Parametric curve function in ConTeXt, using Lua. Specifically, how to input the parametric equations

You have the following line

local fx = hash.fx or math.cos(u)

but you haven't defined u anywhere, so Lua treats it as if it were nil. There's a standard trick that you can use in Lua to prevent you from accessing undefined variables though:

local ignore_nils = table.tohash {
    "actuarian",
    "askedformat",
    "attr",
    "fix_lmmonoregular",
    "mode",
    "recursive",
    "relativeid",
    "width",
}

table.setmetatableindex(_ENV, function(t, k)
    -- ConTeXt itself tries to access the undefined variables listed in
    -- "ignore_nils", so we need to ignore these.
    if ignore_nils[k] then
        return nil
    else
        error("Undefined variable: " .. k, 2)
    end
end)

Adding this to the start of your Lua code gives the following error:

lua error       > lua error on line 96 in file ./context-lua-eval-param.tex:

registered function call [1860]: [ctxlua]:35: Undefined variable: u
stack traceback:
        [C]: in function 'error'
        [ctxlua]:19: in metamethod 'index'
        [ctxlua]:35: in function <[ctxlua]:25>
        (...tail calls...)
25     local function append_curve(hash
26         -- u_start
27         -- ,u_end
28         -- ,u_samples
29         -- ,fx,fy,fz
30     )
31         local hash = settings_to_hash(hash) or {}
32         local u_start = hash.u_start or 0
33         local u_end = hash.u_end or 2*math.pi
34         local u_samples = hash.u_samples or 3
35 >>      local fx = hash.fx or math.cos(u)
36         local fy = hash.fy or math.sin(u)
37         local fz = hash.fz or 1.5 * math.sin(2 * u)
38     
39     
40         local u_step = (u_end - u_start) / (u_samples - 1)
41         local function parametric_curve(u)
42             return {fx(u),fy(u),fz(u),1}
43         end
44         for i = 0, u_samples - 2 do
45             local u = u_start + i * u_step
mtx-context     | fatal error: return code: 1

If you add inspect(hash) to the definition of append_curve, you can see what the error is:

\startluacode
local settings_to_hash = utilities.parsers.settings_to_hash
local function append_curve(hash)
    local hash = settings_to_hash(hash) or {}
    inspect(hash)
end

interfaces.implement {
    name      = "AppendCurve",
    actions   = append_curve,
    public = true,
    arguments = { "optional" },
}
\stopluacode


\starttext
    \startMPcode
        \AppendCurve[
            u_start   = 0,
            u_end     = 1,
            u_samples = 20,
            fx        = "math.cos(u)",
            fy        = "math.sin(u)",
            fz        = "0"
        ]
    \stopMPcode
\stoptext

table={
 ["fx "]=" \"math.cos(u)\"",
 ["fy "]=" \"math.sin(u)\"",
 ["fz "]=" \"0\" ",
 ["u_end "]=" 1",
 ["u_samples "]=" 20",
 ["u_start "]=" 0",
}

This shows that you shouldn't include spaces around the equals sign (=) and you shouldn't use "quotes" around your variables.

You have quite a few other small errors in your code; correcting them all gives the following results:

\startluacode
local segments = {} -- **always** declare all variables as local

local settings_to_hash = utilities.parsers.settings_to_hash

-- Converts a string as a function of "u" to a Lua function with 1 parameter
local function string_function(str)
    if not str or str == "" then
        return nil
    end
    return load(("return function(u) return %s end"):format(str))()
end

local function append_curve(hash)
    local hash = settings_to_hash(hash) or {}

    local u_start   = hash.u_start or 0
    local u_end     = hash.u_end or 2*math.pi
    local u_samples = hash.u_samples or 3
    local fx        = string_function(hash.fx) or math.cos
    local fy        = string_function(hash.fy) or math.sin
    local fz        = string_function(hash.fz) or function(u) return 1.5 * math.sin(2 * u) end
    local color     = string_function(hash.color) or function() return "black" end

    local u_step = (u_end - u_start) / (u_samples - 1)

    local function parametric_curve(u)
        return { fx(u), fy(u), fz(u), 1 }
    end

    for i = 0, u_samples - 2 do
        local u = u_start + i * u_step
        local A = parametric_curve(u)
        local B = parametric_curve(u + u_step)
        -- the tables for surfaces have 5 values
        table.insert(segments, { A, B, color(u) })
    end
end

local observer = { { 0, 0, 1, 1 } }

local function render_segments()
    for _, seg in ipairs(segments) do
        local S3, E3 = seg[1], seg[2]
        local Sx, Sy = S3[1], S3[2]
        local Ex, Ey = E3[1], E3[2]
        local color = seg[3]

        context(
            "draw (%.4fcm,%.4fcm) -- (%.4fcm,%.4fcm) withcolor %s;",
            Sx, Sy, Ex, Ey, color
        )
    end
    segments = {}
end

interfaces.implement {
    name      = "rendersegments",
    actions   = render_segments,
    public = true,
    arguments = {},
}

interfaces.implement {
    name      = "AppendCurve",
    actions   = append_curve,
    public = true,
    arguments = { "optional" },
}
\stopluacode

\startMPpage[offset=1ex, scale=10000]
    \AppendCurve[
        u_start=0,
        u_end=6.28,
        u_samples=20,
        fx=math.cos(u),
        fy=math.sin(u),
        fz=0,
        color={u < 3.14 and "red" or "blue"}
    ]
    \rendersegments
\stopMPpage