beb77175-109d-4a2f-986b-ceb79b50c36c 69 6 5 auto

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

Hermite Cubic Derivative

The cubic polynomial is constrained at both end points p(0) = 0 ... p(1) = 0 The next two conditions are user defined or searched for in case of modeling. These two conditions are: diff(p(0),t) = c ... diff(p(1),t) = b A new form of polynomial emerges on those conditins, after solving the: a0, a1, a2, a3 => "Hermite Cubic Derivative" They have applications in modeling phenomenons known to respond to the derivative of hidden quantity.

x p a0 a1 x * + a2 x 2^* + a3 x 3^* + :

0 p 0 ≡

1 p 0 ≡

0 p t diff c ≡

1 p t diff b ≡

a0 a1 a2 a3 41 mat 0 c 2 c * - 2 b * - c b + 41 mat ≡

x b c spline c x 1 n +^* 2 c * b + (x 2 n +^* - c b + (x 3 n +^* + :

v b c 21 mat :

n 0 :

x v spline v 2 el x 1 n +^* 2 v 2 el * v 1 el + (x 2 n +^* - v 2 el v 1 el + (x 3 n +^* + :

The working experimental constrained "spline (x,v)... n = user x v spline 0 x ≤ (x 1 ≤ (& v 2 el x 1 n +^* 2 v 2 el * v 1 el + (x 2 n +^* - v 2 el v 1 el + (x 3 n +^* + out range if :

v 1121 mat :

Hermite Cubic Deriv, regular

x v spline

v 1 - 121 mat :

Drag up/down to see the 4 basic regular shapes wrt [b,c,n].

v 11 - 21 mat :

v 1 - 1 - 21 mat :

New Experimental shape(s)

This new experiment is interesting in two counts: 1. For modelling as a 3 parameters dependent. 2. It hiddes difficulties in calculating the "arclength. We are left with an unsolvable problem in Smath. Alternately: 1. Legendre Quadrature 2. Discretise, proceed with the "FD Cumulative Integration".

v 0.5 1 - 21 mat :

n 0.5 - :

x Spline x v spline 1 x - v spline + 2 / :

Hermite equalities

0.5 v spline 0.265165 -

The Hermite "chaînette" ... user experimental

x v spline 1 x - v spline x Spline 3 1 sys

1 0.5 - v spline 0.265165 -

0.5 Spline 0.265165 -

[b,c,n]=>[0.5,-1,-0.5]

x λ 231 x - (* - 1 x - (2^+ (1 x - sqrt * 2 / - :

x q x sqrt 2 x 3 - x + (* + (* 2 / - :

x h x q x λ + 2 / :

Next is the Integration difficulty: Can't integrate λ(x), h(x),for their respective arc length, both are singular @ their RHS end point. However, q(x) is regular on the RHS and its arc length exists. The question remains: does the Smath arc length of q(x) equates h(x) ?

x h x q x λ 3 1 sys

u arc_h 1 x h x diff (2^+ sqrt :

z Int u arc_h eval u 0 x int :

x Int z Int :

0.999 Int 8.174

x h x diff

u Arc_q 1 x q x diff (2^+ sqrt :

z Int u Arc_q eval u 0.5 x int :

1 Int 0.559

u Arc_λ 1 x λ x diff (2^+ sqrt :

=> MCAD h(x) 1.233

z Int u Arc_λ eval u 0 x int :

=> MCAD q(x) 1.287

0.5 Int 0.547

0.569 0.547 + 1.116

Integrand q(x) ... ~ regular

x Arc_q x ( Arc_λ 2 1 sys

Integrand h(x)... singular

x arc_h

Alternately: proceed to the discrete inspection. 1. Discretise the h(x). 2. Finite Diff's Derivative. 3. Compose the discrete arc length Integrand. 4. Finite Diff's Cumulative Integration.

