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aplesch

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X3D: behind-the-scenes

23/12/2015 - 14:56:38
more gears, fast

11/11/2015 - 01:28:48
general smoothness

02/11/2015 - 08:18:08
smoother gears

30/10/2015 - 07:33:05
helical gears

26/10/2015 - 04:09:02
3d involute gears

21/10/2015 - 07:30:11
involute gear

16/10/2015 - 22:49:13
circle involute

09/10/2015 - 06:50:27
Flying Boxes

07/10/2015 - 00:07:10
A paper dragon

05/10/2015 - 15:41:53

more gears, fast

Submitted by aplesch on Wed, 11/11/2015 - 01:28

I updated the involute (curved teeth), helical gear generation procedure to use Extrusion rather than IndexedFaceSets. Extrusion automatically supports the back and front faces of the gears, and a smooth appearance by producing a mesh with points shared by triangles. So the logo code could be streamlined and therefore is now much faster generating gears.

However, since extrusions with teeth or other concave corners have some issues with current x3dom, the maximum number of teeth currently is 14. Here is a nice gears mesh with four such helical gears.

The main involute gear profile function remained unchanged but 3d gears are produced as extrusions. Here is the complete logo code for above:

ERALL
; involute: draw involute
; rb: base radius
; limit: max. generating angle
; steps: number of segments on involute to draw
; returns list of positions: top is first
 
TO involute :rb :limit :steps
 
  LOCALMAKE "start POS
  LOCALMAKE "i 0
  LOCALMAKE "step 0.5 * :limit / :steps
  LOCALMAKE "coords []
  PUSH "coords POS
  PU
 
  REPEAT ABS(:steps)
  [
    ;push "coords :start
    MAKE "i :i + :step
    ;move along circle
    LT :step
    FD 2 * :rb * SIN :step
    ;find tangent
    RT 180 - :step
    ;unroll string
    FD 2 * :i * :deg2rad * :rb
    ;make "start pos
    ;record position
    PUSH "coords POS
    BACK 2 * :i * :deg2rad * :rb
    ;turn forward
    LT 180
  ]
  SETPOS :start
  OUTPUT :coords
 
END
 
; gearinvolute: draw involute side of gear tooth
; rb: base radius
; PA: pressure angle
; P: diametral pitch
; steps: number of subdivisions
; returns list of position from landing to base
 
TO gearinvolute :rb :PA :P :steps
 
  LOCALMAKE "a 1 / :P
  LOCALMAKE "ro ( :rb / COS :PA ) + :a
  ;calculate generating angle for top of tooth
  ;(needs drawing)
  LOCALMAKE "l2 :ro ^ 2 - :rb ^ 2
  LOCALMAKE "limit ( SQRT :l2 ) / :rb
  LOCALMAKE "coords involute :rb (:limit / :deg2rad) :steps
  OUTPUT :coords
 
END
 
; geartooth: draw one full tooth
; rp: radius of pitch circle
; PA: pressure angle
; P: diametral pitch
; hdelta: half angular base width (expected)
; segments: number of subdivisions on each side
; returns list of positions from root to landing to root
 
TO geartooth :rp :PA :P :hdelta :segments
 
  LOCALMAKE "rb :rp * COS :PA
  LOCALMAKE "clr (2.157 - 2) / :P ;coarser 20P
  LOCALMAKE "ddif (1/:P + :clr) - (:rp - :rb)
  LOCALMAKE "coords []
  PUSH "coords POS
  ; record straight edge
  RT 90
  FD :ddif PUSH "coords POS
  LT 90
  ; save position
  LOCALMAKE "base POS
  LOCALMAKE "head HEADING
  ; record side
  MAKE "coords SENTENCE ( gearinvolute :rb :PA :P :segments ) :coords
  ;record with negative step
  LOCALMAKE "top POS
  SETPOS :base
  SETHEADING :head
  ;turn and move to other side of tooth
  ;half angular base width
  LT :hdelta
  FD 2 * :rb * ( SIN ( :hdelta ) )
  LT :delta / 2
  ;save pos
  MAKE "base POS
  MAKE "head HEADING
  ;draw other side
  ;prepend reverse
  MAKE "coords SENTENCE REVERSE ( gearinvolute :rb :PA :P :segments * -1 ) :coords
  ;return to base
  SETPOS :base
  SETHEADING :head
  ;record straight edge
  LT 90
  FD :ddif PUSH "coords POS
  RT 90
  OUTPUT :coords
 
END
 
; gear: draw involute gear
; rp: radius at pitch circle
; PA: pressure angle
; P: number of teeth normalized by pitch diameter
; segments: subdivions on each involute side of each tooth
; returns list of positions
 
