Mobius Strip?

[fuccaro]

No offense, my Scottish friends. Just some sense of humor ple(eee)ase, if it is possible!

[Mr T]

No offense, my Scottish friends. Just some sense of humor ple(eee)ase, if it is possible!

Ah it's ok. The funny thing is that on average scots give more to charity than their british counterparts.

Nick

[gcp310]

Lets wait to the Rugby World Cup results for some humour.

Not looking so far for you UK folks??

G

[fuccaro]

For a solid Mobius strip:

;|    Solid Mobius strip
    mfuccaro@hotmail.com
------------------------------October, 2004 ----------|;

(defun c:mobius( / w h r ang0 ang1 ang2 c0 N0 N1 N2 segs lx ly ss)

  (setq w 1        ;width of cross section
    h 60        ;height
    r 75        ;inner radius
    segs 80        ;number of segments
    )
  ; prepare the screen
  (command "vpoint" "1,1,-1")
  (command "box" (list (setq lx (+ r w h)) (setq ly (max w h)) lx)
            (list (- lx) (- ly) (- lx)))
  (command "zoom" "e")
  (entdel (entlast))
  (setq osmode (getvar "osmode"))
  (setvar "osmode" 0)
  (setq ang0 (/ PI segs)
    c0 (list (+ r (/ w 2.0)) 0 0)
    N0 (list
         (list r (/ h -2.0) 0)
         (list (+ r w) (/ h -2.0) 0)
         (list (+ r w) (/ h 2.0) 0)
         (list r (/ h 2.0) 0))
    ang1 (- ang0))
  (setq ss (ssadd))
  (repeat segs
    (setq ang1 (+ ang1 ang0)
      ang2 (+ ang1 ang0)
      N1 (tw_rot N0 c0 ang1)
      N2 (tw_rot N0 c0 ang2))
    (draw_solid N1 N2)
    (ssadd (entlast) ss)
    )
  (command "union" ss "")
  (setvar "osmode" osmode)
  (princ)
  )
  
(defun tw_rot (N c ang)
  ;twists a cross section N around C with angle ANG
  ;than rotates the twisted section around OY with angle 2*ANG
  (mapcar '(lambda (p)
         (list (* (car p) (sin (* 2.0 ang))) (cadr p) (* (car p) (cos (* 2.0 ang)))))
      (mapcar '(lambda (p) (polar c (+ ang (angle p c)) (distance p c))) N))
  )

(defun draw_solid (N1 N2 / maxx maxy maxz minx miny minz i j k)
  ; Draws the bounding box of the two consecutive cross sections N1 and N2
  (setq N12 (append N1 N2)
    maxx (apply 'max (mapcar 'car N12))
    minx (apply 'min (mapcar 'car N12))
    maxy (apply 'max (mapcar 'cadr N12))
    miny (apply 'min (mapcar 'cadr N12))
    maxz (apply 'max (mapcar 'caddr N12))
    minz (apply 'min (mapcar 'caddr N12)))
  (command "box" (list minx miny minz) (list maxx maxy maxz))
  ; The box is sliced according to the two sections
  (command "slice" "l" "" (car N1) (cadr N1) (caddr N1) (car N2))
  (command "slice" "l" "" (car N2) (cadr N2) (caddr N2) (car N1))
  (setq i -1)
  (repeat 4
    (setq i (1+ i))
    (setq j (if (= i 3) 0 (1+ i)))
    (setq k (if (= j 3) 0 (1+ j)))
    (command "slice" "l" "" (nth i N1) (nth i N2) (nth j N1) (nth k N1))
    (command "slice" "l" "" (nth j N1) (nth j N2) (nth i N2) (nth k N2))
    )
  )

*** editing ***
the "clean code":

;|    Solid Mobius strip
    mfuccaro@hotmail.com
------------------------------October, 2004 ----------|;

(defun c:mobius( / w h r ang0 ang1 ang2 c0 N0 N1 N2 segs lx ly ss)

