I make a square around/below a given point with given width (w) and length (l) like shown below
But now I have to induce the direction of the point (or heading so to say). I want to create a rectangle directed towards that direction like shown below. I have never worked with headings before, if anyone can point me in the right directions, that would be helpful.
-----------------EDIT---------------
Thank you #MBo & #HansHirse, I implemented as suggested by your people. for simplicity, I chose the point to be on the top line of the rectangle rather to be away from the rectangle. my code is as shown below:
% Point coordinates
P = [5; 5];
% Direction vector
d = [1; -5];
%normalizing
Len = sqrt(d(1)*d(1)+d(2)*d(2));
cs = d(1)/Len;
sn = d(2)/Len;
% Dimensions of rectangle
l = 200;
w = 100;
% Original corner points of rectangle
x1 = P(1) -w/2;
y1 = P(2);
x2 = P(1)+ w/2;
y2 = P(2);
x3 = P(1)+ w/2;
y3 = P(2) - l;
x4 = P(1) -w/2;
y4 = P(2) - l;
rect = [x1 x2 x3 x4; y1 y2 y3 y4];
%rotated rectangles coordinates:
x1_new = P(1)+ (x1 - P(1))* cs - (y1 - P(2))*sn;
y1_new = P(2) +(x1-P(1))*sn + (y1-P(2))*cs;
x2_new = P(1)+ (x2 - P(1))* cs - (y2 - P(2))*sn;
y2_new = P(2) +(x2-P(1))*sn + (y2-P(2))*cs;
x3_new = P(1)+ (x3 - P(1))* cs - (y3 - P(2))*sn;
y3_new = P(2) +(x3-P(1))*sn + (y3-P(2))*cs;
x4_new = P(1)+ (x4 - P(1))* cs - (y4 - P(2))*sn;
y4_new = P(2) +(x4-P(1))*sn + (y4-P(2))*cs;
new_rect = [x1_new x2_new x3_new x4_new; y1_new y2_new y3_new y4_new];
%plot:
figure(1);
plot(P(1), P(2), 'k.', 'MarkerSize', 21);
hold on;
plot([P(1) P(1)+d(1)*10], [P(2) P(2)+d(2)*10], 'b');
patch(new_rect(1,:),new_rect(2,:), 'b', 'FaceAlpha', 0.2);
patch(rect(1,:),rect(2,:),'b')
hold off
The way it is rotating is not what I wanted: I am not able to upload a pic, imgur is acting weird. You can run the exact code, you will get the output I am getting.
So you have point P, and rectangle R (defined by coordinates).
Now you want to rotate rectangle by angle A around point P (as far as I understand)
New coordinates for every vertex are:
NewV[i].X = P.X + (V[i].X - P.X) * Cos(A) - (V[i].Y - P.Y) * Sin(A)
NewV[i].Y = P.Y + (V[i].X - P.X) * Sin(A) + (V[i].Y - P.Y) * Cos(A)
If you have direction vector D = [d1, d2], you don't need to operate with sin and cos functions: just exploit components of normalized vector (perhaps matlab contains function for normalizing):
Len = magnitude(D) = sqrt(d1*d1+d2*d2)
//normalized (unit vector)
dx = d1 / Len
dy = d2 / Len
NewV[i].X = P.X + (V[i].X - P.X) * dx - (V[i].Y - P.Y) * dy
NewV[i].Y = P.Y + (V[i].X - P.X) * dy + (V[i].Y - P.Y) * dx
Also you can omit creating coordinates of axis-aligned rectangle and calculate vertices immediately (perhaps more possibilities for mistakes)
//unit vector perpendicula to direction
perpx = - dy
perpy = dx
V[0].X = P.X - dist * dx + w/2 * perpx
V[1].X = P.X - dist * dx - w/2 * perpx
V[2].X = P.X - dist * dx - w/2 * perpx - l * dx
V[3].X = P.X - dist * dx + w/2 * perpx - l * dx
and similar for y-components
The general concept was already mentioned in MBo's answer. Nevertheless, since I was working on some code, here is my solution:
% Point coordinates
P = [5; 5];
