NOTES about integration

90147c1b-e465-4e27-900b-d46735242a2f 54 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

Si(x), Ci(x) ... Integral representation ... acc 500

datum π 2 / :

Euler constant γ 0.57721566490153286 :

▼ Transit ▼

Si(x)... free of freak z Ci ε 1015 -^: z Ci γ x ln + t cos 1 - t / t ε x int + : 21 line :

Si(x)... free of freak z Si ε 1015 -^: z Si t sin t / t ε x int : 21 line :

x Ci z Ci :

x Si z Si :

1 Ci 0.3374

1 Si 0.9461

x plot x Si x Ci datum 41 sys :

Complete Integral representation

x plot

sanity Abramowitz  p. 242
...  acc 500 7 Si 7 Ci 21 mat 1.4545966138 0.0766952775 21 mat

NOTES about integration

ε 1015 -^:

16.3658005743936
very accurate
... acc 1000 x Si x ε 11 int 16.3658005743936

Does NOT integrate
... acc 2000 [33 min]
0.014795716100422 MCD x Ci x ε 11 int 0.00914 -

Integrate over Σ(n)
∫kernel(f,ε,11,20000) 0.014795722454334

Si(x), Ci(x) ... Σ(x) ... Σ(x,n) series representation

◄ CosineIntegral x Ci γ x ln + x 2^- (k^(2 k * 2 k * (! * / k 140 sum + :

SineIntegral ► x Si 1 - (m^x 12 m * +^* 12 m * + (12 m * + (! (* / (m 040 sum :

x Ci 0 ≡ x 0.5 0.75 solve 0.6165

optional ON/OFF shift 0.6165 :

Nielsen_0 x 0.1 20 0.2 range : i 1 x rows range Ux i el x i el Si eval : Vy i el x i el Ci eval : 21 line for Ux Vy augment 31 line :

Nielsen_1 x 0.6165 20 0.7 range : i 1 x rows range Nx i el x i el Si eval : Ny i el x i el Ci eval : 21 line for Nx shift - Ny augment 31 line :

usable length x 20 Si 16 ≤ x 40 Si 32 ≤ x 80 Si 40 ≤ 31 mat

Nielsen_0 Nielsen_0 :

Nielsen_1 Nielsen_1 :

lookup(value,table,col)

value table col lookup k 1 : j k table rows range table j col el value ≡ index k el j : k k 1 + : 21 line continue if for k 1 ≡ index NO match : index if 31 line :

ROW Nielsen_0 2 col Max Nielsen_0 2 lookup : 1611 mat

Nielsen_0 ROW row 1.3892 0.4717 12 mat

Nielsen_0 1.3892 0.4717 + 30 red 1 5 mat 2 1 sys Nielsen_1 2 1 sys

...  Abramowitz  p. 242 7 Si 7 Ci 21 mat 1.4545966142 0.0766952785 21 mat

Approximation

http://www.dgp.toronto.edu/~mccrae/projects/clothoid/sbim2008mccrae.pdf

t R 0.506 t * 1 + 1.79 t 2^* 2.054 t * + 2 sqrt + / :

t A 1 0.803 t 3^* 1.886 t 2^* + 2.524 t * + 2 + / :

t C 12 / t R 12 / π * t A t 2^- (* sin * - :

t S 12 / t R 12 / π * t A t 2^- (* cos * - :

x 0 ≥ x C Ignored if x 0 ≥ x S Ignored if lookup(value,table,col)

value table col lookup k 1 : j k table rows range table j col el value ≡ index k el j : k k 1 + : 21 line continue if for k 1 ≡ index NO match : index if 31 line :

Clothoid t 0 π 0.005 range : i 1 t rows range U i el t i el C eval : V i el t i el S eval : 21 line for V U augment 31 line :

Clothoid 0.4381 0.7796 + 40 red 1 5 mat 2 1 sys

ROW Clothoid 2 col Max Clothoid 2 lookup : 20111 mat

Clothoid ROW row 0.4381 0.7796 12 mat

Tools ► Options ► acc[100]

Fresnel integrals

They arise in the description of far field Fresnel diffraction phenomena, and are defined through the following integral representations: The simultaneous parametric plot of S(x) and C(x) is the Euler spiral (also known as the Cornu spiral or clothoid). Clothoid and arcs of clothoid have applications in the construction of railway, freeway and roller coaster. Im this Smath document, we propose an exact view and export of these two integrals as well as the Cornu spiral as you drive your car in advanced maths style.

