# 3D Solids ### 1 collaborator Uri Wilensky (Author)

### Tags

(This model has yet to be categorized with any tags)
Model group CCL | Visible to everyone | Changeable by group members (CCL)
Model was written in NetLogo 5.0.4 • Viewed 842 times • Downloaded 28 times • Run 0 times Download this modelEmbed this model

## WHAT IS IT?

This model creates 3D shapes out of 2D turtles by mapping turtles between cartesian and spherical three-dimensional coordinates.

## HOW IT WORKS

To create the 3D shapes the program randomly generates turtles about the shell of the shape in either cartesian (x, y, z) or spherical (theta, phi, z) coordinates, depending on which is easier to accomplish but always stores the information converting when necessary in spherical coordinates. To render the sphere in the NetLogo view, it translates the turtles from spherical to cartesian coordinates using color to simulate depth. The positions of the turtles are always stored as spherical coordinates because they are rotated on the z-axis, and the simplest way to do so is to increase theta in spherical coordinates.

Converting from cartesian to spherical coordinates:

x = r * cos(theta) = p * sin(phi) * cos(theta)
y = r * sin(theta) = p * sin(phi) * sin(theta)
z = p * cos(theta)

theta: angle of the turtle's projection on the x-y plane.
phi: turtles angle of incidence to the z axis.
p: distance of the turtle from the origin.

## HOW TO USE IT

Click the different setup-SHAPE buttons to generate different 3D shapes. The turtles are randomly distributed about the surface of the shape. Click the go (forever) button to run the model.

GO starts rotating the model.

COLOR determines the color that is used to simulate depth in generating the various shapes (uses predefined NetLogo color constants).

NUM-TURTLES determines the number of turtles that are used to generate the various shapes.

SHAPE-SIZE determines the overall size of the shape. Most often it is radius or edge length.

THETA-VELOCITY determines the speed at which the turtles are rotated.

(Rotating turtles in the rotate-turtles procedure is implemented simply by increasing each turtle's theta variable by theta-velocity! Rotating turtles (around the z-axis) is easy in spherical coordinates. However it's far easier to transpose turtles in cartesian coordinates.)

## THINGS TO NOTICE

Notice that turtles closer (positive) on the y-axis appear lighter in shade and turtles further away (negative) appear darker in shade.

## THINGS TO TRY

Try adjusting the theta-vel or render-color slider as the model is running. This will provide real time feedback to your adjustments.

## EXTENDING THE MODEL

[EASY] Adjust the `setup-square` procedure to generate a rectangle.

Create a procedure to transpose turtle coordinates. Remember that it is easier to transpose in cartesian coordinates.

Create a procedure to generate new 3D geometries.

Try animating the phi variable. Conceptually why does this not make sense?

Create a procedure to rotate the geometries on a different axis.

[VERY DIFFICULT] Create a procedure to view the geometries at ANY angle instead of the present three.

## NETLOGO FEATURES

Notice the use of `scale-color` to show the depth of turtles and thereby simulate 3D.

## HOW TO CITE

If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:

• Wilensky, U. (1998). NetLogo 3D Solids model. http://ccl.northwestern.edu/netlogo/models/3DSolids. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.
• Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL. This model was created as part of the project: CONNECTED MATHEMATICS: MAKING SENSE OF COMPLEX PHENOMENA THROUGH BUILDING OBJECT-BASED PARALLEL MODELS (OBPML). The project gratefully acknowledges the support of the National Science Foundation (Applications of Advanced Technologies Program) -- grant numbers RED #9552950 and REC #9632612.

This model was converted to NetLogo as part of the projects: PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT. The project gratefully acknowledges the support of the National Science Foundation (REPP & ROLE programs) -- grant numbers REC #9814682 and REC-0126227. Converted from StarLogoT to NetLogo, 2001.

