# Hi! I have a function, let's say \`f(ϕ) = sin(ϕ), ϕ = 0...10\`. Now I want to break

**URL:** <https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826>\
**Category:** General Usage\
**Created:** [February 8, 2021, 6:56am UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826 "2021-02-08T06:56:40Z")\
**Posts on this page:** 20\
**Page:** 1

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**Author:** ![BridgeBot](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bridgebot/32/21491_2.png) [@BridgeBot](https://discourse.julialang.org/u/BridgeBot)\
**Post date:** [February 8, 2021, 6:56am UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/1 "2021-02-08T06:56:40Z")

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Hi! I have a function, let’s say `f(ϕ) = sin(ϕ), ϕ = 0...10`. Now I want to break that function down in a list of lines that approximates the real function well (maybe with a maximum or minimum line length, angle between lines, or number of lines). Is there a package that can help with that? Bonus points if it also works in polar coordinates.

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**Author:** ![Karajan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/karajan/32/8545_2.png) [@Karajan](https://discourse.julialang.org/u/Karajan)\
**Post date:** [February 8, 2021, 8:39am UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/2 "2021-02-08T08:39:09Z")

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A little more information based on the discussion on Slack:

I have a function that only works on lines (let’s say `(0.0, 0.0) -> (0.1, 0.3)`), that’s why I want to turn my `sin` (for example) into a piecewise linear function/list of lines.

I know that some combination of inputs (e.g. number of lines and maximum length of lines) might give trouble, we can ignore that for now.

I imagined that adaptive algorithms of integration/differentiation equations might do something close to what I’m looking for, but 1) I’m not sure about that being true, 2) I don’t know any package that does just that.

PS: Is there any way for me to change the title of the post? At the moment it’s quite … nondescript.

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**Author:** ![kdyrhage](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/kdyrhage/32/2326_2.png) [@kdyrhage](https://discourse.julialang.org/u/kdyrhage)\
**Post date:** [February 8, 2021, 9:02am UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/3 "2021-02-08T09:02:53Z")

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Are you looking for something like this?

```julia
julia> function lines(f, r)
           step = Float64(r.step)
           map(x -> ((x, f(x)), (x+step, f(x+step))), r[1:end-1])
       end
lines (generic function with 2 methods)

julia> lines(sin, 0.0:0.1:1.0)
10-element Vector{Tuple{Tuple{Float64, Float64}, Tuple{Float64, Float64}}}:
 ((0.0, 0.0), (0.1, 0.09983341664682815))
 ((0.1, 0.09983341664682815), (0.2, 0.19866933079506122))
 ((0.2, 0.19866933079506122), (0.30000000000000004, 0.2955202066613396))
 ((0.3, 0.29552020666133955), (0.4, 0.3894183423086505))
 ((0.4, 0.3894183423086505), (0.5, 0.479425538604203))
 ((0.5, 0.479425538604203), (0.6, 0.5646424733950354))
 ((0.6, 0.5646424733950354), (0.7, 0.644217687237691))
 ((0.7, 0.644217687237691), (0.7999999999999999, 0.7173560908995227))
 ((0.8, 0.7173560908995228), (0.9, 0.7833269096274834))
 ((0.9, 0.7833269096274834), (1.0, 0.8414709848078965))

```

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**Author:** ![Karajan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/karajan/32/8545_2.png) [@Karajan](https://discourse.julialang.org/u/Karajan)\
**Post date:** [February 8, 2021, 9:41am UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/4 "2021-02-08T09:41:56Z")

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Thank you but this isn’t what I’m looking for.  
If we stay with the example of `sin`, around `0` this would be a good approximation but around `π/2` it might be to coarse. So I’m looking for an package/algorithm that picks the steps so that I can set “maximum or minimum line length, angle between lines, or number of lines” (or at least more than just number of lines). If I have the points, turning them into lines should be fine.

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**Author:** ![kdyrhage](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/kdyrhage/32/2326_2.png) [@kdyrhage](https://discourse.julialang.org/u/kdyrhage)\
**Post date:** [February 8, 2021, 9:47am UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/5 "2021-02-08T09:47:50Z")

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I see. In that case I can’t help you, unfortunately.

