<p> hatch </p>

6912a6ab-8f49-4172-b601-c35a816d497e 92 15 12 decimal

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

<p> hatch </p>

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und 
g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*N vertical lines (hatching) 
between functions f(x) and g(x) in x-range from x1 to x2 f g x.1 x.2 N hatch dx x.2 x.1 - N / (: h x.1 x.1 f x.1 x.1 g 22 mat : x x.1 x.2 dx - x.1 dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

Graphing the Analytics of a function

3 appVersion 0.98.6179 ≡

The function a x f a x 3^* :

The curvature a x K a x f x 2 diff 1 a x f x diff (2^+ (32 /^/ :

The normal a x N a xo K xo diff (1 -^x xo - (* - a xo K + :

The tangent a x T a xo K xo diff (x xo - (* a xo K + :

a 0.5 : (user parameter ≡ xo 0.45 - : (point of normalcy ≡ 21 line

x f a x K :

x g a x N :

X a u N u 2 - 0 solve : T a u T u 01 solve : 21 mat

X T 21 mat 1.9828 - 0.4625 21 mat

a 0 N a 0 T 21 mat 1.5298 - 0.5994 - 21 mat

a x K a x N a x T xo a xo K . 15 black 1 5 mat a x f x f x g xo X 20 hatch 6 1 sys

Notes:

1. This document is demo about tangent/normal.

2. Normalcy xo is user.

3. There is a way to determine normalcy xo such

that [f(a,x), K(a,x), T(a,x)] join @ same point.

Not aware of such application ...

Solution is given below.

]]>

Intersection of "normal/tangent" with curvature a u N a u K - 0 ≡ u 1 - 0 solve a u T a u K - 0 ≡ u 0.5 1.5 solve 21 mat 0.7913 - 0.45 - 21 mat 0.9391 21 mat

Solving for the "normalcy" of the triplet junctions

specific jonction of "f & K" is unique fK a u f a u K - 0 ≡ u 0.9 1.1 solve : 1.0039

curvature @ 'fK' a fK K 0.505819995750104

determine a solving intersection a fK Φ 0.505819995750424 :

The tangent @ xo a xo x T a xo K xo diff (x xo - (* a xo K + :

normalcy of the triplet a xo fK T a fK Φ - xo 0.47 - 0.45 - solve 0.4607 -

a t x a t * : a t y a 2 / t 2^* : 12 mat

a 1 :

t 2 - 2 1.99 - range :

a t X a t x a t y t diff a t x t diff 2^a t y t diff 2^+ a t x t 2 diff a t y t diff * a t y t 2 diff a t x t diff * - / * + :

a t Y a t y a t x t diff a t x t diff (2^a t y t diff (2^+ a t y t 2 diff a t x t diff * a t x t 2 diff a t y t diff * - / * + :

a t parabola a t x vectorize a t y vectorize augment : a t evolute a t X vectorize a t Y vectorize augment : 21 sys

evolute of the parabola ...

a t parabola a t evolute 2 1 sys

parabola
...
normal
...
tangent a x f a x 2^* : a x N a u f u diff (1 -^x u - (* - a u f + : a x T a u f u diff (x u - (* a u f + : 31 sys

a 0.5 :

L H N u U 0 : dx H L - N / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

U 1.25 - 1.25 100 u : Q i 1 U rows range vx i el U i el : vy i el a U i el f : 21 line for vx vy augment 21 line : 21 line

range/populate tangents/normals lines x 1.5 - 1.5 50 u :

x x : u 0.75 - 0.75 0.5 - range : 12 mat i 1 u rows range tan i el a u i el f u i el diff (x u i el - (* a u i el f + eval : for 21 line

Not used or
transparent x y axes x y / - :

Observe . . .

For Normal escape divide 0 by ranging u wisely. Proposal: acceptable [ u:= -1,-0.801 .. 1]

]]>

x x : u 1 - 1 0.801 - range : 12 mat i 1 u rows range norm i el a u i el f u i el diff (1 -^x u i el - (* - a u i el f + eval : for 21 line

j tan x tan j el augment : j norm x norm j el augment : 21 sys

Tan Q axes 1 tan 2 tan 3 tan 4 tan 5 tan 6 tan 7 tan 91 sys :

Norm Q 1 norm 2 norm 3 norm 4 norm 5 norm 6 norm 7 norm 8 norm 9 norm 10 norm 11 norm 121 sys :

7 nests [tan]
11 nests [norm]
50 values per nest tan rows norm rows tan 1 el rows 31 mat 7115031 mat

Mathcad creates a ONE col of nests. Each nest of the ranged Tangents/Normals. The plot explodes each corresponding nest. Done otherwise Smath X_Y plot by collection.

Tan Norm