  (setq w 1        ;width of cross section
    h 60        ;height
    r 75        ;inner radius
    segs 80        ;number of segments
    )
  ; prepare the screen
  (command "vpoint" "1,1,-1")
  (command "box" (list (setq lx (+ r w h)) (setq ly (max w h)) lx)
            (list (- lx) (- ly) (- lx)))
  (command "zoom" "e")
  (entdel (entlast))
  (setq osmode (getvar "osmode"))
  (setvar "osmode" 0)
  (setq ang0 (/ PI segs)
    c0 (list (+ r (/ w 2.0)) 0 0)
    N0 (list
         (list r (/ h -2.0) 0)
         (list (+ r w) (/ h -2.0) 0)
         (list (+ r w) (/ h 2.0) 0)
         (list r (/ h 2.0) 0))
    ang1 (- ang0))
  (setq ss (ssadd))
  (repeat segs
    (setq ang1 (+ ang1 ang0)
      ang2 (+ ang1 ang0)
      N1 (tw_rot N0 c0 ang1)
      N2 (tw_rot N0 c0 ang2))
    (draw_solid N1 N2)
    (ssadd (entlast) ss)
    )
  (command "union" ss "")
  (setvar "osmode" osmode)
  (princ)
  )
  
(defun tw_rot (N c ang)
  ;twists a cross section N around C with angle ANG
  ;than rotates the twisted section around OY with angle 2*ANG
  (mapcar '(lambda (p)
         (list (* (car p) (sin (* 2.0 ang))) (cadr p) (* (car p) (cos (* 2.0 ang)))))
      (mapcar '(lambda (p) (polar c (+ ang (angle p c)) (distance p c))) N))
  )

(defun draw_solid (N1 N2 / maxx maxy maxz minx miny minz i j k)
  ; Draws the bounding box of the two consecutive cross sections N1 and N2
  (setq N12 (append N1 N2)
    maxx (apply 'max (mapcar 'car N12))
    minx (apply 'min (mapcar 'car N12))
    maxy (apply 'max (mapcar 'cadr N12))
    miny (apply 'min (mapcar 'cadr N12))
    maxz (apply 'max (mapcar 'caddr N12))
    minz (apply 'min (mapcar 'caddr N12)))
  (command "box" (list minx miny minz) (list maxx maxy maxz))
  ; The box is sliced according to the two sections
  (command "slice" "l" "" (car N1) (cadr N1) (caddr N1) (car N2))
  (command "slice" "l" "" (car N2) (cadr N2) (caddr N2) (car N1))
  (setq i -1)
  (repeat 4
    (setq i (1+ i))
    (setq j (if (= i 3) 0 (1+ i)))
    (setq k (if (= j 3) 0 (1+ j)))
    (command "slice" "l" "" (nth i N1) (nth i N2) (nth j N1) (nth k N1))
    (command "slice" "l" "" (nth j N1) (nth j N2) (nth i N2) (nth k N2))
    )
  )

[stupens]

Hello everyone!

I've just joined the CADTutor community and I must thank everyone for their efforts in passing on AutoCAD knowledge. Excellent website!

In particular, I want to thank Fuccaro for his Mobius Surface routine which has proven extremely useful.

However, I don´t know a thing about the AutoLisp language; and I want to modify this routine to generate a Mobius Strip with an elliptical section instead of a circular one.

I need to create strips with both major and minor axis perpendicular to the folding point and be able to modify this dimensions. Any thoughts on how can this be accomplished?

Thanks in advance!

[fuccaro]

Hello Stupens!
I thought that the Mobius strip is a theoretical thing -I am surprised to see this big interest. Well, I wrote the programs for fun and I provided no options for changing the shape, the radius and the other parameters. Changes must be operated in the program.
I will look what I can do but not these days. Sorry!

[fuccaro]

OK, here it is!
The cross section is eliptycal.
But let's call it "Twisted thorus" or so because it is not a Mobius strip any more.
It is more complex as the previous one. As solid it would take forever to be generated, so I created it as a surface. It looks smoothr too.



