### 3.17.3 Animation of a sequence of graphic objects : `animation`

`animation` animates the representation of a sequence of graphic objects with a given display time. The sequence of objects depends most of the time of a parameter and is defined using the `seq` command but it is not mandatory.
`animation` takes as argument the sequence of graphic objects.
To define a sequence of graphic objects with `seq`, enter the definition of the graphic object (depending on the parameter), the parameter name, it’s minimum value, it’s maximum value maximum and optionnaly a step value.
Input :

`animation(seq(plotfunc(cos(a*x),x),a,0,10))`

Output :

`The sequence of the curves defined by` `y=``cos``(ax)``, for` `a=0,1,2..10`

Input :

`animation(seq(plotfunc(cos(a*x),x),a,0,10,0.5))`
or
`animation(seq(plotfunc(cos(a*x),x),a=0..10,0.5))`

Output :

`The sequence of the curves defined by` `y=``cos``(ax)``, for` `a=0,0.5,1,1.5..10```

Input :

`animation(seq(plotfunc([cos(a*x),sin(a*x)],x=0..2*pi/a), a,1,10))`

Output :

`The sequence of two curves defined by` `y=``cos``(ax)``and` `y=``sin``(ax)``, for` `a=1..10` `and for` `x=0..2π/a```

Input :

`animation(seq(plotparam([cos(a*t),sin(a*t)], t=0..2*pi),a,1,10))`

Output :

`The sequence of the parametric curves defined by` `x=``cos``(at)``and` `y=``sin``(at)``, for` `a=1..10` `and for` `t=0..2π```

Input :

`animation(seq(plotparam([sin(t),sin(a*t)], t,0,2*pi,tstep=0.01),a,1,10))`

Output :

`The sequence of the parametric curves defined by` `x=``sin``(t),y=``sin``(at)``, for` `a=0..10` `and` `t=0..2π`

Input :

`animation(seq(plotpolar(1-a*0.01*t```^```2, t,0,5*pi,tstep=0.01),a,1,10))`

Output :

`The sequence of the polar curves defined by` `ρ=1-a*0.01*t``2``, for` `a=0..10` `and` `t=0..5π`

Input :

`plotfield(sin(x*y),[x,y]); animation(seq(plotode(sin(x*y),[x,y],[0,a]),a,-4,4,0.5))`

Output :

`The tangent field of y’=sin(xy) and the sequence of the integral curves crossing through the point` `(0,a)``for` `a``=-4,-3.5...3.5,4`

Input :

`animation(seq(display(square(0,1+i*a),filled),a,-5,5))`

Output :

`The sequence of the squares defined by the points 0 and 1+i*``a` `for` `a=-5..5`

Input :

`animation(seq(droite([0,0,0],[1,1,a]),a,-5,5))`

Output :

`The sequence of the lines defined by the points [0,0,0] and [1,1,``a``] for` `a=-5..5`

Input :

`animation(seq(plotfunc(x```^```2-y```^```a,[x,y]),a=1..3))`

Output :

`The sequence of the "3D" surface defined by` `x``2``-y``a``, for` `a=1..3` `with rainbow colors`

Input :

`animation(seq(plotfunc((x+i*y)```^```a,[x,y], display=filled),a=1..10)`

Output :

`The sequence of the "4D" surfaces defined by` `(x+i*y)``a``, for` `a=0..10` `with rainbow colors`

Remark We may also define the sequence with a program, for example if we want to draw the segments of length 1,√2...√20 constructed with a right triangle of side 1 and the previous segment (note that there is a `c:=evalf(..)` statement to force approx. evaluation otherwise the computing time would be too long) :

    seg(n):={
     local a,b,c,j,aa,bb,L;
     a:=1;
     b:=1;
     L:=[point(1)];
     for(j:=1;j<=n;j++){
      L:=append(L,point(a+i*b));
      c:=evalf(sqrt(a^2+b^2));
      aa:=a;
      bb:=b;
      a:=aa-bb/c;
      b:=bb+aa/c;
     }
     L;
    }

Then input :

`animation(seg(20))`

We see, each point, one to one with a display time that depends of the `animate` value in `cfg`.
Or :

`L:=seg(20); s:=segment(0,L[k])$(k=0..20)`

We see 21 segments.
Then, input :

`animation(s)`

We see, each segment, one to one with a display time that depends of the `animate` value in `cfg`.