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On 2/23/25 09:07, Bald Eagle wrote:
> William F Pokorny <ano### [at] anonymous org> wrote:
>
>> This a general helper function / technique to match any two curves at
>> their endpoints given you have the means to calculate the first and
>> second derivatives for both curves being glued together at (t).
>
> Are f' and f" arguments to the function?
Could be, but see below.
>
>> The function will calculate and return 'k(t)' or the 'radius of
>> curvature' (1.0/k(t)).
>>
>> Attaching images showing on the left k(t), and on the right the 'radius
>> of curvature' at t, as the length of the normal vectors for a
>> f_bezier_2d_cubic() curve.
>
...
>
> Do you plug f_bezier_2d_cubic() into your function and it calculates the
> derivatives? Can you plug in, say, Perlin Noise, and have that work?
>
Sort of.
No. The capability isn't that general - and limited to 2D (*).
With the f_bezier_2d_* functions and, for example, f_catenary() from a
few releases back, the function which generates the curve has too
calculation modes inbuilt for the first and second derivatives - and
sometimes the third as well.
So the typical set up would be something like:
//...
#declare FnBez_2d = function (ro_t,ro_mode) {
f_bezier_2d_cubic(
f_enc2x_f32(-1.0,0),
f_enc2x_f32(-1/2,0.5),
f_enc2x_f32(+1/2,-0.5),
f_enc2x_f32(1.0,0),
ro_t,ro_mode
)
}
#declare U02 = union {
#for (I,0,SphCnt,4)
...
#declare D1 = FnBez_2d(I/SphCnt,3); // Calc raw 1st derivative
#declare D2 = FnBez_2d(I/SphCnt,4); // Calc raw 2nd derivative
#declare Kappa = f_kappa_2d(D1,D2,0); // Pass the values
...
#end
texture { pigment { Frostbite } finish {emission 1.0} }
}
//...
Where you have a univariate polynomial for the curve, it's not too bad
to calculate the derivative polynomials as Alain pointed out.
Bill P.
(*) Lying some as by measuring local deltas we could probably come up
with D1 & D2 values which would be in the ballpark to some sampling
accuracy.
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