Animate a pendulum
You are encouraged to solve this task according to the task description, using any language you may know.
One good way of making an animation is by simulating a physical system and illustrating the variables in that system using a dynamically changing graphical display. The classic such physical system is a simple gravity pendulum.
For this task, create a simple physical model of a pendulum and animate it.
C
<lang c>#include <stdio.h>
- include <stdlib.h>
- include <stdbool.h>
- include <math.h>
- include <cairo.h>
- include <SDL.h>
- define WIDTH 320.0
- define HEIGHT 240.0
- ifndef M_PI
- define M_PI 3.14159265358979323846
- endif
const double L = HEIGHT*150.0/200.0; // length double theta = M_PI/4.0; double dtheta = 0.0; int homeX = WIDTH / 2; int homeY = HEIGHT*25.0/200.0;
static uint32_t update_anim(uint32_t iv, void *p) {
double scaling = 30.0 / (L*L); // this seems better for this impl. double firstDDtheta = -sin(theta) * scaling; double midDtheta = dtheta + firstDDtheta; double midtheta = theta + (dtheta + midDtheta)/2.0;
double midDDtheta = -sin(midtheta) * scaling; midDtheta = dtheta + (firstDDtheta + midDDtheta)/2.0; midtheta = theta + (dtheta + midDtheta)/2.0;
midDDtheta = -sin(midtheta) * scaling; double lastDtheta = midDtheta + midDDtheta; double lasttheta = midtheta + (midDtheta + lastDtheta)/2.0;
double lastDDtheta = -sin(lasttheta) * scaling; lastDtheta = midDtheta + (midDDtheta + lastDDtheta)/2.0; lasttheta = midtheta + (midDtheta + lastDtheta)/2.0;
dtheta = lastDtheta; theta = lasttheta;
return iv;
}
void show_pendolum(SDL_Surface *s) {
cairo_surface_t *surf = cairo_image_surface_create_for_data( s->pixels,
CAIRO_FORMAT_RGB24, s->w, s->h, s->pitch);
cairo_t *ct = cairo_create(surf);
double x = homeX + L*sin(theta); double y = homeY + L*cos(theta);
cairo_set_source_rgba(ct, 0, 0, 0, 1); cairo_new_path(ct); cairo_rectangle(ct, 0, 0, WIDTH, HEIGHT); cairo_fill(ct);
cairo_set_source_rgba(ct, 1, 1, 1, 1); // rod cairo_new_path(ct); cairo_move_to(ct, homeX, homeY); cairo_line_to(ct, x, y); cairo_stroke(ct);
cairo_set_source_rgba(ct, 1, 1, 0, 1); // bob cairo_new_path(ct); cairo_move_to(ct, x, y); cairo_arc(ct, x, y, 20.0, 0, 2*M_PI); cairo_fill(ct);
cairo_set_source_rgba(ct, 0.7, 0.7, 0.7, 1);
// plate cairo_new_path(ct); cairo_move_to(ct, 0, homeY); cairo_line_to(ct, WIDTH, homeY); cairo_stroke(ct);
// pivot cairo_new_path(ct); cairo_move_to(ct, homeX, homeY); cairo_arc(ct, homeX, homeY, 5.0, 0, 2*M_PI); cairo_fill(ct);
cairo_surface_destroy(surf); cairo_destroy(ct);
}
int main() {
SDL_Surface *scr, *t; SDL_Event event[1]; bool quit = false;
if ( SDL_Init(SDL_INIT_VIDEO | SDL_INIT_TIMER) >= 0 ) { atexit(SDL_Quit); if ( SDL_SetVideoMode(WIDTH, HEIGHT, 32, SDL_DOUBLEBUF) != NULL ) { scr = SDL_GetVideoSurface(); SDL_AddTimer(30, update_anim, NULL);
event->type = SDL_VIDEOEXPOSE; SDL_PushEvent(event);
while(SDL_WaitEvent(event) && !quit) {
switch(event->type) { case SDL_VIDEOEXPOSE: while(SDL_LockSurface(scr) != 0) SDL_Delay(1);
show_pendolum(scr);
SDL_UnlockSurface(scr); SDL_Flip(scr); event->type = SDL_VIDEOEXPOSE; SDL_PushEvent(event); break; case SDL_KEYDOWN: if (event->key.keysym.sym == SDLK_q) quit = true; break; }
}
SDL_FreeSurface(scr); } }
return 0;
}</lang>
E
(Uses Java Swing for GUI. The animation logic is independent, however.)