% Direction vector
d = [1; -5];
% Dimensions of rectangle
l = 200;
w = 100;
% Distance P <-> rectangle
dist = 20;
% Determine rotation angle from direction vector
% (Omitting handling of corner/extreme cases, e.g. d(2) = 0)
theta = atand(d(1) / d(2));
if ((d(1) > 0) && (d(2) < 0))
theta = theta + 180;
elseif ((d(1) < 0) && (d(2) < 0))
theta = theta - 180;
end
% Original corner points of rectangle
xMin = P(1) - w/2;
xMax = P(1) + w/2;
yMin = P(2) - dist;
yMax = P(2) - dist - l;
rect = [xMin xMax xMax xMin; yMin yMin yMax yMax];
% Auxiliary matrix for rotation
center = repmat(P, 1, 4);
% Rotation matrix
R = [cosd(-theta) -sind(-theta); sind(-theta) cosd(-theta)];
% Rotation
rotrect = (R * (rect - center)) + center;
% Plot
figure(1);
hold on;
plot(P(1), P(2), 'k.', 'MarkerSize', 21); % Point
plot([P(1) P(1)+d(1)*10], [P(2) P(2)+d(2)*10], 'b'); % Direction vector
patch(rect(1, :), rect(2, :), 'b', 'FaceAlpha', 0.2); % Original rectangle
patch(rotrect(1, :), rotrect(2, :), 'b'); % Rotated rectangle
hold off;
xlim([-250 250]);
ylim([-250 250]);
This will produce an output image like this:
Caveat: In my rotation matrix R, you'll find -theta, since the way, how theta was determined is just some simple approach. This might be improved, so that R could also be set up properly.
I want to move a red star marker along the spiral trajectory with an equal distance of 5 units between the red star points on its circumference like in the below image.
vertspacing = 10;
horzspacing = 10;
thetamax = 10*pi;
% Calculation of (x,y) - underlying archimedean spiral.
b = vertspacing/2/pi;
theta = 0:0.01:thetamax;
x = b*theta.*cos(theta)+50;
y = b*theta.*sin(theta)+50;
% Calculation of equidistant (xi,yi) points on spiral.
smax = 0.5*b*thetamax.*thetamax;
s = 0:horzspacing:smax;
thetai = sqrt(2*s/b);
xi = b*thetai.*cos(thetai);
yi = b*thetai.*sin(thetai);
plot(x,y,'b-');
hold on
I want to get a figure that looks like the following:
This is my code for the circle trajectory:
% Initialization steps.
format long g;
format compact;
fontSize = 20;
r1 = 50;
r2 = 35;
r3= 20;
xc = 50;
yc = 50;
% Since arclength = radius * (angle in radians),
% (angle in radians) = arclength / radius = 5 / radius.
deltaAngle1 = 5 / r1;
deltaAngle2 = 5 / r2;
deltaAngle3 = 5 / r3;
theta1 = 0 : deltaAngle1 : (2 * pi);
theta2 = 0 : deltaAngle2 : (2 * pi);
theta3 = 0 : deltaAngle3 : (2 * pi);
x1 = r1*cos(theta1) + xc;
y1 = r1*sin(theta1) + yc;
x2 = r2*cos(theta2) + xc;
y2 = r2*sin(theta2) + yc;
x3 = r3*cos(theta3) + xc;
y3 = r3*sin(theta3) + yc;
plot(x1,y1,'color',[1 0.5 0])
hold on
plot(x2,y2,'color',[1 0.5 0])
hold on
plot(x3,y3,'color',[1 0.5 0])
hold on
% Connecting Line:
plot([70 100], [50 50],'color',[1 0.5 0])
% Set up figure properties:
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0, 0, 1, 1]);
drawnow;
axis square;
for i = 1 : length(theta1)
plot(x1(i),y1(i),'r*')
pause(0.1)
end
for i = 1 : length(theta2)
plot(x2(i),y2(i),'r*')
pause(0.1)
end
for i = 1 : length(theta3)
plot(x3(i),y3(i),'r*')
pause(0.1)
end
I can't think of a way to compute distance along a spiral, so I'm approximating it with circles, in hopes that it will still be useful.