TO gear :rp :PA :P :segments
 
  LOCALMAKE "cpitch :PI / :P
  LOCALMAKE "apitch (:cpitch / :rp) / :deg2rad
  LOCALMAKE "N :P * :rp * 2
  LOCALMAKE "rb :rp * COS :PA
  LOCALMAKE "clr (2.157 - 2) / :P ;coarser 20P
  LOCALMAKE "ded (1/:P + :clr)
  LOCALMAKE "rr :rp - :ded
  ;this is the main calculation
  ;for angular base width delta
  ;(needs drawing)
  LOCALMAKE "alpha ( TAN :prA ) / :deg2rad
  LOCALMAKE "beta :alpha - :prA
  LOCALMAKE "delta ( 0.5 * :apitch ) + ( 2 * :beta )
  LOCALMAKE "coords []
  ; move from center to root
  PU
  RT 90 FD :rr LT 90
  REPEAT :N
  [
    ;tooth from root to root
    MAKE "coords FPUT ( geartooth :rp :PA :P :delta / 2 :segments ) :coords
    ;advance to root of next tooth
    LT (:apitch - :delta) / 2
    FD 2 * :rr * SIN ((:apitch - :delta) / 2)
    LT (:apitch - :delta) / 2
  ]
  ;close up
  ;make "coords fput pos :coords
  LT 90 FD :rr RT 90
  OUTPUT :coords
 
END
 
; helicalGear: create a helical gear as an extrusion
;              with its internal center        
;              at the position and orientation of the turtle
; PR pitch radius
; PA pressure angle
; P pitch
; iS number of segments defining involute
; W depth
; HA helical angle
; fS number of segments across depth extent
 
TO helicalGear :PR :PA :P :iS :W :HA :fS
 
  ;save turtle
  LOCALMAKE "tpos POS
  LOCALMAKE "tori HEADING
  ; setup turtle for generating gear
  ; needs this starting configuration due to extrusion
  HOME RU 90
  ; produce gear in xz plane at origin
  LOCALMAKE "gear3d gearExtrusionY (gear :PR :PA :P :iS) :W :HA :PR :fS
  ; put at turtle position and orientation
  SET :gear3d "rotation :tori
  SET :gear3d "translation :tpos
  SETPOS :tpos
  SETHEADING :tori
  ;return object for further manipulation
  OUTPUT :gear3d
 
END
 
TO gearExtrusionY :positions :width :helix :r :fsegments
 
  ;positions are expected in the xy plane
  EXTRUSION
  LOCALMAKE "extrusion WORD OBJECT "_extrusion
  ;reset any initial rotation
  SET OBJECT "rotation [1 0 0 0]
  ;make cross-section
  ; just xy and close by repeating last point
  LOCALMAKE "cs flatten :positions
  SET :extrusion "crossSection flattenxy FPUT (LAST :cs) :cs
  ;set :extrusion "crossSection flattenxy reverse :cs
  ; make spine, scale, orientation per segment
  ; spine centered at 0
  LOCALMAKE "sp []
  LOCALMAKE "sc []
  LOCALMAKE "rot []
 
  LOCALMAKE "rothA (TAN :helix) * :width / :r
  FOR [s 0 :fsegments] [
    ; make spine
    ; needs to be in Y or X direction, use Y
    QUEUE "sp ( LIST 0 (:s * :width/:fsegments - :width/2) 0 )
    ; make scale, constant
    QUEUE "sc [1 1 1]
    ;make orientation for helix
    QUEUE "rot ( LIST 0 1 0 :s * :rothA / :fsegments )
    ]
  SET :extrusion "spine flatten :sp
  SET :extrusion "orientation flatten :rot
  SET :extrusion "scale flatten :sc
  SET :extrusion "creaseAngle 1.57
  SET :extrusion "endCap TRUE
  SET :extrusion "beginCap TRUE
 
  OUTPUT OBJECT
 
END
 
TO combxy :xyz1 :xyz2
  OUTPUT SENTENCE BL :xyz1 :xyz2
END
 
TO flattenxy :xyzpos
  OUTPUT BL REDUCE "combxy :xyzpos
END
 
 
TO flatten :listoflists
  OUTPUT REDUCE "sentence :listoflists
END
 
;involute (for gears)
;comprehensive guide
;http://bostongear.com/pdf/gear_theory.pdf
RESET
CLEARSCREEN
SETBACKGROUND 34
;pointlight
 
MAKE "PI 4 * RADARCTAN(1)
MAKE "deg2rad :PI / 180
MAKE "prA 20 ; pressure angle
MAKE "piR 0.5 ; pitch radius 1
MAKE "piR2 0.25; pitch radius 2
MAKE "pitch 14 ; pitch for all gears
MAKE "isegments 6 ; involute segments
MAKE "facesegments 4 ; number of cross-sections
MAKE "hAngle 45 ; helical angle
MAKE "depth 0.2 ; gear depth
 