The angle of a pendulum with length and acceleration due to gravity with all its mass at the end and no friction/air resistance has an acceleration at any given moment of
- Failed to parse (syntax error): {\displaystyle \frac{d^2θ}{dt^2}\theta = -\frac{g}{L} \sin \theta}
This simulation uses this formula directly, updating the velocity from the acceleration and the position from the velocity; inaccuracy results from the finite timestep.
The event flow works like this: The clock object created by the simulation steps the simulation on the specified in the interval. The simulation writes its output to angle
, which is a Lamport slot which can notify of updates. The whenever set up by makeDisplayComponent
listens for updates and triggers redrawing as long as interest has been expressed, which is done whenever the component actually redraws, which happens only if the component's window is still on screen. When the window is closed, additionally, the simulation itself is stopped and the application allowed to exit. (This logic is more general than necessary; it is designed to be suitable for a larger application as well.)
<lang e>#!/usr/bin/env rune pragma.syntax("0.9")
def pi := (-1.0).acos() def makeEPainter := <unsafe:com.zooko.tray.makeEPainter> def makeLamportSlot := <import:org.erights.e.elib.slot.makeLamportSlot> def whenever := <import:org.erights.e.elib.slot.whenever> def colors := <import:java.awt.makeColor>
- --------------------------------------------------------------
- --- Definitions
def makePendulumSim(length_m :float64,
gravity_mps2 :float64, initialAngle_rad :float64, timestep_ms :int) { var velocity := 0 def &angle := makeLamportSlot(initialAngle_rad) def k := -gravity_mps2/length_m def timestep_s := timestep_ms / 1000 def clock := timer.every(timestep_ms, fn _ { def acceleration := k * angle.sin() velocity += acceleration * timestep_s angle += velocity * timestep_s }) return [clock, &angle]
}
def makeDisplayComponent(&angle) {
def c def updater := whenever([&angle], fn { c.repaint() }) bind c := makeEPainter(def paintCallback { to paintComponent(g) { try { def originX := c.getWidth() // 2 def originY := c.getHeight() // 2 def pendRadius := (originX.min(originY) * 0.95).round() def ballRadius := (originX.min(originY) * 0.04).round() def ballX := (originX + angle.sin() * pendRadius).round() def ballY := (originY + angle.cos() * pendRadius).round()
g.setColor(colors.getWhite()) g.fillRect(0, 0, c.getWidth(), c.getHeight()) g.setColor(colors.getBlack()) g.fillOval(originX - 2, originY - 2, 4, 4) g.drawLine(originX, originY, ballX, ballY) g.fillOval(ballX - ballRadius, ballY - ballRadius, ballRadius * 2, ballRadius * 2) updater[] # provoke interest provided that we did get drawn (window not closed) } catch p { stderr.println(`In paint callback: $p${p.eStack()}`) } } }) c.setPreferredSize(<awt:makeDimension>(300, 300)) return c
}
- --------------------------------------------------------------
- --- Application setup
def [clock, &angle] := makePendulumSim(1, 9.80665, pi*99/100, 10)
- Initialize AWT, move to AWT event thread
when (currentVat.morphInto("awt")) -> {