Just for fun:

; Twisted ELiptycal Thorus
;    mfuccaro@hotmail.com
;------------------------------ November 2004 ----
(defun c:telto( / r0 r1 r2 sg a0 a1 l0 l1 osmode)
  (setq r0 (getreal "\nbase radius R0 ")
    r1 (getreal "    Major radius R1 ")
    r2 (getreal "    minor radius R2 "))
  (setq sg 40    ;surface quality
    a0 0.0 a1 (/ PI (1- sg) 0.5) st (list r0 0) l0 nil)
  (mapcar 'setvar '("osmode" "cmdecho") '(0 0))
  (command "3DMESH" sg sg)
  (setq osmode (getvar "OSMODE"))
  (repeat sg
    (setq l0 (cons (list (+ r0 (* r1 (cos (setq a0 (+ a0 a1))))) (* r2 (sin a0))) l0)))
; l0 contains the points of the ellipse
  (setq a0 0.0)
  (repeat sg
    (setq a0 (+ a0 a1) l1 nil)
    (foreach p l0
    (setq l1 (cons (polar st (+ (* 0.5 a0) (angle st p)) (distance st p)) l1))
      )
; l1 is the l0 cross section twised
    (foreach p l1
      (command (list (* (sin a0) (car p)) (cadr p) (* (cos a0) (car p))))
; l1 is rotated around OZ and sent to AutoCAD
      )
    )
  (setvar "OSMODE" osmode)
  )

*** editing ***
The cleaned code:

; Twisted ELiptycal Thorus
;    mfuccaro@hotmail.com
;------------------------------ November 2004 ----
(defun c:telto( / r0 r1 r2 sg a0 a1 l0 l1 osmode)
  (setq r0 (getreal "\nbase radius R0 ")
    r1 (getreal "    Major radius R1 ")
    r2 (getreal "    minor radius R2 "))
  (setq sg 40    ;surface quality
    a0 0.0 a1 (/ PI (1- sg) 0.5) st (list r0 0) l0 nil)
  (mapcar 'setvar '("osmode" "cmdecho") '(0 0))
  (command "3DMESH" sg sg)
  (setq osmode (getvar "OSMODE"))
  (repeat sg
    (setq l0 (cons (list (+ r0 (* r1 (cos (setq a0 (+ a0 a1))))) (* r2 (sin a0))) l0)))
; l0 contains the points of the ellipse
  (setq a0 0.0)
  (repeat sg
    (setq a0 (+ a0 a1) l1 nil)
    (foreach p l0
    (setq l1 (cons (polar st (+ (* 0.5 a0) (angle st p)) (distance st p)) l1))
      )
; l1 is the l0 cross section twised
    (foreach p l1
      (command (list (* (sin a0) (car p)) (cadr p) (* (cos a0) (car p))))
; l1 is rotated around OZ and sent to AutoCAD
      )
    )
  (setvar "OSMODE" osmode)
  )

[hyposmurf]

Nice work Fuccaro

[stupens]

Hello Fuccaro!

First of all I must say I look at your talent in awe and wonder...
How have you acquired all of this knowledge?

I thank you very much for your time and effort, but I must admit I didn't explain myself accurately:

...and I want to modify this routine to generate a Möbius Strip with an elliptical section instead of a circular one...

I used the term "section" meaning the plan of the strip instead of the cross section. Sorry, my mistake.

The kind of surfaces I need to model have an elliptical plan such as these:




You might wonder why all the interest in Möbius Bands?

I'm an architecture student currently analizing the Max Reinhardt Haus project by Peter Eisenman. This project designed in 1992, unbuilt; is based upon the development of surfaces based on a Mobius Strip.



And this is the corresponding geometry at the core, in plan and elevation...