My solution relies on the InterX function from FEX, to find the intersection of circles with the spiral. I am providing an animation so it is easier to understand.
The code (tested on R2017a):
function [x,y,xi,yi] = q44916610(doPlot)
%% Input handling:
if nargin < 1 || isempty(doPlot)
doPlot = false;
end
%% Initialization:
origin = [50,50];
vertspacing = 10;
thetamax = 5*(2*pi);
%% Calculation of (x,y) - underlying archimedean spiral.
b = vertspacing/(2*pi);
theta = 0:0.01:thetamax;
x = b*theta.*cos(theta) + origin(1);
y = b*theta.*sin(theta) + origin(2);
%% Calculation of equidistant (xi,yi) points on spiral.
DST = 5; cRes = 360;
numPts = ceil(vertspacing*thetamax); % Preallocation
[xi,yi] = deal(NaN(numPts,1));
if doPlot && isHG2() % Plots are only enabled if the MATLAB version is new enough.
figure(); plot(x,y,'b-'); hold on; axis equal; grid on; grid minor;
hAx = gca; hAx.XLim = [-5 105]; hAx.YLim = [-5 105];
hP = plot(xi,yi,'r*');
else
hP = struct('XData',xi,'YData',yi);
end
hP.XData(1) = origin(1); hP.YData(1) = origin(2);
for ind = 2:numPts
P = InterX([x;y], makeCircle([hP.XData(ind-1),hP.YData(ind-1)],DST/2,cRes));
[~,I] = max(abs(P(1,:)-origin(1)+1i*(P(2,:)-origin(2))));
if doPlot, pause(0.1); end
hP.XData(ind) = P(1,I); hP.YData(ind) = P(2,I);
if doPlot, pause(0.1); delete(hAx.Children(1)); end
end
xi = hP.XData(~isnan(hP.XData)); yi = hP.YData(~isnan(hP.YData));
%% Nested function(s):
function [XY] = makeCircle(cnt, R, nPts)
P = (cnt(1)+1i*cnt(2))+R*exp(linspace(0,1,nPts)*pi*2i);
if doPlot, plot(P,'Color',lines(1)); end
XY = [real(P); imag(P)];
end
end
%% Local function(s):
function tf = isHG2()
try
tf = ~verLessThan('MATLAB', '8.4');
catch
tf = false;
end
end
function P = InterX(L1,varargin)
% DOCUMENTATION REMOVED. For a full version go to:
% https://www.mathworks.com/matlabcentral/fileexchange/22441-curve-intersections
narginchk(1,2);
if nargin == 1
L2 = L1; hF = #lt; %...Avoid the inclusion of common points
else
L2 = varargin{1}; hF = #le;
end
%...Preliminary stuff
x1 = L1(1,:)'; x2 = L2(1,:);
y1 = L1(2,:)'; y2 = L2(2,:);
dx1 = diff(x1); dy1 = diff(y1);
dx2 = diff(x2); dy2 = diff(y2);
%...Determine 'signed distances'
S1 = dx1.*y1(1:end-1) - dy1.*x1(1:end-1);
S2 = dx2.*y2(1:end-1) - dy2.*x2(1:end-1);
C1 = feval(hF,D(bsxfun(#times,dx1,y2)-bsxfun(#times,dy1,x2),S1),0);
C2 = feval(hF,D((bsxfun(#times,y1,dx2)-bsxfun(#times,x1,dy2))',S2'),0)';
%...Obtain the segments where an intersection is expected
[i,j] = find(C1 & C2);
if isempty(i), P = zeros(2,0); return; end
%...Transpose and prepare for output
i=i'; dx2=dx2'; dy2=dy2'; S2 = S2';
L = dy2(j).*dx1(i) - dy1(i).*dx2(j);
i = i(L~=0); j=j(L~=0); L=L(L~=0); %...Avoid divisions by 0
%...Solve system of eqs to get the common points
P = unique([dx2(j).*S1(i) - dx1(i).*S2(j), ...
dy2(j).*S1(i) - dy1(i).*S2(j)]./[L L],'rows')';
function u = D(x,y)
u = bsxfun(#minus,x(:,1:end-1),y).*bsxfun(#minus,x(:,2:end),y);
end
end
Result:
Note that in the animation above, the diameter of the circle (and hence the distance between the red points) is 10 and not 5.