# Create the window def frame := <unsafe:javax.swing.makeJFrame>("Pendulum") frame.setContentPane(def display := makeDisplayComponent(&angle)) frame.addWindowListener(def mainWindowListener { to windowClosing(_) { clock.stop() interp.continueAtTop() } match _ {} }) frame.setLocation(50, 50) frame.pack()
# Start and become visible frame.show() clock.start()
}
interp.blockAtTop()</lang>
Factor
Approximation of the pendulum for small swings : theta = theta0 * cos(omega0 * t) <lang factor>USING: accessors alarms arrays calendar colors.constants kernel locals math math.constants math.functions math.rectangles math.vectors opengl sequences system ui ui.gadgets ui.render ; IN: pendulum
CONSTANT: g 9.81 CONSTANT: l 20 CONSTANT: theta0 0.5
- current-time ( -- time ) nano-count -9 10^ * ;
- T0 ( -- T0 ) 2 pi l g / sqrt * * ;
- omega0 ( -- omega0 ) 2 pi * T0 / ;
- theta ( -- theta ) current-time omega0 * cos theta0 * ;
- relative-xy ( theta l -- xy )
[ [ sin ] [ cos ] bi ] [ [ * ] curry ] bi* bi@ 2array ;
- theta-to-xy ( origin theta l -- xy ) relative-xy v+ ;
TUPLE: pendulum-gadget < gadget alarm ;
- O ( gadget -- origin ) rect-bounds [ drop ] [ first 2 / ] bi* 0 2array ;
- window-l ( gadget -- l ) rect-bounds [ drop ] [ second ] bi* ;
- gadget-xy ( gadget -- xy ) [ O ] [ drop theta ] [ window-l ] tri theta-to-xy ;
M: pendulum-gadget draw-gadget*
COLOR: black gl-color [ O ] [ gadget-xy ] bi gl-line ;
M:: pendulum-gadget graft* ( gadget -- )
[ gadget relayout-1 ] 20 milliseconds every gadget (>>alarm) ;
M: pendulum-gadget ungraft* alarm>> cancel-alarm ;
- <pendulum-gadget> ( -- gadget )
pendulum-gadget new { 500 500 } >>pref-dim ;
- pendulum-main ( -- )
[ <pendulum-gadget> "pendulum" open-window ] with-ui ;
MAIN: pendulum-main </lang>
J
<lang j>require 'gl2 trig' coinsert 'jgl2'
DT =: %30 NB. seconds ANGLE=: 0.25p1 NB. radians L =: 1 NB. metres G =: 9.80665 NB. ms_2 VEL =: 0 NB. ms_1
PEND=: noun define pc pend;pn "Pendulum"; xywh 0 0 320 200;cc isi isigraph rightmove bottommove; pas 0 0;pcenter; rem form end; )
pend_run =: verb def ' wd PEND,;pshow;timer ,":DT * 1000 ' pend_close =: verb def ' wd timer 0; pclose ' pend_isi_paint=: verb def ' drawPendulum ANGLE '
sys_timer_z_=: verb define
recalcAngle wd 'psel pend; setinvalid isi'
)
recalcAngle=: verb define
accel=. - (G % L) * sin ANGLE VEL =: VEL + accel * DT ANGLE=: ANGLE + VEL * DT
)
drawPendulum=: verb define
width=. {. glqwh ps=. (-: width) , 40 pe=. ps + 280 <.@* (cos , sin) 0.5p1 + y NB. adjust orientation glrgb 91 91 91 glbrush gllines ps , pe glellipse (,~ ps - -:) 40 15 glellipse (,~ pe - -:) 20 20 glrect 0 0 ,width, 40
)
pend_run NB. run animation</lang>
Logo
<lang logo>make "angle 45 make "L 1 make "bob 10
to draw.pendulum
clearscreen seth :angle+180 ; down on screen is 180 forward :L*100-:bob penup forward :bob pendown arc 360 :bob
end
make "G 9.80665 make "dt 1/30 make "acc 0 make "vel 0
to step.pendulum
make "acc -:G / :L * sin :angle make "vel :vel + :acc * :dt make "angle :angle + :vel * :dt wait :dt*60 draw.pendulum
end
hideturtle until [key?] [step.pendulum]</lang>
Oz
Inspired by the E and Ruby versions.