N 1000 :

ArcSpline t 011 N / range : i 1 t rows range arc_H i el t i el h eval : 11 line for t arc_H augment 31 line :

ArcSpline

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - range deriv i el 0.5 Y i 1 - el Y i el - X i 1 - el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows 1 - el : X deriv augment 101 line :

D ArcSpline Deriv :

D 0 2 1 sys

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

ArcIntegrand δx D 2 col : xx D 1 col : i 1 δx rows range yy i el 1 δx i el (2^+ sqrt : 11 line for xx yy augment 41 line :

ArcIntegrand 32 el 6.829941

ArcIntegrand N 2 el 6.829941

The merit of the FD procedure

We have no verdict for this project. The closet answer is the Legendre Quadrature. The Finite Diff's looks reasonably asymptotic, Slightly in eccess of Legendre Quadrature. Thisis a known defect of FD resulting from the rectangular infinitesimal elements.

sub ArcIntegrand 1 N 12 submatrix :

ΣA sub Σ :

Discrete Integrand h(x)

ArcIntegrand

Cumulative Arc Length of h(x)

ΣA

ΣA N 2 el 1.221

N 1000

Σlegendre15 1.21 ≡

Asymptote 1 1.194 2 1.206 3 1.211 4 1.214 5 1.216 6 1.218 7 1.219 8 1.22 9 1.221 10 1.221 102 mat :

Asymptotic convergence. The graph "x" is in 100 points.

Asymptote

The curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform gravitational force. The word catenary is derived from the Latin word for "chain." In 1669, Jungius disproved Galileo's claim that the curve of a chain hanging under gravity would be a parabola (MacTutor Archive). The curve is also called the alysoid and chainette. The equation was obtained by Leibniz, Huygens, and Johann Bernoulli in 1691 in response to a challenge by Jakob Bernoulli. Huygens was the first to use the term catenary in a letter to Leibniz in 1690, and David Gregory wrote a treatise on the catenary in 1690 (MacTutor Archive). If you roll a parabola along a straight line, its focus traces out a catenary. As proved by Euler in 1744, the catenary is also the curve which, when rotated, gives the surface of minimum surface area (the catenoid) for the given bounding circle.

chainette a t a / cosh * ≡

... a suspended bridge is not "chaînette". It is strictly a parabola due to the distributed weight of the tablier. Hermite is another gyzma.

Нахождение нулей полиномов Лежандра

Penyelesaian Polinomial Legendre

Legendre Quadrature

Степень полинома Pangkat polinomial Polynomial order n 5 :

Формула Родрига Formula Rodrigues Rodrigues formula z n P 12 n^n! (* / z 2^1 - (n^z n diff * :

T 1 el z n P :

k 1 : k n ≤ k k 1 + : T k 1 + el T k el (z diff k / : for

z D T :

Коэффициенты полинома Лежандра Koefisien polinomial Legendre Legendre polynomial coefficients 0 D 0158 / 0354 / - 0638 / 61 mat

Корни полинома Лежандра Akar polinomial Legendre Roots of Legendre polynomial
 ------- order 'n' -------- 0 D polyroots 0 0.538469310105683 - 0.538469310105683 0.906179845938664 - 0.906179845938664 51 mat

r 0 D polyroots :

ii 1 r rows range p ii el . : for

ii 1 r rows range zero ii el 0 : for

collect x 0 P x 1 P x 2 P x 3 P x 4 P x 5 P r zero p augment 71 sys :

collect

https://pomax.github.io/bezierinfo/legendre-gauss.html

0. prepare the "wr", n rows x 2 cols 1. Select/copy numbers in web site, one at time ... paste in Smath. 2. very easy: points repeat.

n 20 ≡

n 15 ≡

wr 0.1527533871307258 0.0765265211334973 - 0.1527533871307258 0.0765265211334973 0.1491729864726037 0.2277858511416451 - 0.1491729864726037 0.2277858511416451 0.1420961093183820 0.3737060887154195 - 0.1420961093183820 0.3737060887154195 0.1316886384491766 0.5108670019508271 - 0.1316886384491766 0.5108670019508271 0.1181945319615184 0.6360536807265150 - 0.1181945319615184 0.6360536807265150 0.1019301198172404 0.7463319064601508 - 0.1019301198172404 0.7463319064601508 0.0832767415767048 0.8391169718222188 - 0.0832767415767048 0.8391169718222188 0.0626720483341091 0.9122344282513259 - 0.0626720483341091 0.9122344282513259 0.0406014298003869 0.9639719272779138 - 0.0406014298003869 0.9639719272779138 0.0176140071391521 0.9931285991850949 - 0.0176140071391521 0.9931285991850949 202 mat :