;ru 90
;make first gear horizontally, at origin
TRANSFORM
MAKE "g1spin OBJECT
SETPARENT :g1spin
SETMATERIAL 3 33
MAKE "g1 helicalGear :piR :prA :pitch :isegments :depth :hAngle :facesegments
;use inner trafo for initial rotation
MAKE "beta (TAN :prA) - (:prA * :deg2rad)
SET :g1 "rotation (LIST 0 1 0 :beta)
SETPARENT "root
 
;make second gear horizontally, at origin
TRANSFORM
MAKE "g2spin OBJECT
SETPARENT :g2spin
SETMATERIAL 3 18
MAKE "g2 helicalGear :piR2 :prA :pitch :isegments :depth -:hAngle :facesegments
;use inner trafo for initial rotation
SET :g2 "rotation (LIST 0 1 0 :PI/:pitch/:piR2/2 + :beta)
SETPARENT "root
 
;make third gear horizontally, at origin
TRANSFORM
MAKE "g3pos OBJECT
SETPARENT :g3pos
TRANSFORM
MAKE "g3spin OBJECT
SETPARENT :g3spin
SETMATERIAL 3 26
MAKE "g3 helicalGear :piR2 :prA :pitch :isegments :depth -:hAngle :facesegments
;use inner trafo for initial rotation
SET :g3 "rotation (LIST 0 1 0  :PI/:pitch/:piR2/2 + :beta)
SETPARENT "root
 
;make fourth gear horizontally, at origin
TRANSFORM
MAKE "g4pos OBJECT
SETPARENT :g4pos
TRANSFORM
MAKE "g4spin OBJECT
SETPARENT :g4spin
SETMATERIAL 3 30
MAKE "g4 helicalGear :piR :prA :pitch :isegments :depth :hAngle :facesegments
;use inner trafo for initial rotation
SET :g4 "rotation (LIST 0 1 0  :PI/:pitch/:piR/2 + :beta)
SETPARENT "root
 
; place second gear
 
RT 90
FD :piR + :piR2
 
SET :g2spin "translation POS
 
; rotate and place third gear
 
FD 2*:piR2
 
SET :g3pos "rotation [1 0 0 -1.57]
SET :g3pos "translation POS
 
; rotate and place fourth gear
 
FD :piR2 + :piR
 
SET :g4pos "rotation [1 0 0 -1.57]
SET :g4pos "translation POS
 
;spin
MAKE "period1 10
 
SPIN :g1spin "lt :period1
SPIN :g2spin "rt :period1 * :piR2/:piR
SPIN :g3spin "rt :period1 * :piR2/:piR
SPIN :g4spin "lt :period1
 
PU
SHOWALL

And here is mesh with pitch 42 which requires a fix to x3dom which can be dynamically applied, see below.

If the front and back face of the large gears do not show, please use this user script with Greasemonkey (install here):

https://gist.githubusercontent.com/andreasplesch/9c7369583dc9f23d6fe7/raw/x3domEarClippingFixer.user.js

Reload the page after enabling the user script.

aplesch's blog
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Comments

ear clipping

Submitted by Andy on Wed, 11/11/2015 - 11:27.

Hope x3dom can pull this request soon. I am aware of the concave polygon issue for a long time and very glad to see it is fixed by your earclipping script.

The involute gear has much smoother gear teeth than my 4-circle gears. Thanks for enriching the contents of VRMath2.

I noticed that Blender and MeshLab cannot not import extrusion so that could be a down side for extrusion gears. The IndexedFaceSet gears will be still very useful... and I need to add script to remove the duplicate coordinates too.

merged

Submitted by aplesch on Thu, 12/11/2015 - 08:41.

My PR was merged. Since the PR was a bit premature though functional, I submitted a second PR with cleaned up code and some other improvements. It is available as a Greasemonkey user script as well, linked from the PR, and above.

It unfortunate that Blender and Meshlab cannot import extrusion. I guess they do not have a native corresponding object or concept. Since x3dom in effect produces an IndexedFaceSet from the the Extrusion it may be possible to come up with a converter.

The facet optimizing function should pretty much do what you need.

x3dom updated

Submitted by Andy on Thu, 12/11/2015 - 21:27.

VRMath2 is now using the x3dom with #576, therefore, the gears above are rendering correctly with both caps.

I have also tested your improved EarClipping in https://github.com/x3dom/x3dom/pull/577 and left a comment there.

remove duplicate coordinates

Submitted by Andy on Thu, 12/11/2015 - 15:43.

I have some options to do so. I can follow your codes after the coordinates of the indexedfaceset was created, or I can check for duplicate coordinates during the creation of indexedfactset (PENDOWN). I am thinking of the later. It may slow down the creation because of checking, but once checked it actually saved time because I only need to update the indicies and not the coordinate points and colors.

Do you know the internal structure of the extrusion in x3dom? There are probably some converters (at least this commercial one http://www.okino.com/conv/imp_vrml.htm) but It will be good if there is an api in x3dom to call.

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more gears, fast

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ear clipping

merged

x3dom updated

remove duplicate coordinates

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