<lang oz>declare
[QTk] = {Link ['x-oz://system/wp/QTk.ozf']}
Pi = 3.14159265
class PendulumModel feat
K
attr
angle velocity
meth init(length:L <= 1.0 %% meters
gravity:G <= 9.81 %% m/s² initialAngle:A <= Pi/2.) %% radians self.K = ~G / L angle := A velocity := 0.0
end
meth nextAngle(deltaT:DeltaTMS %% milliseconds
?Angle) %% radians
DeltaT = {Int.toFloat DeltaTMS} / 1000.0 %% seconds Acceleration = self.K * {Sin @angle} in velocity := @velocity + Acceleration * DeltaT angle := @angle + @velocity * DeltaT Angle = @angle end end
%% Animates a pendulum on a given canvas. class PendulumAnimation from Time.repeat feat Pend Rod Bob home:pos(x:160 y:50) length:140.0
delay
meth init(Pendulum Canvas delay:Delay <= 25) %% milliseconds
self.Pend = Pendulum self.delay = Delay %% plate and pivot
{Canvas create(line 0 self.home.y 320 self.home.y width:2 fill:grey50)} {Canvas create(oval 155 self.home.y-5 165 self.home.y+5 fill:grey50 outline:black)}
%% the pendulum itself self.Rod = {Canvas create(line 1 1 1 1 width:3 fill:black handle:$)}
self.Bob = {Canvas create(oval 1 1 2 2 fill:yellow outline:black handle:$)} %% {self setRepAll(action:Animate delay:Delay)} end
meth Animate
Theta = {self.Pend nextAngle(deltaT:self.delay $)} %% calculate x and y from angle X = self.home.x + {Float.toInt self.length * {Sin Theta}} Y = self.home.y + {Float.toInt self.length * {Cos Theta}}
in
%% update canvas try {self.Rod setCoords(self.home.x self.home.y X Y)} {self.Bob setCoords(X-15 Y-15 X+15 Y+15)} catch system(tk(alreadyClosed ...) ...) then skip end
end end Pendulum = {New PendulumModel init}
Canvas GUI = td(title:"Pendulum" canvas(width:320 height:210 handle:?Canvas) action:proc {$} {Animation stop} {Window close} end ) Window = {QTk.build GUI}
Animation = {New PendulumAnimation init(Pendulum Canvas)}
in
{Window show} {Animation go}</lang>
Ruby
This does not have the window resizing handling that Tcl does -- I did not spend enough time in the docs to figure out how to get the new window size out of the configuration event. Of interest when running this pendulum side-by-side with the Tcl one: the Tcl pendulum swings noticibly faster.
<lang ruby>require 'tk'
$root = TkRoot.new("title" => "Pendulum Animation") $canvas = TkCanvas.new($root) do
width 320 height 200 create TkcLine, 0,25,320,25, 'tags' => 'plate', 'width' => 2, 'fill' => 'grey50' create TkcOval, 155,20,165,30, 'tags' => 'pivot', 'outline' => "", 'fill' => 'grey50' create TkcLine, 1,1,1,1, 'tags' => 'rod', 'width' => 3, 'fill' => 'black' create TkcOval, 1,1,2,2, 'tags' => 'bob', 'outline' => 'black', 'fill' => 'yellow'
end $canvas.raise('pivot') $canvas.pack('fill' => 'both', 'expand' => true)
$Theta = 45.0 $dTheta = 0.0 $length = 150 $homeX = 160 $homeY = 25
def show_pendulum
angle = $Theta * Math::PI / 180 x = $homeX + $length * Math.sin(angle) y = $homeY + $length * Math.cos(angle) $canvas.coords('rod', $homeX, $homeY, x, y) $canvas.coords('bob', x-15, y-15, x+15, y+15)
end
def recompute_angle
