Sliders and animation

A variable defined as a number is already a slider in Desmos. The slider property gives it a range, and playing sets it moving. For anything a slider cannot do - a count that steps once a second, a clock that keeps real time - there is the ticker.

Ranges

A slider’s range is written lo..hi, with an optional step after it:

a = 1 @ slider: -5..5 step 0.5
b = 0 @ slider: -3..3 step 0.1
c = 1 @ slider: 0.5..4
y = a * (x - b) ^ 2 + c

Open in playground →

.., two dots, is the range literal, and it is only ever a whole property value. Three dots, [1...10], are something else: a list.

Either end may be left off to keep Desmos’ own default for it, so 0.. runs from 0 up to Desmos’ usual top of 10. Every end, and the step, may be any expression, so one slider can set the reach of another:

p = 2 @ slider: 0.. step 1
lim = 3 @ slider: 1..6 step 1
s = 1 @ slider: -lim..lim step lim / 10
y = p + s * sin(x)

Open in playground →

A stepped slider snaps its value to the step’s grid, counted from its min, not from zero. a = 1 @ slider: -3..3 step 0.3 starts at 0.9, the nearest point on that grid to 1.

Soft ends

Both ends are hard by default: the slider stops there, and a value typed into the expression list is held to them too. soft makes both ends soft - the slider still spans the range, but a value typed in may go past it - and soft min or soft max softens just the one. step comes before soft when both are written.

q = 1 @ slider: -2..2 soft
k = 1 @ slider: 0..3 step 0.5 soft max
y = q sin(k x)

Open in playground →

Animating a slider

playing starts a slider moving the moment the graph opens. Three more properties say how:

loopMode is what it does at the end of its range: LOOP_FORWARD_REVERSE (the default) bounces back and forth, LOOP_FORWARD jumps back to the start, PLAY_ONCE stops, and PLAY_INDEFINITELY carries on past the end.
animationPeriod is how long one sweep takes, in milliseconds. The default is 8000.
playDirection is 1 to run forwards and -1 to run backwards.

t = 0 @ slider: 0..2pi, playing, loopMode: LOOP_FORWARD, animationPeriod: 4000
(t, sin(t)) @ color: BLUE, pointSize: 18
y = sin(x) @ color: BLUE, lineOpacity: 0.4

Open in playground →

animationPeriod and playDirection are plain numbers rather than expressions, because Desmos keeps them as numbers.

Curves drawn over a range

A parametric curve - a point written in terms of t - is drawn over the range its domain gives, and a polar one over its polarDomain. Desmos’ own default runs t from 0 to 1, not over a whole period, so a curve that should close up needs a domain:

(cos(t), sin(t)) @ color: RED, domain: 0..tau
(0.3t * cos(t), 0.3t * sin(t)) @ color: GREEN, domain: 0..6pi

Open in playground →

A domain is its two ends and nothing else; step or soft on one is an error. Like a slider’s, either end may be an expression, so a slider can draw a curve out as it moves.

The ticker

A ticker runs one action over and over for as long as the graph is open. It is how a graph animates something a slider cannot: a counter, a simulation, a clock.

n = 0
clock = 0

ticker n -> mod(n + 1, 60), clock -> clock + dt @ minStep: 50, playing

(4cos(n * tau / 60), 4sin(n * tau / 60)) @ color: ORANGE, pointSize: 16
(2cos(clock * tau / 3000), 2sin(clock * tau / 3000)) @ color: PURPLE, pointSize: 12

Open in playground →

The handler is an action, target -> new value, or a run of them separated by commas, which happen together on each tick. Three properties go on it:

minStep is the shortest gap between two ticks, in milliseconds. Left out, Desmos ticks once a frame.
playing starts it when the graph opens. Without it the ticker is there but stopped, to be started from the expression list.
open shows it expanded in Desmos’ expression list.

dt

Inside the handler, dt is the milliseconds since the last tick. It is not always minStep, since a busy page ticks late, which is why clock above adds up dt rather than counting ticks: it keeps real time however late they come, and goes round once every three seconds whatever the frame rate. Divide by 1000 for seconds.

dt means nothing anywhere else, so written outside the handler it is an error:

y = dt

It is always spelt dt. In Desmos’ own latex it has to be the single operator \operatorname{dt}; written any other way it would be d times t, and the ticker would run and change nothing, silently. The compiler spells it for you.

One per graph

A graph has exactly one ticker, and Desmos keeps it beside the expression list rather than in it - so ticker goes at the top level, outside every folder, and a second one is an error. An imported file’s ticker is replaced by the importing file’s.

Desmos decides whether actions are allowed at all (the actions setting, left at auto) by looking at the expression list, and the ticker is not in it. A graph whose only action is its ticker would never tick, so Axis switches actions on for any file with a ticker.