Functions for line intersections

8756e0b0-4786-49cc-b282-8d7a293f7675 122 Functions for line intersections Marco Bernacca 4 5 decimal

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

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.

Guess that the submatrix version must to be faster

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 :

V mdiff V rows 1 > V 0 stack 0 V stack - 2 V rows 11 submatrix V 0 augment 0 V augment - 112 V cols submatrix if eval :

z2 4273151 mat :

z2 mdiff 2.0000 - 5.0000 4.0000 - 2.0000 - 41 mat

z1 13589416 mat :

V 2 V length range el V 1 V length 1 - (range el -

Examples:

z1 mdiff 2.0000 2.0000 3.0000 1.0000 5.0000 - 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.

Index isn't used in this version of InterX

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 j 0 : R 0 : c cond vectorize : 13 mat i 1 V length range c i el j j 1 + : R j el V i el : 21 line continue if for V rows 1 > R R transpose if eval 31 line :

Examples:

V rows 1 > V cond vectorize mwfind 2 el el V cond vectorize mwfind 2 el el transpose if eval

z1 z1 4 > index 5.0000 8.0000 9.0000 13 mat

z2 z1 mdiff 0 > index 4.0000 2.0000 7.0000 3.0000 41 mat

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

x y char size color plotG plot 05 matrix : i 1 x length range plot plot x i el y i el char size color augment stack : for plot 31 line :

Notice the use of matrix(0,5) as NULL.

x y char size color plotG i 1 x length range : plot i 1 el x i el : plot i 2 el y i el : plot i 3 el char : plot i 4 el size : plot i 5 el color : plot 71 line :

x y char size color plotG template 00 char size color 15 mat : r 1 x length range : c 15 range : plot r c el template 1 c el : plot r 1 el x r el : plot r 2 el y r el : plot 71 line :

xy char size color plotG xy cols 1 > xy 1 col xy 2 col char size color plotG xy Re xy Im char size color plotG if :

xy char size color plotG xy cols 1 > xy 1 col xy 2 col char size color plotG xy Re xy Im char size color plotG if 11 line :

"Scalar" (loop based) version of mwfind. Can be improved using findrow?

M find rm M rows : cm M cols : lm cm rm * : gtzM M 0 ≠ vectorize : nzM gtzM (sum : R nzM 1 matrix : C R : V R : 13 mat jj 1 : ii 1 lm range gtzM ii el t ii cm mod : R jj el ii cm / Ceil : C jj el t 0 ≡ cm t if : V jj el M ii el : 13 mat jj jj 1 + : 31 line 0 if 11 line for R C V 31 mat 91 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

Complete version, with error checking, but the other it's just a little faster.

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 & X x : Y y : 12 mat xh yh < κ 2 yh range X X x augment : for 121 line xh yh > κ 2 xh range Y Y y augment : for 121 line 1 if if xv yv < κ 2 yv range X X x stack : for 121 line xv yv > κ 2 xv range Y Y y stack : for 121 line 1 if if 12 mat X Y fun vectorize 31 line Incompatible sizes error if 41 line :

xc x cols : xr x rows : yc y cols : yr y rows : 22 mat X x : Y y : 12 mat r xr yr Max : c xc yc Max : 12 mat X cols c < X X x augment : while Y cols c < Y Y y augment : while X rows r < X X x stack : while Y rows r < Y Y y stack : while X eval Y eval fun vectorize eval 81 line

x y 2 fun map xc x cols : xr x rows : yc y cols : yr y rows : 22 mat X x : Y y : 12 mat xc yc ≡ 0 xc yc < c 1 xc yc * (range : r 1 xr range : X r c el X r 1 c 1 - xc mod + el : 31 line c 1 xc yc * (range : r 1 yr range : Y r c el Y r 1 c 1 - yc mod + el : 31 line if if xr yr ≡ 0 xr yr < c 1 X cols range : r 1 xr yr * (range : X r c el X 1 r 1 - xr mod + c el : 31 line c 1 Y cols range : r 1 xr yr * (range : Y r c el Y 1 r 1 - yr mod + c el : 31 line if if X Y fun vectorize eval 51 line :

a b plus a b + :

a b times a b * :

a b minus a b - :

a b divide a b / :

X Y fun vectorize r xr yr Max : c xc yc Max : 12 mat ρ 1 r range χ 1 c range Z ρ χ el X ρ χ el Y ρ χ el fun : for for 21 line ≡

