Functions for line intersections

8756e0b0-4786-49cc-b282-8d7a293f7675 123 Functions for line intersections Marco Bernacca 3 5 decimal

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These functions have been written to implement in Smath some features that are natively provided by either Scilab or Matlab in order to make available in Smath a function originally written in Matlab to find intersections between lines. The function in subject is P = InterX(L1, L2), written by NS. See below link: https://de.mathworks.com/matlabcentral/fileexchange/22441-curve-intersections

Function mdiff(V) Input: a vector (either a row or a column) Output: a vector of the same nature of input (a row or a column) containing the differences V(i+1)-V(i). Number of output elements are lenght(V)-1.

Function mdiff(V)
Input: a vector (either a row or a column)
Output: a vector of the same nature of input (a row or a column)
                  containing the differences V(i+1)-V(i).
                   Number of output elements are lenght(V). V mdiff R V rows V rows 1 > (- V cols V cols 1 > (- matrix : ii 1 R length range R ii el V ii 1 + el V ii el - : for R 31 line :

Examples:

z2 4273151 mat :

z2 mdiff 2 - 54 - 2 - 41 mat

z1 13589416 mat :

z1 mdiff 22315 - 15 mat

Function index(x; cond) Input: x: a vector (row or column). cond: expression evaluating to an array of T/F values. Output: a vector of same nature of input array whose components are those corresponding to a TRUE value for the given condition.

Function index(x; cond)
Input: x: a vector (row or column). cond: expression evaluating to an array of T/F values.
Output: a vector of same nature of input array whose components are those corresponding
                to a TRUE value for the given condition. V cond index R V rows V cols matrix : j 0 : V.end V length : 12 mat i 1 V.end range cond vectorize i el j j 1 + : R j el V i el : 21 line continue if for R rows 1 > (R 1 j 11 submatrix R 111 j submatrix if 41 line :

Examples:

z1 z1 4 > index 58913 mat

z2 z1 mdiff 0 > index 427341 mat

Function plotG(x; y; char; size; color) Thanks to Omorr!! :-)

data char size clr plotG k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

Function map(x; y; fun(2)) Input: x, y: vectors or matrices with compatible sizes. fun: a function of 2 argumens Output: vector or matrix whose elements are fun(el(x);el(y)) Apply a given function to corresponding elements of compatible arrays. For info on the Matlab concept of "arrays of compatible sizes" please refer to: https://it.mathworks.com/help/matlab/matlab_prog/compatible-array-sizes-for-basic-operations.html

a b plus a b + :

a b times a b * :

a b minus a b - :

a b divide a b / :

x y 2 fun map xh x cols : xv x rows : yh y cols : yv y rows : 22 mat okx xh yh ≡ (xh 1 ≡ (yh 1 ≡ (| (| (: oky xv yv ≡ (xv 1 ≡ (yv 1 ≡ (| (| (: okx oky & xx x : yy y : xh yh < ii 2 yh range xx xx x augment : for xh yh > ii 2 xh range yy yy y augment : for 1 if if xv yv < ii 2 yv range xx xx x stack : for xv yv > ii 2 xv range yy yy y stack : for 1 if if hv xv yv Max : hx xh yh Max : 12 mat ii 1 hv range jj 1 hx range zz ii jj el xx ii jj el yy ii jj el fun : for for zz 71 line Incompatible sizes error if 41 line :

z1 z2 A B times map 4122032361626101618872135566328391524271213589456 mat

Examples:

z1 z1 A B plus map 26101618816 mat

z2 z2 A B plus map 84146251 mat

Function D(x; y) Helper function for InterX. See below.

x y D x 1 x rows 1 x cols 1 - submatrix y A B minus map x 1 x rows 2 x cols submatrix y A B minus map A B times map :

Function InterX(L1; L2) Input: L1; L2 : 2-rows matrices of coords (Row1 = x-coord; Row2 = y-coord) Output: 2-rows matricx of coords of intersecting points