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**Author:** ![rafael.guerra](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rafael.guerra/32/216610_2.png) [@rafael.guerra](https://discourse.julialang.org/u/rafael.guerra)\
**Post date:** [February 8, 2021, 11:00am UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/6 "2021-02-08T11:00:10Z")

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If you write the derivative f’(t) of your function and plug it in the [DifferentialEquations.jl](https://github.com/SciML/DifferentialEquations.jl) package, you can define an error tolerance and you will get as solution f(t), sampled at irregular times. Use the resulting array of irregular times to break down your original function.  
_PS: Yes, it sounds like using a sledgehammer to crack a nut._

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**Author:** ![Karajan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/karajan/32/8545_2.png) [@Karajan](https://discourse.julialang.org/u/Karajan)\
**Post date:** [February 8, 2021, 11:44am UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/7 "2021-02-08T11:44:51Z")

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Well, a sledgehammer indeed, but I prefer that over using my fingernails so I’ll give it a try if I don’t find anything better fitting. Thanks!

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**Author:** ![rafael.guerra](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rafael.guerra/32/216610_2.png) [@rafael.guerra](https://discourse.julialang.org/u/rafael.guerra)\
**Post date:** [February 8, 2021, 12:11pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/8 "2021-02-08T12:11:55Z")

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Another approach might be piecewise linear interpolation. I have never tried the [PiecewiseLinearOpt.jl](https://github.com/joehuchette/PiecewiseLinearOpt.jl) package but have a look at it, also the associated [presentation](https://www.juliaopt.org/meetings/mit2017/huchette.pdf) by the author.

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**Author:** ![josuagrw](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/josuagrw/32/1015_2.png) [@josuagrw](https://discourse.julialang.org/u/josuagrw)\
**Post date:** [February 8, 2021, 12:13pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/9 "2021-02-08T12:13:54Z")

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It sounds like geometrically, you want to define a small circle around a point and then find the point where the function intersects the circle, then draw a circle around that point and find the next intersection and so on.

Fundamentally, this is a root finding problem in two dimensions. It should be possible to use Roots.jl or nlsolve, for example.

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**Author:** ![Karajan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/karajan/32/8545_2.png) [@Karajan](https://discourse.julialang.org/u/Karajan)\
**Post date:** [February 8, 2021, 1:23pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/10 "2021-02-08T13:23:17Z")

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> [@rafael.guerra](#):
>
> the [PiecewiseLinearOpt.jl](https://github.com/joehuchette/PiecewiseLinearOpt.jl) package

Hm, I’m not sure if I misunderstand the package but to me it sounds like optimization of a function that already is piecewise?

> [@josuagrw](#):
>
> It sounds like geometrically, you want to define a small circle around a point and then find the point where the function intersects the circle, then draw a circle around that point and find the next intersection and so on.

I guess that comes close to what I’m looking for but isn’t really that “adaptive”?

I’ve coded something crude up:

```julia
using Optim

f(x) = sin(x)
x = 0.0:0.01:3

function pwl(x, pnts)
    idx = length(pnts)
    for i in 2:length(pnts)
        if x <= pnts[i]
            idx = i
            break
        end
    end
    p1 = pnts[idx-1]
    p2 = pnts[idx]
    f1 = f(p1)
    f2 = f(p2)
    return f1 + (f2 - f1) * (x - p1) / (p2 - p1)
end

function cost(p)
    pnts = copy(p)
    push!(pnts, first(x))
    push!(pnts, last(x))
    sort!(pnts)

    pwlf = x -> pwl(x, pnts)
    
    sum(abs2, f.(x) .- pwlf.(x))
end

N = 19
x0 = range(0, 3, length = N)[2:end-1] |> collect
lower = zeros(N-2)
upper = ones(N-2) .* 3
result = optimize(cost, lower, upper, x0)

```

In the linear part of the `sin` there are very few points, around `π/2` there are a lot.

Of course this won’t work in a number of cases (most of them, probably). This also only considers the number of points/lines, not their length or angle.

I guess with enough time I would be able to code something properly but of course I’d prefer to reuse something existing.

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**Author:** ![josuagrw](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/josuagrw/32/1015_2.png) [@josuagrw](https://discourse.julialang.org/u/josuagrw)\
**Post date:** [February 8, 2021, 1:36pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/11 "2021-02-08T13:36:33Z")

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> [@Karajan](#):
>
> In the linear part of the `sin` there are very few points, around `π/2` there are a lot.