Examples:

z1 z2 A B times map 4.0000 12.0000 20.0000 32.0000 36.0000 16.0000 2.0000 6.0000 10.0000 16.0000 18.0000 8.0000 7.0000 21.0000 35.0000 56.0000 63.0000 28.0000 3.0000 9.0000 15.0000 24.0000 27.0000 12.0000 1.0000 3.0000 5.0000 8.0000 9.0000 4.0000 56 mat

z2 z1 A B times map 4.0000 12.0000 20.0000 32.0000 36.0000 16.0000 2.0000 6.0000 10.0000 16.0000 18.0000 8.0000 7.0000 21.0000 35.0000 56.0000 63.0000 28.0000 3.0000 9.0000 15.0000 24.0000 27.0000 12.0000 1.0000 3.0000 5.0000 8.0000 9.0000 4.0000 56 mat

z2 z2 A B times map 16.0000 4.0000 49.0000 9.0000 1.0000 51 mat

z1 z1 A B times map 1.0000 9.0000 25.0000 64.0000 81.0000 16.0000 16 mat

z1 z1 A B plus map 2.0000 6.0000 10.0000 16.0000 18.0000 8.0000 16 mat

z2 z2 A B plus map 8.0000 4.0000 14.0000 6.0000 2.0000 51 mat

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 : 14 mat dx1 x1 mdiff : dy1 y1 mdiff : dx2 x2 mdiff : dy2 y2 mdiff : 14 mat k 1 dx1 length range : S1 k el dx1 k el y1 k el * dy1 k el x1 k el * - : k 1 dx2 length range : S2 k el dx2 k el y2 k el * dy2 k el x2 k el * - : a b t a b A B times map : a b c d T a b t c d t - eval : x y D r x rows : c x cols : 12 mat X x 1 r range 1 c 1 - (range el y A B minus map : Y x 1 r range 2 c range el y A B minus map : X Y t 41 line : C1 dx1 y2 dy1 x2 T S1 D : C2 y1 dx2 x1 dy2 T transpose S2 D : C1 C1 0 ≤ vectorize : C2 C2 0 ≤ vectorize transpose : 12 mat m n vals 13 mat C1 C2 & vectorize find transpose eval : m m transpose : dx2 dx2 transpose : dy2 dy2 transpose : 13 mat L dy2 n el dx1 m el dy1 m el dx2 n el T : Lo L find 1 el eval : m m Lo el eval : n n Lo el eval : L L Lo el eval : 13 mat P1 dx2 n el S1 m el dx1 m el S2 n el T : P2 dy2 n el S1 m el dy1 m el S2 n el T : lc L cols : P1 P2 augment L 1 L rows range 11 lc range 1 - lc mod vectorize + el A B divide map 91 line 21 Zeros try 131 line :

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

L mwfind 1 el eval L 0 ≠ vectorize mwfind 1 el eval ≡

to 0 time :

Let's test it with a simple example:

L1 123456890112203 0.5 28 mat :

L2 34566579701332021429 mat :

tt L1 L2 InterX : 4.5000 2.0000 5.5000 1.0000 5.6667 0.6667 7.2500 1.8750 7.8333 2.7500 8.7500 1.1250 8.5000 1.7500 72 mat

plot1 L1 L2 InterX o 12 Green plotG :

L1 transpose L2 transpose plot1 3 1 sys

and a bit more complicate one:

t 02 π * (2 π 20 / * range transpose :

r1 4 t * sin vectorize 2 + :

r2 8 t * sin vectorize 2 + :

c t cos vectorize :

s t sin vectorize :

xy1 r1 c * vectorize r1 s * vectorize stack :

xy2 r2 c * vectorize r2 s * vectorize stack :

XY xy1 xy2 InterX :

plot2 XY o 12 Green plotG :

xy1 transpose xy2 transpose plot2 3 1 sys

0 time to - 1.3990 s *

CWD C:\SmathFile CurrentDirectory :

T1 CWD TrifoliumBlue rfile :

T2 CWD TrifoliumRed rfile :

T1 T2 2 1 sys

Ω T1 transpose T2 transpose InterX :

Intersect Ω 1 col Ω 2 col o 6 black plotG :

1 time to - min 2

T1 T2 Intersect 3 1 sys