Click to Run Model

```turtles-own [
x-pos  ;; x-pos, y-pos, and z-pos are the cartesian coordinates
y-pos  ;; don't confuse them with xcor and ycor, which are predefined
z-pos  ;;   NetLogo variables for turtles
p      ;; p, theta, and phi are the spherical coordinates
theta
phi
]

to setup
clear-all
set-default-shape turtles "circle"
end

to setup-sphere
setup
;; generate a sphere of radius SHAPE-SIZE
crt num-turtles
[
set p shape-size            ; all points are equal distance from the center
set theta random-float 360  ; however, random distribution on the surface of the sphere
set phi random-float 180
render-turtle
]
reset-ticks
end

; a filled 3D cube

to setup-cube-filled
setup
;; generate a square with edge of length radius
;; placing a point randomly anywhere inside the square
crt num-turtles
[
cartesian ((- shape-size) + 2 * (random-float shape-size))
((- shape-size) + 2 * (random-float shape-size))
((- shape-size) + 2 * (random-float shape-size))
render-turtle
]
reset-ticks
end

; cube with turtles only on its surface

to setup-cube-surface
setup
crt num-turtles
[
let temp-alpha shape-size * (1 - 2 * (random 2))   ; +-shape-size
; random distribution bounded by +-shape-size
let temp-beta shape-size - 2 * (random-float shape-size)
let temp-gamma (random 2)                          ; front/back or left/right?
ifelse temp-gamma = 0                              ; generate front & back surfaces
[
cartesian (temp-alpha)
(temp-beta)
(shape-size - (2 * (random-float shape-size)))
]
[
cartesian (temp-beta)                             ; generating the side surfaces
(temp-alpha)
(shape-size - (2 * (random-float shape-size)))
]
render-turtle
]
reset-ticks
end

; 3D cone

to setup-cone
setup
crt num-turtles
[
set theta (random-float 360)        ; points have a random angle
set p (random-float shape-size)
cartesian (p * (cos theta))     ; x = r*cos(theta)
(p * (sin theta))     ; y = r*sin(theta)
(shape-size - 2 * p)  ; this centers the cone at the origin
; instead of it only being in positive space
render-turtle
]
reset-ticks
end

; vertical cylinder

to setup-cylinder-v
setup
;the code is almost the same as the setup-cone code
;except the xy-plane radius remains constant
crt num-turtles
[
let temp-alpha (random 3) - 1         ; which surface (left, right, or body?)
set theta (random-float 360)
ifelse temp-alpha = 0
[
cartesian (shape-size * (cos theta))
(shape-size * (sin theta))
((- shape-size) + 2 * (random-float shape-size))
]
[
cartesian ((random-float shape-size) * (cos theta))
((random-float shape-size) * (sin theta))
(temp-alpha * shape-size)
]
render-turtle
]
reset-ticks
end

; horizontal cylinder

to setup-cylinder-h
setup
;generates a cylinder in a horizontal position with capped ends
crt num-turtles
[
let temp-alpha (random 3) - 1      ; which surface (left, right, or body?)
set theta (random-float 360)
ifelse temp-alpha = 0              ; generating the actual cylinder
[
cartesian ((- shape-size) + 2 * (random-float shape-size))
(shape-size * (cos theta))
(shape-size * (sin theta))
]
[
cartesian (temp-alpha * shape-size)
((random-float shape-size) * (cos theta))
((random-float shape-size) * (sin theta))
]
render-turtle
]
reset-ticks
end

to setup-pyramid
setup
crt num-turtles
[
let temp-alpha (- shape-size) + 2 * (random-float shape-size)  ; z coordinate
set theta (random 2)                         ; front/back or side?
let temp-beta (shape-size - temp-alpha) / 2
let temp-gamma -1 + 2 * (random 2)           ; left/right or front/back (-1 or 1)
ifelse theta = 0
[
cartesian (temp-beta * temp-gamma)          ;  left & right surfaces
((- temp-beta) + 2 * (random-float temp-beta))
(temp-alpha)
]
[
cartesian ((- temp-beta) + 2 * (random-float temp-beta)) ;  front & back surfaces
(temp-beta * temp-gamma)
(temp-alpha)
]
render-turtle
]
reset-ticks
end

;; convert from cartesian to spherical coordinates

to cartesian [x y z]
set p sqrt((x ^ 2) + (y ^ 2) + (z ^ 2))
set phi (atan sqrt((x ^ 2) + (y ^ 2)) z)
set theta (atan y x)
end

to go
; rotate-turtles on z axis
[
set theta (theta + theta-velocity) mod 360  ; increment angle to simulate rotation
render-turtle
]
tick
end

to render-turtle
calculate-turtle-position
set-turtle-position
end

;; convert from spherical to cartesian coordinates

to calculate-turtle-position
set y-pos p * (sin phi) * (sin theta)
set x-pos p * (sin phi) * (cos theta)
set z-pos p * (cos phi)
end

;; set the turtle's position and color

to set-turtle-position
ifelse view = "side"                                     ; sideview
[
setxy x-pos z-pos
set color scale-color display-color y-pos (- shape-size) shape-size
]
[
ifelse view = "top"                                  ; topview
[
setxy x-pos y-pos
set color scale-color display-color z-pos (- shape-size) shape-size
]
[
setxy (p * (sin phi) * (cos theta))              ; bottomview
(- (p * (sin phi) * (sin theta)))
set color scale-color display-color (- z-pos) (- shape-size) shape-size
]
]
end

```

There are 10 versions of this model.

Uri Wilensky over 7 years ago Updated to NetLogo 5.0.4 Download this version
Uri Wilensky almost 8 years ago Updated version tag Download this version
Uri Wilensky almost 9 years ago Updated to NetLogo 5.0 Download this version
Uri Wilensky over 10 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 10 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 10 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 10 years ago Updated from NetLogo 4.1 Download this version
Uri Wilensky over 10 years ago Model from NetLogo distribution Download this version