Functions for line intersections

8756e0b0-4786-49cc-b282-8d7a293f7675 98 Functions for line intersections Marco Bernacca 4 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

V Δ i 2 V rows range d i 1 - el V i el V i 1 - el - : for d 21 line :

z1 13589416 mat transpose :

z2 4273115 mat transpose :

z1 Δ 22315 - 51 mat

z2 Δ 2 - 54 - 2 - 41 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!! :-)

x y char size color plotG n x length : plot x 1 el y 1 el char size color augment : i 2 n range plot plot x i el y i el char size color augment stack : for plot 41 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 :

tt 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 tt 1 col tt 2 col o 12 Green plotG :

L1 transpose L2 transpose plot1 3 1 sys

Intersection of two vectors

a 13298216 mat transpose : b 30781516 mat transpose : 21 line

v x in v x ≡ vectorize Max :

Test for intersection of two vectors v w intersect u 0 : i 1 : x v w x in u i el x : i i 1 + : 21 line if for u 41 line :

vectors aren't parallel a b in 0

a b intersect transpose 13813 mat

i 05 range : ii 05 range : 21 line

U ii a augment : V i b augment : 21 line

U V 2 1 sys

1. vectors are NOT parallel 2. they will collide sooner or later i.e: in the sense of 'intersect' 3. because of their direction: 'b' will collide 'a' @ 1 'b' will collide 'a' @ 3 'b' will collide 'a' @ 8

The counter loop scans for partnership between [a,b]. Mutual partners are valued and collected.

Partnership in vectors

is 'x' a component of 'y'
in(x,y) answers the question x y in x y ≡ vectorize Max :

312345678933 mat in 1

n 12313 mat transpose :

2.5 n in 0

a 1329815 mat transpose :

b 3078115 mat transpose :

a b ≡ vectorize (transpose 0000015 mat

b 3079115 mat transpose :

a b ≡ vectorize (transpose 0001015 mat