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

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^* sin t int maple t FresnelS

x FC π 2 / t 2^* cos t 0 x int :

1 FC 0.7799

x FS π 2 / t 2^* sin t 0 x int :

1 FS 0.4383

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 : 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 : 21 line for u F augment 31 line :

FresnelC

FresnelS

Cornu s 2 : x s - s 0.99 s * - range : i 1 x rows range U i el x i el FS : V i el x i el FC : 21 line for U V augment 41 line :

XY 0.714 - 0.78 0.714 0.78 0.714 0.78 - 0.714 - 0.78 - 0.714 - 0.78 52 mat :

Cornu spiral [clothoid]

Cornu XY 2 1 sys

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 Invalid if x 0 ≥ x S Invalid if

Clothoid t 01 π * 0.01 range : i 1 t rows range U i el t i el C : V i el t i el S : 21 line for V U augment 31 line :

Clothoid

Project notes:

1. FC, FS are presumed to run faster using the RK integrator. RemToDo for the volunteers as home project. 2. 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 de la clothoïde

b 2 sqrt :

N 12 :

Snath limit N 12 ≤ ≡

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 0.001 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 :

Clothoïde: highway access ramp

Clothoid 1 2 1 sys