% segment2_num.m - numerical resolution
%{
Assigned two points in the real plane, P1, P2; we want to establish
if a third point Pj belongs, or not, to the segment between P1 and P2.
%}

clear; clc; %close all

%% line and segment between these two points
rng('default') % Set the random number generator to the default seed(0)
P1=rand(2,1);
P2=2*rand(2,1);

%% parametric equation of the line through P1 and P2 (of the segment if lambda in [0,1])
% Symbolic equation is used only to draw the segment and the line.
syms lambda real
P = P1 + lambda*(P2 - P1); % parametric vector eq of the line
d = (P2 - P1);             % director vector

%% Other points
P3=[2 0]'; % not on the line
P4=P1 + pi/2*(P2 - P1); % P4 on the line, but outside of the segment
P5=P1 + pi/4*(P2 - P1); % P5 on the line, and inside the segment

%% Display segment and points
figure(2); clf
h=ezplot(P(1),P(2),[-2 3]); set(h,'Color','r','LineStyle','--')
axis([-1.5 3 -1 3])
axis equal; hold on; grid on; box on
h=ezplot(P(1),P(2),[0 1]); set(h,'Color','r','LineWidth',3);
plot([P1(1) P2(1)],[P1(2) P2(2)],'Color','r','Marker','.','MarkerSize',24)
HT=text(P1(1),P1(2),' P_1','FontSize',14,'Color','k','FontWeight','Bold','VerticalAlignment','top','HorizontalAlignment','right');
HT=text(P2(1),P2(2),' P_2','FontSize',14,'Color','k','FontWeight','Bold','VerticalAlignment','top','HorizontalAlignment','right');
title('Segment between P_1 and P_2','FontSize',14)

h=plot(P3(1),P3(2),'om','MarkerSize',10,'MarkerFaceColor','m');
HT=text(P3(1),P3(2),' P_3','FontSize',14,'Color','m','FontWeight','Bold','VerticalAlignment','bottom');

h=plot(P4(1),P4(2),'og','MarkerSize',10,'MarkerFaceColor','g');
HT=text(P4(1),P4(2),' P_4','FontSize',14,'Color','g','FontWeight','Bold','VerticalAlignment','top','HorizontalAlignment','right');

h=plot(P5(1),P5(2),'ob','MarkerSize',10,'MarkerFaceColor','b');
HT=text(P5(1),P5(2),' P_5','FontSize',14,'Color','b','FontWeight','Bold','VerticalAlignment','bottom');

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% P3 is not on the line: 2) without solve(), as a linear system
S=(P3-P1)./d

line = S(1) == S(2)          % check for point on the line
segm = S(1) > 0  &  S(1) < 1 % if on the line, check for point on the segment

if line && segm % if on the line, check for point on the segment
    disp(['P3 = (' num2str(P3') ') belongs to the segment.']); disp('')
else
    disp(['P3 = (' num2str(P3') ') does not belong to the segment,'])
    if line
        fprintf('\tbut it lies on the line.\n\n')
    else
        fprintf('\tand it does not lie on the line neither.\n\n')
    end
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% P4 on the line, but outside of the segment: 2) without solve(), as a linear system
S=(P4-P1)./d

%line = S(1) == S(2)            % THIS CODE DOES NOT WORK IN GENERAL!
 line = abs(S(1) - S(2)) < 1e-8 % check for point on the line
segm = S(1) > 0  &  S(1) < 1    % if on the line, check for point on the segment

if line && segm % if on the line, check for point on the segment
    disp(['P4 = (' num2str(P4') ') belongs to the segment.']); disp('')
else
    disp(['P4 = (' num2str(P4') ') does not belong to the segment,'])
    if line
        fprintf('\tbut it lies on the line.\n\n')
    else
        fprintf('\tand it does not lie on the line neither.\n\n')
    end
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% P5 on the segment: 2) without solve(), as a linear system
S=(P5-P1)./d

%line = S(1) == S(2)            % THIS CODE DOES NOT WORK IN GENERAL!
 line = abs(S(1) - S(2)) < 1e-8 % check for point on the line
segm = S(1) > 0  &  S(1) < 1    % if on the line, check for point on the segment

if line && segm % if on the line, check for point on the segment
    disp(['P5 = (' num2str(P5') ') belongs to the segment.']); disp('')
else
    disp(['P5 = (' num2str(P5') ') does not belong to the segment,'])
    if line
        fprintf('\tbut it lies on the line.\n\n')
    else
        fprintf('\tand it does not lie on the line neither.\n\n')
    end
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%