π 2 / t 2^* cos t int maple t FresnelC

π 2 / t 2^* sin t int maple t FresnelS

x FC π 2 / t 2^* cos t 0 x int :

x FS π 2 / t 2^* sin t 0 x int :

C x 1.5 π * - 1.5 π * (1.5 π * - 0.1 + (range : i 1 x rows range u i el x i el : F i el x i el FC eval : 21 line for u F augment 31 line :

S x 1.5 π * - 1.5 π * (1.5 π * - 0.1 + (range : i 1 x rows range u i el x i el : F i el x i el FS eval : 21 line for u F augment 31 line :

FresnelC

FresnelS

Cornu x π - π 0.99 π * - range : i 1 x rows range U i el x i el FS eval : V i el x i el FC eval : 21 line for U V augment 31 line :

box 0.714 - 0.78 0.714 0.78 0.714 0.78 - 0.714 - 0.78 - 0.714 - 0.78 52 mat :

Cornu_Observer π - π 0.99 π * - range π - π 2 / 0.99 π * - range 21 sys ≡

Cornu spiral [clothoid]

Cornu box 0.5 0.5 + 50 black 1 5 mat 3 1 sys Approximations

Approximation

http://www.dgp.toronto.edu/~mccrae/projects/clothoid/sbim2008mccrae.pdf

t R 0.506 t * 1 + 1.79 t 2^* 2.054 t * + 2 sqrt + / :

t A 1 0.803 t 3^* 1.886 t 2^* + 2.524 t * + 2 + / :

t C 12 / t R 12 / π * t A t 2^- (* sin * - :

t S 12 / t R 12 / π * t A t 2^- (* cos * - :

x 0 ≥ x C Ignored if x 0 ≥ x S Ignored if lookup(value,table,col)

value table col lookup k 1 : j k table rows range table j col el value ≡ index k el j : k k 1 + : 21 line continue if for k 1 ≡ index NO match : index if 31 line :

Clothoid t 0 π 0.01 range : i 1 t rows range U i el t i el C eval : V i el t i el S eval : 21 line for V U augment 31 line :

Clothoid 0.4381 0.7796 + 40 red 1 5 mat 2 1 sys

ROW Clothoid 2 col Max Clothoid 2 lookup : 10111 mat

Clothoid ROW row 0.4381 0.7796 12 mat

Project notes:

For high accuracy of point compute FC(x) FS(x) set the numerical integrator at higher accuracy than the default. This work sheet is in default accuracy 100.

Formule approximative de la clothoïde

b 2 sqrt :

convenient 'N' N 10 :

x f 1 b / 1 - (n^x 4 n * 3 +^* 2 n * 1 + (! (4 n * 3 + (* b 4 n * 2 +^* / (n 0 N sum * :

x g 1 b / 1 - (n^x 4 n * 1 +^* 2 n * (! (4 n * 1 + (* b 4 n *^* / (n 0 N sum * :

Clothoid t 0 π sqrt 0.01 range : i 1 t rows range U i el t i el f eval : V i el t i el g eval : 21 line for U V augment 31 line :

Clothoid Clothoid :

user u 1.25 : u f u g 13 mat : 1.25 0.2203 0.8314 13 mat

Clothoïde: highway access ramp

Clothoid 1 user 1 2 el user 1 3 el + 20 black 1 5 mat 3 1 sys

n Clothoid rows : 178

Clothoid n row 0.5475 0.9774 12 mat

τ 1178 range :

t X Clothoid 1 col t el : t Y Clothoid 2 col t el : 12 mat

route 0.05 1.125 TransCanada Highway ► 14 blue 15 mat :

t 1 ≡ path t X t Y 1 2 mat : path path t X t Y 1 2 mat stack : if 0.9774 route 0.5475 0.9774 + 20 black 1 5 mat 4 1 sys