Isn’t that exactly what you want? Near the zeros (0, pi, 2pi, etc) the sine can be approximate for a large interval with a linear function. so you would expect one long piece there

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**Author:** ![Karajan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/karajan/32/8545_2.png) [@Karajan](https://discourse.julialang.org/u/Karajan)\
**Post date:** [February 8, 2021, 1:40pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/12 "2021-02-08T13:40:26Z")

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Yes it is. I meant that the circle approach doesn’t seem to do this so I’m not sure it’s going to work for me.

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**Author:** ![josuagrw](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/josuagrw/32/1015_2.png) [@josuagrw](https://discourse.julialang.org/u/josuagrw)\
**Post date:** [February 8, 2021, 1:49pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/13 "2021-02-08T13:49:25Z")

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no problem, I’m glad you found a solution that works for you

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**Author:** ![rafael.guerra](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rafael.guerra/32/216610_2.png) [@rafael.guerra](https://discourse.julialang.org/u/rafael.guerra)\
**Post date:** [February 8, 2021, 1:50pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/14 "2021-02-08T13:50:43Z")

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@Karajan, you are probably right. I thought that package would allow optimizing an initial set of N-breakpoints of the piecewise linear function. May be someone can confirm.

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**Author:** ![Karajan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/karajan/32/8545_2.png) [@Karajan](https://discourse.julialang.org/u/Karajan)\
**Post date:** [February 8, 2021, 1:55pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/15 "2021-02-08T13:55:42Z")

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Well, I’m pretty certain this would break down with more complicated functions or more points so I’m not really happy with the `Optim` approach. But if there doesn’t exist anything else I’ll try my luck 😃

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**Author:** ![rafael.guerra](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rafael.guerra/32/216610_2.png) [@rafael.guerra](https://discourse.julialang.org/u/rafael.guerra)\
**Post date:** [February 8, 2021, 3:56pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/16 "2021-02-08T15:56:23Z")

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Dierckx linear splines in its free knots option, produces good results by tuning the admissible error tolerance parameter `s`. You can use the array `xk` with the knots along the x-axis.

```julia
using Dierckx, Plots; gr()
x = LinRange(0, π, 100)
y = sin.(x)
spl1 = Spline1D(x, y; k=1, bc="nearest", s= 1e-5)
xk = get_knots(spl1)
plot(x,y,label="sin(x)", legend =:bottom)
scatter!(xk,spl1(xk),ms=3, label="Dierckx linear spline free knots")

```

![Dierckx_linear_spline_free_knots](https://global.discourse-cdn.com/julialang/original/3X/4/0/407a6dcb3269fa8fcba30887c070339d80e01c9a.png)

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**Author:** ![Karajan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/karajan/32/8545_2.png) [@Karajan](https://discourse.julialang.org/u/Karajan)\
**Post date:** [February 8, 2021, 4:18pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/17 "2021-02-08T16:18:23Z")

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This look good, great find. I’ll have a play with it, thank you so much!

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**Author:** ![Iulian.Cioarca](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/iulian.cioarca/32/30166_2.png) [@Iulian.Cioarca](https://discourse.julialang.org/u/Iulian.Cioarca)\
**Post date:** [February 8, 2021, 5:43pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/18 "2021-02-08T17:43:46Z")

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Does this help?

> **[GitHub - iuliancioarca/AdaptiveSampling.jl: Adaptive sampling rate algorithms...](https://github.com/iuliancioarca/AdaptiveSampling.jl)**
>
> Adaptive sampling rate algorithms for waveform compression - GitHub - iuliancioarca/AdaptiveSampling.jl: Adaptive sampling rate algorithms for waveform compression

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**Author:** ![Karajan](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/karajan/32/8545_2.png) [@Karajan](https://discourse.julialang.org/u/Karajan)\
**Post date:** [February 8, 2021, 5:57pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/19 "2021-02-08T17:57:08Z")

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Looks also interesting! However, there don’t seem to be any docs so I’ll need to work through it to see if helps.

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**Author:** ![Iulian.Cioarca](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/iulian.cioarca/32/30166_2.png) [@Iulian.Cioarca](https://discourse.julialang.org/u/Iulian.Cioarca)\
**Post date:** [February 8, 2021, 7:16pm UTC](https://discourse.julialang.org/t/hi-i-have-a-function-lets-say-f-sin-0-10-now-i-want-to-break/54826/20 "2021-02-08T19:16:42Z")

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Just take a look at the demo in the example folder. If you have issues please ping me since I’m the author.

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