The below code is something that I am cooking up. I am plotting the orbits of the Sun, Mercury, Earth and the Moon. I have gotten this far into the project but the orbit of Mercury is terrifyingly wrong. This is seen by typing "SunEarthMoon(2,50)" at the command prompt and viewing the bottom left plot. The logic behind the project is utilizing Newton's Second Law toggled with the command "ode45" to find the positions of the bodies during a given time interval. I've been staring at this for far too long. Can anyone help to fix the orbit of Mercury?
function [] = SunEarthMoon(years,framerate)
%% Clean Up
close all
clc
%% Initializaion
x_earth = 147300000000; % [m]
x_mercury = 57.91e9; % [m]
v_earth = 30257; % [m/s]
v_mercury = 47362; % [m/s]
r_sat = 384748000; % earth surface [m]
r_earth = 6367000; % earth radius [m]
v_sat = 1023; % relative velocity from earth [m/s]
a = 5.145; % Angle to vertical (y) axis
b = 90; % Angle to horizontal (x) axis in xz plane
x_earth_o = [x_earth; 0; 0];
x_sun_o = [0; 0; 0];
x_mercury_o = [x_mercury; 0; 0];
x_sat_o = [x_earth+r_sat+r_earth; 0; 0];
v_earth_o = [0; v_earth; 0];
v_sun_o = [0; 0; 0];
v_mercury_o = [0; v_mercury; 0];
v_sat_o = v_sat*[cos(pi/180*b)*sin(pi/180*a); cos(pi/180*a); sin(pi/180*b)*sin(pi/180*a)] + v_earth_o;
interval = years*[0 31536000];
%% Error Control
h = [0.01 36000];
tol = 100000;
Options.AbsTol = tol;
Options.MaxStep = h(2);
Options.InitialStep = h(1);
%% Analysis
ao = [x_earth_o; v_earth_o; x_sun_o; v_sun_o; x_sat_o; v_sat_o; x_mercury_o; v_mercury_o];
[t, x] = ode45(#earthfinal,interval,ao,Options);
for i = 1:length(t)
R1(i) = (x(i,13)-x(i,1));
R2(i) = (x(i,14)-x(i,2));
R3(i) = (x(i,15)-x(i,3));
R(i) = sqrt(R1(i)^2+R2(i)^2+R3(i)^2);
end
T_index_earth = find([1; x(:,4)].*[x(:,4);1]<=0);
T_index_moon = find([1; R2(:)].*[R2(:); 1]<=0);
for i = 4:length(T_index_earth)
T_earth_semi(i-3) = (t(T_index_earth(i)-1)-t(T_index_earth(i-2)-1))/24/60/60;
end
T_earth = mean(T_earth_semi);
for i = 4:length(T_index_moon)
T_moon_semi(i-3) = (t(T_index_moon(i)-1)-t(T_index_moon(i-2)-1))/24/60/60;
end
T_moon = mean(T_moon_semi);
D_earth = 0;
for i = 2:(T_index_earth(4)-1)
D_earth = D_earth + sqrt((x(i,1)-x(i-1,1))^2+(x(i,2)-x(i-1,2))^2+(x(i,3)-x(i-1,3))^2);
end
D_moon = 0;
for i = 2:(T_index_moon(4)-1)
D_moon = D_moon + sqrt((R1(i)-R1(i-1))^2+(R2(i)-R2(i-1))^2+(R3(i)-R3(i-1))^2);
end
%% Plots
q = framerate;
scrsz = get(0,'ScreenSize');