scaling = 3000.0 / ($length ** 2) # first estimate firstDDTheta = -Math.sin($Theta * Math::PI / 180) * scaling midDTheta = $dTheta + firstDDTheta midTheta = $Theta + ($dTheta + midDTheta)/2 # second estimate midDDTheta = -Math.sin(midTheta * Math::PI / 180) * scaling midDTheta = $dTheta + (firstDDTheta + midDDTheta)/2 midTheta = $Theta + ($dTheta + midDTheta)/2 # again, first midDDTheta = -Math.sin(midTheta * Math::PI / 180) * scaling lastDTheta = midDTheta + midDDTheta lastTheta = midTheta + (midDTheta + lastDTheta)/2 # again, second lastDDTheta = -Math.sin(lastTheta * Math::PI/180) * scaling lastDTheta = midDTheta + (midDDTheta + lastDDTheta)/2 lastTheta = midTheta + (midDTheta + lastDTheta)/2 # Now put the values back in our globals $dTheta = lastDTheta $Theta = lastTheta
end
def animate
recompute_angle show_pendulum $after_id = $root.after(15) {animate}
end
show_pendulum $after_id = $root.after(500) {animate}
$canvas.bind('<Destroy>') {$root.after_cancel($after_id)}
Tk.mainloop</lang>
Tcl
and
<lang tcl>package require Tcl 8.5 package require Tk
- Make the graphical entities
pack [canvas .c -width 320 -height 200] -fill both -expand 1 .c create line 0 25 320 25 -width 2 -fill grey50 -tags plate .c create line 1 1 1 1 -tags rod -width 3 -fill black .c create oval 1 1 2 2 -tags bob -fill yellow -outline black .c create oval 155 20 165 30 -fill grey50 -outline {} -tags pivot
- Set some vars
set points {} set Theta 45.0 set dTheta 0.0 set pi 3.1415926535897933 set length 150 set homeX 160
- How to respond to a changing in size of the window
proc resized {width} {
global homeX .c coords plate 0 25 $width 25 set homeX [expr {$width / 2}] .c coords pivot [expr {$homeX-5}] 20 [expr {$homeX+5}] 30 showPendulum
}
- How to actually arrange the pendulum, mapping the model to the display
proc showPendulum {} {
global Theta dTheta pi length homeX set angle [expr {$Theta * $pi/180}] set x [expr {$homeX + $length*sin($angle)}] set y [expr {25 + $length*cos($angle)}] .c coords rod $homeX 25 $x $y .c coords bob [expr {$x-15}] [expr {$y-15}] [expr {$x+15}] [expr {$y+15}]
}
- The dynamic part of the display
proc recomputeAngle {} {
global Theta dTheta pi length set scaling [expr {3000.0/$length**2}]
# first estimate set firstDDTheta [expr {-sin($Theta * $pi/180)*$scaling}] set midDTheta [expr {$dTheta + $firstDDTheta}] set midTheta [expr {$Theta + ($dTheta + $midDTheta)/2}] # second estimate set midDDTheta [expr {-sin($midTheta * $pi/180)*$scaling}] set midDTheta [expr {$dTheta + ($firstDDTheta + $midDDTheta)/2}] set midTheta [expr {$Theta + ($dTheta + $midDTheta)/2}] # Now we do a double-estimate approach for getting the final value # first estimate set midDDTheta [expr {-sin($midTheta * $pi/180)*$scaling}] set lastDTheta [expr {$midDTheta + $midDDTheta}] set lastTheta [expr {$midTheta + ($midDTheta + $lastDTheta)/2}] # second estimate set lastDDTheta [expr {-sin($lastTheta * $pi/180)*$scaling}] set lastDTheta [expr {$midDTheta + ($midDDTheta + $lastDDTheta)/2}] set lastTheta [expr {$midTheta + ($midDTheta + $lastDTheta)/2}] # Now put the values back in our globals set dTheta $lastDTheta set Theta $lastTheta
}
- Run the animation by updating the physical model then the display
proc animate {} {
global animation
recomputeAngle showPendulum
# Reschedule set animation [after 15 animate]
} set animation [after 500 animate]; # Extra initial delay is visually pleasing
- Callback to handle resizing of the canvas
bind .c <Configure> {resized %w}
- Callback to stop the animation cleanly when the GUI goes away
bind .c <Destroy> {after cancel $animation}</lang>