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## Uniform point plot

When we plot a curve with the option  style=point  , symbols go evenly not along the length of this curve, but along the range of the independent variable. For this reason the plot often looks unattractive. Here are two examples. In the first example, the default option  adaptive=true  is used, in which Maple adds points in some places.

```restart;
plot(surd(x,3), x=-2.5..2.5, style=point, scaling=constrained, symbol=solidcircle, symbolsize=8, numpoints=30, size=[800,300]);
plot(surd(x,3), x=-2.5..2.5, style=point, scaling=constrained, symbol=solidcircle, symbolsize=8, numpoints=30, adaptive=false, size=[800,300]);
```

The  UniformPointPlot  procedure allows you to plot curves by symbols (as for  style=point), and these symbols go from each other at equal distances, measured along this curve. The procedure uses a well-known formula for the length of a curve in two and three dimensions. The procedure parameters are clear from the three examples below.

```UniformPointPlot:=proc(F::{algebraic,list},eq::`=`,n::posint:=40,Opt::list:=[symbol=solidcircle, symbolsize=8, scaling=constrained])
local t, R, P, g, L, step, L1, L2;
uses plots;
Digits:=4:
t:=lhs(eq); R:=rhs(eq);
P:=`if`(type(F,algebraic),[t,F],F);
g:=x->`if`(F::algebraic or nops(F)=2,evalf(Int(sqrt(diff(P[1],t)^2+diff(P[2],t)^2), t=lhs(R)..x, epsilon=0.001)),evalf(Int(sqrt(diff(P[1],t)^2+diff(P[2],t)^2+diff(P[3],t)^2), t=lhs(R)..x, epsilon=0.001))):
L:=g(rhs(R)); step:=L/(n-1);
L1:=[lhs(R),seq(fsolve(g-k*step, fulldigits),k=1..n-2),rhs(R)];
L2:=map(s->`if`(type(F,algebraic),[s,eval(F,t=s)],eval(F,t=s)), L1):
`if`(F::algebraic or nops(F)=2,plot(L2, style=point, Opt[]),pointplot3d(L2, Opt[]));
end proc:
```

Examples of use:

`UniformPointPlot(surd(x,3), x=-2.5..2.5, 30);`

`UniformPointPlot([5*cos(t),3*sin(t)], t=0..2*Pi, [color=red,symbol=solidcircle,scaling=constrained, symbolsize=8,  size=[800,400]]);`

`UniformPointPlot([cos(t),sin(t),2-2*cos(t)], t=0..2*Pi, 41, [color=red,symbol=solidsphere, symbolsize=8,scaling=constrained, labels=[x,y,z]]);`

Here's another example of using the same technique as in the procedure. In this example, we are plotting Archimedean spiral uniformly colored with 7 rainbow colors:

```f:=t->[t*cos(t),t*sin(t)]:
g:=t->evalf(Int(sqrt(diff(f(s)[1],s)^2+diff(f(s)[2],s)^2), s=0..t)):
h:=s->fsolve(s=g(t), t):
L:=evalf(g(2*Pi)): step:=L/7:
L1:=[0,seq(h(k*step), k=1..6),2*Pi]:
Colors:=convert~([Red,Orange,Yellow,Green,Blue,Indigo,Violet], string):
plots:-display(seq(plot([f(t)[], t=L1[i]..L1[i+1]], color=Colors[i], thickness=12), i=1..7), scaling=constrained, size=[500,400]);
```

Uniform_Point_Plot.mw

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