L1 L2 InterX x1 L1 1 row transpose : x2 L2 1 row : y1 L1 2 row transpose : y2 L2 2 row : dx1 x1 mdiff eval : dy1 y1 mdiff eval : dx2 x2 mdiff eval : dy2 y2 mdiff eval : ii 1 dx1 length range S1 ii el dx1 ii el y1 ii el * dy1 ii el x1 ii el * - : for ii 1 dx2 length range S2 ii el dx2 ii el y2 ii el * dy2 ii el x2 ii el * - : for S2 S2 transpose eval : C1 dx1 y2 A B times map dy1 x2 A B times map - S1 D 0 mwle eval : C2 y1 dx2 A B times map x1 dy2 A B times map - (transpose S2 transpose D 0 mwle transpose eval : ff C1 C2 mwand mwfind eval : ii ff 1 el eval : jj ff 2 el eval : ii length 0 ≡ 21 Zeros ii ii transpose eval : dx2 dx2 transpose eval : dy2 dy2 transpose eval : S2 S2 transpose eval : L dy2 jj el dx1 ii el A B times map dy1 ii el dx2 jj el A B times map - eval : ii ii L 0 ≠ index eval : jj jj L 0 ≠ index eval : L L L 0 ≠ index eval : PP1 dx2 jj el S1 ii el A B times map dx1 ii el S2 jj el A B times map - eval : PP2 dy2 jj el S1 ii el A B times map dy1 ii el S2 jj el A B times map - eval : PP PP1 PP2 augment L L augment A B divide map eval : PP 121 line if 171 line :

Let's test it with a simple example:

L1 123456890112203 0.5 28 mat :

L2 34566579701332021429 mat :

t0 1 time :

ω L1 L2 InterX : 4.5 2 5.5 1 5.6667 0.6667 7.25 1.875 8.75 1.125 7.8333 2.75 8.5 1.75 72 mat

plot1 ω o 6 black plotG :

1 time t0 - 0.8 s *

L1 transpose L2 transpose plot1 3 1 sys

Engineering examples

T1, T2

T1 00 0.038 - 0.022 0.075 - 0.046 0.112 - 0.072 0.15 - 0.099 0.188 - 0.128 0.225 - 0.16 0.262 - 0.193 0.3 - 0.229 0.338 - 0.269 0.375 - 0.312 0.412 - 0.359 0.45 - 0.413 0.488 - 0.476 0.525 - 0.555 0.525 - 0.846 0.488 - 0.872 0.45 - 0.88 0.412 - 0.876 0.375 - 0.864 0.338 - 0.844 0.3 - 0.817 0.262 - 0.783 0.225 - 0.74 0.188 - 0.69 0.15 - 0.629 0.112 - 0.554 0.075 - 0.46 0.038 - 0.33 00 0.067 0.036 0.133 0.068 0.2 0.096 0.267 0.121 0.333 0.141 0.4 0.158 0.467 0.17 0.533 0.179 0.6 0.184 0.667 0.184 0.733 0.178 0.8 0.166 0.867 0.145 0.933 0.109 1010 0.933 0.109 - 0.867 0.145 - 0.8 0.166 - 0.733 0.178 - 0.667 0.184 - 0.6 0.184 - 0.533 0.179 - 0.467 0.17 - 0.4 0.158 - 0.333 0.141 - 0.267 0.121 - 0.2 0.096 - 0.133 0.068 - 0.067 0.036 - 00 0.038 - 0.022 - 0.075 - 0.046 - 0.112 - 0.072 - 0.15 - 0.099 - 0.188 - 0.128 - 0.225 - 0.16 - 0.262 - 0.193 - 0.3 - 0.229 - 0.338 - 0.269 - 0.375 - 0.312 - 0.412 - 0.359 - 0.45 - 0.413 - 0.488 - 0.476 - 0.525 - 0.555 - 0.525 - 0.846 - 0.488 - 0.872 - 0.45 - 0.88 - 0.412 - 0.876 - 0.375 - 0.864 - 0.338 - 0.844 - 0.3 - 0.817 - 0.262 - 0.783 - 0.225 - 0.74 - 0.188 - 0.69 - 0.15 - 0.629 - 0.112 - 0.554 - 0.075 - 0.46 - 0.038 - 0.33 - 00902 mat :

T2 00 0.044 - 0.001 0.088 - 0.003 0.133 - 0.006 0.179 - 0.011 0.227 - 0.017 0.275 - 0.026 0.324 - 0.036 0.374 - 0.049 0.427 - 0.064 0.481 - 0.082 0.537 - 0.105 0.596 - 0.133 0.66 - 0.168 0.732 - 0.219 0.878 - 0.47 0.858 - 0.512 0.83 - 0.537 0.795 - 0.553 0.757 - 0.561 0.714 - 0.562 0.668 - 0.558 0.619 - 0.546 0.565 - 0.529 0.507 - 0.504 0.444 - 0.469 0.374 - 0.424 0.295 - 0.361 0.198 - 0.267 00 0.04 0.065 0.081 0.126 0.125 0.183 0.171 0.238 0.218 0.289 0.268 0.337 0.319 0.381 0.372 0.422 0.428 0.459 0.485 0.493 0.546 0.521 0.61 0.544 0.678 0.559 0.754 0.561 0.866 0.5 0.866 0.5 0.863 0.372 0.823 0.308 0.776 0.256 0.724 0.212 0.669 0.174 0.612 0.141 0.552 0.111 0.489 0.086 0.425 0.063 0.359 0.045 0.291 0.029 0.221 0.017 0.15 0.008 0.076 0.002 00 0.021 - 0.038 - 0.042 - 0.078 - 0.062 - 0.118 - 0.08 - 0.161 - 0.098 - 0.205 - 0.115 - 0.251 - 0.131 - 0.298 - 0.145 - 0.349 - 0.158 - 0.401 - 0.169 - 0.457 - 0.178 - 0.517 - 0.183 - 0.583 - 0.184 - 0.656 - 0.177 - 0.744 - 0.032 - 0.995 - 0.014 0.999 - 0.05 0.987 - 0.081 0.965 - 0.107 0.936 - 0.13 0.9 - 0.149 0.858 - 0.164 0.809 - 0.175 0.754 - 0.182 0.691 - 0.184 0.619 - 0.18 0.536 - 0.165 0.436 - 0.133 0.305 - 00902 mat :