figure('position', [0.05*scrsz(3) 0.05*scrsz(4) 0.75*scrsz(3) 0.85*scrsz(4)])
set(gcf,'name','Sun, Earth, and Moon Orbits')
for i = 1:length(t)/q
subplot(2,2,1)
plot3(x(1:i*q,1),x(1:i*q,2),x(1:i*q,3),'g',x(1:i*q,7),x(1:i*q,8),x(1:i*q,9),'r',x(1:i*q,13),x(1:i*q,14),x(1:i*q,15),'b',x(1:i*q,19),x(1:i*q,20),x(1:i*q,21),'black')
axis(1.1*[min(x(:,1)) max(x(:,1)) min(x(:,2)) max(x(:,2)) 2*min(x(:,15)) 2*max(x(:,15))])
xlabel('Universal X Coordinate (m)')
ylabel('Universal Y Coordinate (m)')
zlabel('Universal Z Coordinate (m)')
title('Relative Orbits')
legend('Earth','Sun','Moon')
hold on
plot3(x(i*q,1),x(i*q,2),x(i*q,3),'g-o',x(i*q,7),x(i*q,8),x(i*q,9),'r-o',x(i*q,13),x(i*q,14),x(i*q,15),'b-o',x(i*q,19),x(i*q,20),x(i*q,21),'black-o')
hold off
subplot(2,2,2)
plot3(R1(1:i*q),R2(1:i*q),R3(1:i*q),'b',zeros(1,i*q),zeros(1,i*q),zeros(1,i*q),'g')
axis(1.5*[min(R1) max(R1) min(R2) max(R2) min(R3) max(R3)])
xlabel('Universal X Coordinate (m)')
ylabel('Universal Y Coordinate (m)')
zlabel('Universal Z Coordinate (m)')
title('Relative Moon Orbit About Earth')
hold on
plot3(R1(i*q),R2(i*q),R3(i*q),'b-o',0,0,0,'g-o')
text(0,1.45*max(R2),1.40*max(R3),sprintf('Orbital Period, T = %3.5g days',T_moon))
text(0,1.45*max(R2),1.15*max(R3),sprintf('Orbital Circumference, D = %3.5g gigameters',D_moon*1e-9))
hold off
subplot(2,2,3)
plot(x(1:i*q,1),x(1:i*q,2),'g',x(1:i*q,7),x(1:i*q,8),'r', x(1:i*q,19),x(1:i*q,20),'black')
axis(1.5*[min(x(:,1)) max(x(:,1)) min(x(:,2)) max(x(:,2))])
xlabel('Universal X Coordinate (m)')
ylabel('Universal Y Coordinate (m)')
title('Relative Earth Orbit About Sun')
hold on
plot(x(i*q,1),x(i*q,2),'g-o',x(i*q,7),x(i*q,8),'r-o',x(i*q,19),x(i*q,20),'black-o')
text(1.45*min(x(:,1)),1.40*max(x(:,2)),sprintf('Orbital Period, T = %3.5g days',T_earth))
text(1.45*min(x(:,1)),1.25*max(x(:,2)),sprintf('Orbital Circumference, D = %3.5g gigameters',D_earth*1e-9))
text(1.45*min(x(:,1)),1.40*min(x(:,2)),sprintf('Time, t = %3.3g days',round(t(q*i)/24/60/60)))
hold off
subplot(2,2,4)
plot(t(1:i*q)/(60*60*24),R(1:i*q)/1000,'b')
axis([t(1)/24/60/60 t(end)/24/60/60 0.999*min(R)/1000 1.001*max(R)/1000])
xlabel('Time,t (days)')
ylabel('Orbit Radius, R (km)')
title('Moon-Earth Distance')
hold on
plot(t(i*q)/(60*60*24),R(i*q)/1000,'b-o')
hold off
drawnow
end
end
%% Differential Equation Function
function [udot]= earthfinal(t,u)
m_earth = 5.9742e24; % [kg]
m_mercury = 3.285e23; % [kg]
m_sun = 1.98892e30; % [kg]