Any spiral [c,k,r] ...

lookup(value,table,col)

value table col lookup k 1 : j k table rows range table j col el value ≡ index k el j : k k 1 + : 21 line continue if for k 1 ≡ index NO match : index if 31 line :

◄  The composite 'any' coefficients c 0.125 : k 0.125 : 12 mat

i (2^1 -

◄  The governing function x any c x * k + exp :

start 0 : end 5 π * : N 1000 : 13 mat u v N var u v u v u - N / + (range : x start end N var : 31 sys

± svg the spiral r 1.25 :

r x Ψ r e i x *^* (x any * i (2^* :

U r x Ψ Re vectorize :

V r x Ψ Im vectorize :

ANY U V augment eval :

Extremas Extr ANY 1 row ANY ANY rows row 21 sys : 1.4164 - 012 mat 10.0699 0.1582 - 12 mat 21 sys

ROW_1 ANY 2 col Max ANY 2 lookup : 70911 mat

ANY ROW_1 row 0.7129 - 5.6428 12 mat

ROW_2 ANY 1 col Min ANY 1 lookup : 80911 mat

ANY ROW_2 row 6.867 - 0.8675 - 12 mat

The spiral system ... is fully automated wrt the command line [c,k,r] 'r' is + in this demo.

ROW_3 ANY 2 col Min ANY 2 lookup : 90911 mat

ANY ROW_3 row 1.0557 8.3568 - 12 mat

data ANY ROW_1 row ANY ROW_2 row ANY ROW_3 row 31 sys : 0.713 - 5.643 12 mat 6.867 - 0.868 - 12 mat 1.056 8.357 - 12 mat 31 sys

plot ANY data 1 el 11 el data 1 el 12 el + 40 red 15 mat data 2 el 11 el data 2 el 12 el + 40 red 15 mat data 3 el 11 el data 3 el 12 el + 40 red 15 mat Extr 1 el 11 el Extr 1 el 12 el . 16 black 15 mat Extr 2 el 11 el Extr 2 el 12 el . 16 black 15 mat 61 sys :

t X U t el : t Y V t el : 12 mat

τ 1 U rows 5 range :

3 parameters command line spiral

plot t 1 ≡ path t X t Y 1 2 mat : path path t X t Y 1 2 mat stack : if Extr 2 el 1 1 el Extr 2 el 1 2 el . 16 black 1 5 mat 2 1 sys

Golden spiral ... red Logarithmic spiral ... green

Bézier spiral ...

X 1 - 0113 mat transpose :

Y 1 - 0113 mat transpose :

xc 0 : zoom 10 : 12 mat

a ≠ b ... elliptic a 0.5 : b 0.5 : 12 mat

t x t cos :

t y t sin :

t Q0 12 / 1 t xc - (- (2^* : t Q1 t xc - (2^t xc - (+ 12 / + : t Q2 12 / t xc - (2^* : 31 sys

t X a t Q0 X 1 el * t Q1 X 2 el * + t Q2 X 3 el * + (* : t Y b t Q0 Y 1 el * t Q1 Y 2 el * + t Q2 Y 3 el * + (* : 21 sys

t h t x t X * :

t v t y t Y * :

t 06 π * (0.005 range :

t0 1 time :

Bx t h vectorize eval :

By t v vectorize eval :

Bezier Bx By augment :

Bezier 6.115 4 6.115 4 - 2 2 mat 2 1 sys

1 time t0 - 24 s *

Bezier 1 row Bezier Bezier rows row 21 sys 0.25 - 012 mat 9.1724 0.0418 - 12 mat 21 sys

9.1724 9.1724 3 / - 6.1149

Bezier rows 3770