0.2025782419255613 0.0000000000000000 0.1984314853271116 0.2011940939974345 - 0.1984314853271116 0.2011940939974345 0.1861610000155622 0.3941513470775634 - 0.1861610000155622 0.3941513470775634 0.1662692058169939 0.5709721726085388 - 0.1662692058169939 0.5709721726085388 0.1395706779261543 0.7244177313601701 - 0.1395706779261543 0.7244177313601701 0.1071592204671719 0.8482065834104272 - 0.1071592204671719 0.8482065834104272 0.0703660474881081 0.9372733924007060 - 0.0703660474881081 0.9372733924007060 0.0307532419961173 0.9879925180204854 - 0.0307532419961173 0.9879925180204854 152 mat

weight, wi w wr 1 col :

abscissea, xi x wr 2 col :

Legendre-Gauss Quadrature,Arbitrary interval

x b c spline c x 1 n +^* 2 c * b + (x 2 n +^* - c b + (x 3 n +^* + :

The working experimental constrained "spline (x,v)... n = user x v spline 0 x ≤ (x 1 ≤ (& v 2 el x 1 n +^* 2 v 2 el * v 1 el + (x 2 n +^* - v 2 el v 1 el + (x 3 n +^* + out range if :

v 0.5 1 - 21 mat :

n 0.5 - :

y #

x Spline x v spline 1 x - v spline + 2 / :

x q x sqrt 2 x 3 - x + (* + (* 2 / - :

Unary User UnitSystem

x λ 231 x - (* - 1 x - (2^+ (1 x - sqrt * 2 / - :

UnitGraphArcLength 1 :

x h x q x λ + 2 / :

UnitGraphChainetteArea 1 :

Project Notes

a 0 :

b 1 :

m wr rows :

Smath did not digest Legendre 10. For + wi, xi: visit the link above. ----------------------------------- Mathcad 11.2a Quadratures: 1. Legendre 2. Radau 3. Hermite 4. Laguerre 5. Lobato 6. Jacobi 7. ChebyShev

y i 1 m range y i el b a - 2 / (x i el * b a + 2 / (+ : for y 21 line :

1 x q x diff (2^+ sqrt x int 1 b a - 2 / (w i el y i el h * (i 1 m sum eval * abs + ≡

ArcLength 1 b a - 2 / (w i el y i el h * (i 1 m sum eval * abs + :

Disambiguation/Adequation: Revisit your book on Calculus

ArcLength 1.21

1.20961417817708

n 10 ≡

1.20955165911396

n 15 ≡

Area under the chaînette

1.20953580985747

n 20 ≡

b a - (w i el y i el h * (i 1 m sum eval * 0.42 -

0.419228356354155 -

n 10 ≡

Arc length FD 1000 pts 1.221 ≡

0.419103318227913 -

n 15 ≡

0.419071619714936 -

n 20 ≡

The Hermite "chaînette" ... user experimental arc length

x q x λ x h 3 1 sys

Roots of HermiteDeriv

v 0.5 1 - 21 mat :

n 0.5 - :

The working experimental constrained "spline (x,v)... n = user x v spline v 2 el x 1 n +^* 2 v 2 el * v 1 el + (x 2 n +^* - v 2 el v 1 el + (x 3 n +^* + :

x spline x v spline :

r x spline x 5 - 5 solve :

r 01231 mat

ρ1 r 1 el 0 + 20 red 15 mat :

x Lagrange x r 1 el - (x r 2 el - (* x r 3 el - (* :

ρ2 r 2 el 0 + 20 red 15 mat :

ρ3 r 3 el 0 + 20 red 15 mat :

x HerLag x spline x Lagrange + :

x spline ρ1 ρ2 ρ3 4 1 sys x Lagrange ρ1 ρ2 ρ3 4 1 sys x HerLag

r x HerLag x 5 - 5 solve :

r 0 0.25 1241 mat