
Location: Omsk, Russian Federation (UTC+6)

### Re: Cubinder's net

Can anybody give me a picture where the hypermeridians of a glome are highlighted? It should help me to make the equator-first cross sections. The hypersphere must be in such projection: http://hi.gher.space/thumb/439.png. UPD It must be orthographic, not perspective.
icebreaker
Dionian

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Joined: Thu Sep 14, 2017 8:07 am
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