Notice how the "interior ellipse" is off-centered from the "exterior" one?
If I can configure the strip, i'll be able to recreate the volume that defines the surface of the building. (if I can rotate the planes according to the constant turn of the band, that is...)

Anyway, this work is due next tuesday and I'm running short. I'm going to focus on finishing the rest of the analysis first and i'll leave the 3d model as a bonus if there's some extra time left.

Once again, thank you for your time and patience.
I'm sure the Twisted Torus will become famous...

Keep up the good work!

[fuccaro]

These are with rectangular cross section and eliptycal path:

(defun c:project( / r1 r2 w h sg l0 lt lr ld a0 a1 orig osmode)
; by Fuccaro Miklos -November 2005
;
  (setq r1 (getreal "\n minor radius ")
    r2 (getreal "   MAJOR radius ")
    w (* 0.5 (getreal "\nCROSS SECTION: width "))
    h (* 0.5 (getreal "   height")))
  (setq sg 100 tw0 (/ PI 4))
  ; sg = 3...256
  (setq l0 (list (list (- w) (- h))
         (list w (- h))
         (list w h)
         (list (- w) h))
; l0= point list; cross section in "neutral position"
    a1 (/ pi (1- sg)) a0 (- a1)
    orig '(0 0))
  (setq osmode (getvar "OSMODE"))
  (mapcar 'setvar '("cmdecho" "osmode") '(0 0))
  (command "3dmesh" sg 5)
  (repeat sg
    (setq a0 (+ a0 a1))
    (setq lt (mapcar '(lambda (p)
            (polar orig (+ (angle orig p) (* 0.5 a0) tw0) (distance orig p)))
             l0))
; lt=point list; the cross section is twisted
    (setq lr (mapcar '(lambda (p)
            (list (* (car p) (cos a0))
                  (cadr p)
                  (* (car p) (sin a0))))
             lt))
; lr=point list; the twisted cross section is rotated around OY
    (setq ld (mapcar '(lambda (p)
            (list (+ (car p) (* r1 (cos a0)))
                  (cadr p)
                  (+ (caddr p) (* r2 (sin a0)))))
             lr))
; ld=point list; lr moved in place
    (foreach p ld (command p))
    (command (car ld))
    )
  (setvar "osmode" osmode)
  )

*** editing ***
The cleaned code:

(defun c:project( / r1 r2 w h sg l0 lt lr ld a0 a1 orig osmode)
; by Fuccaro Miklos -November 2005
;
  (setq r1 (getreal "\n minor radius ")
    r2 (getreal "   MAJOR radius ")
    w (* 0.5 (getreal "\nCROSS SECTION: width "))
    h (* 0.5 (getreal "   height")))
  (setq sg 100 tw0 (/ PI 4))
  ; sg = 3...256
  (setq l0 (list (list (- w) (- h))
         (list w (- h))
         (list w h)
         (list (- w) h))
; l0= point list; cross section in "neutral position"
    a1 (/ pi (1- sg)) a0 (- a1)
    orig '(0 0))
  (setq osmode (getvar "OSMODE"))
  (mapcar 'setvar '("cmdecho" "osmode") '(0 0))
  (command "3dmesh" sg 5)
  (repeat sg
    (setq a0 (+ a0 a1))
    (setq lt (mapcar '(lambda (p)
            (polar orig (+ (angle orig p) (* 0.5 a0) tw0) (distance orig p)))
             l0))
; lt=point list; the cross section is twisted
    (setq lr (mapcar '(lambda (p)
            (list (* (car p) (cos a0))
                  (cadr p)
                  (* (car p) (sin a0))))
             lt))
; lr=point list; the twisted cross section is rotated around OY
    (setq ld (mapcar '(lambda (p)
            (list (+ (car p) (* r1 (cos a0)))
                  (cadr p)
                  (+ (caddr p) (* r2 (sin a0)))))
             lr))
; ld=point list; lr moved in place
    (foreach p ld (command p))
    (command (car ld))
    )
  (setvar "osmode" osmode)
  )