Strip paired 0's

data Strip2 nest striped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows range u i el v i el * 0 ≠ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

The data set must be clean of multiple intersecting junctions

T1 T2 2 1 sys

t0 1 time :

Ω T1 transpose T2 transpose InterX :

1 time t0 - min 11

Ω Ω :

Ω Ω Strip2 :

XY Ω o 6 black plotG :

T1 T2 XY 3 1 sys

Ω 0.499 - 0.5 0.682 0.183 0.182 - 0.68 - 32 mat

D1, D2

D1 4 - 1.593 3.027 - 1.473 2.338 - 1.34 1.878 - 1.204 1.581 - 1.076 1.393 - 0.958 1.277 - 0.846 1.207 - 0.74 1.168 - 0.637 1.147 - 0.536 1.137 - 0.435 1.133 - 0.336 1.131 - 0.236 1.131 - 0.137 1.131 - 0.037 1.131 - 0.062 - 1.131 - 0.162 - 1.131 - 0.262 - 1.129 - 0.361 - 1.124 - 0.46 - 1.112 - 0.559 - 1.088 - 0.657 - 1.044 - 0.752 - 0.973 - 0.843 - 0.864 - 0.928 - 0.712 - 1.002 - 0.512 - 1.061 - 0.271 - 1.1 - 0.006 - 1.114 - 0.259 1.101 - 0.501 1.064 - 0.703 1.005 - 0.858 0.932 - 0.969 0.847 - 1.042 0.756 - 1.086 0.661 - 1.111 0.564 - 1.124 0.465 - 1.129 0.366 - 1.131 0.266 - 1.131 0.167 - 1.131 0.067 - 1.131 0.032 1.131 0.132 1.131 0.231 1.133 0.331 1.136 0.431 1.146 0.531 1.166 0.632 1.205 0.735 1.273 0.841 1.386 0.952 1.57 1.071 1.861 1.198 2.312 1.333 2.989 1.467 3.947 1.588 5.241 1.715 7.171 1.951 10.869 2.327 19.032 2.924 612 mat :

D2 0.1 2.175 - 0.507 2.103 - 0.848 1.929 - 1.089 1.66 - 1.228 1.317 - 1.288 0.932 - 1.306 0.529 - 1.308 0.122 - 1.307 0.284 1.302 0.69 1.271 1.088 1.184 1.461 1.006 1.778 0.722 2.01 0.35 2.143 0.069 - 2.177 0.478 - 2.111 0.826 - 1.945 1.075 - 1.683 1.221 - 1.345 1.286 - 0.961 1.305 - 0.559 1.308 - 0.153 1.307 - 0.254 - 1.303 - 0.66 - 1.275 - 1.059 - 1.193 - 1.435 - 1.023 - 1.757 - 0.747 - 1.996 - 0.38 - 2.137 - 0.037 2.177 - 312 mat :

The data set must be clean of multiple intersecting junctions

D1 D2 2 1 sys

t0 1 time :

ω D1 transpose D2 transpose InterX :

1 time t0 - min 1

ω ω :

ω ω Strip2 :

XY ω o 6 black plotG :

D1 D2 XY 3 1 sys

ω 1.289 0.857 1.291 - 0.859 22 mat

A 2 - 81 - 122 mat :

A dn_LinAlgEigenvalues 0.5 - 2.398 i * + 0.5 - 2.398 i * - 21 mat

t Y D A Y 1 el Y 2 el 21 mat * :

Original:very slow 1 - 1 0.99 - range

N 400 :

focus U0 00013 mat : p 01 0.25 range W 1121 mat : U W 02 π * N t Y D rkfixed : U0 U0 U stack : 31 line for U0 3 col U0 2 col augment 31 line :

Focus focus 2 N 1 + (12 submatrix :

Focus