m_sat = 11110; % [kg]
G = 6.67300e-11; %[(m)^3(kg)^-1(s)^-2];
d_earth_sun = sqrt((u( 7,1)-u(1,1))^2+(u( 8,1)-u(2,1))^2+(u( 9,1)-u(3,1))^2);
d_earth_sat = sqrt((u(13,1)-u(1,1))^2+(u(14,1)-u(2,1))^2+(u(15,1)-u(3,1))^2);
d_sun_sat = sqrt((u(13,1)-u(7,1))^2+(u(14,1)-u(8,1))^2+(u(15,1)-u(9,1))^2);
d_mercury_sun = sqrt((u(7,1) -u(19,1))^2 + (u(8,1) - u(20,1))^2 + (u(9,1)-u(21,1))^2);
d_mercury_earth = sqrt((u(1,1) -u(19,1))^2 + (u(2,1) - u(20,1))^2 + (u(3,1)-u(21,1))^2);
d_mercury_sat = sqrt((u(13,1) -u(19,1))^2 + (u(14,1) - u(20,1))^2 + (u(15,1)-u(21,1))^2);
% Earth motion
udot( 1,1) = u(4,1);
udot( 2,1) = u(5,1);
udot( 3,1) = u(6,1);
udot( 4,1) = m_sun*G*(u(7,1)-u(1,1))/d_earth_sun^3 + m_sat*G*(u(13,1)-u(1,1))/d_earth_sat^3 + m_mercury*G*(u(19,1)-u(1,1))/d_mercury_earth^3;
udot( 5,1) = m_sun*G*(u(8,1)-u(2,1))/d_earth_sun^3 + m_sat*G*(u(14,1)-u(2,1))/d_earth_sat^3 + m_mercury*G*(u(20,1)-u(2,1))/d_mercury_earth^3;
udot( 6,1) = m_sun*G*(u(9,1)-u(3,1))/d_earth_sun^3 + m_sat*G*(u(15,1)-u(3,1))/d_earth_sat^3 + m_mercury*G*(u(21,1)-u(3,1))/d_mercury_earth^3;
% Sun Motion
udot( 7,1) = u(10,1);
udot( 8,1) = u(11,1);
udot( 9,1) = u(12,1);
udot(10,1) = m_earth*G*(u(1,1)-u(7,1))/d_earth_sun^3 + m_sat*G*(u(13,1)-u(7,1))/d_sun_sat^3 + m_mercury*G*(u(19,1)-u(7,1))/d_mercury_sun^3;
udot(11,1) = m_earth*G*(u(2,1)-u(8,1))/d_earth_sun^3 + m_sat*G*(u(14,1)-u(8,1))/d_sun_sat^3 + m_mercury*G*(u(20,1)-u(8,1))/d_mercury_sun^3;
udot(12,1) = m_earth*G*(u(3,1)-u(9,1))/d_earth_sun^3 + m_sat*G*(u(15,1)-u(9,1))/d_sun_sat^3 + m_mercury*G*(u(21,1)-u(9,1))/d_mercury_sun^3;
% Satellite Motion
udot(13,1) = u(16,1);
udot(14,1) = u(17,1);
udot(15,1) = u(18,1);
udot(16,1) = m_earth*G*(u(1,1)-u(13,1))/d_earth_sat^3 + m_sun*G*(u(7,1)-u(13,1))/d_sun_sat^3 + m_mercury*G*(u(19,1)-u(13,1))/d_mercury_sat^3;
udot(17,1) = m_earth*G*(u(2,1)-u(14,1))/d_earth_sat^3 + m_sun*G*(u(8,1)-u(14,1))/d_sun_sat^3 + m_mercury*G*(u(20,1)-u(14,1))/d_mercury_sat^3;
udot(18,1) = m_earth*G*(u(3,1)-u(15,1))/d_earth_sat^3 + m_sun*G*(u(9,1)-u(15,1))/d_sun_sat^3 + m_mercury*G*(u(21,1)-u(15,1))/d_mercury_sat^3;
% Mercury Motion
udot(19,1) = u(22,1);
udot(20,1) = u(23,1);
udot(21,1) = u(24,1);
udot(22,1) = m_sun*G*(u(7,1)-u(19,1))/d_mercury_sun^3 + m_earth*G*(u(1,1)-u(19,1))/d_mercury_earth^3 + m_sat*G*(u(13,1)-u(19,1))/d_mercury_sat^3;
udot(23,1) = m_sun*G*(u(8,1)-u(20,1))/d_mercury_sun^3 + m_earth*G*(u(2,1)-u(20,1))/d_mercury_earth^3 + m_sat*G*(u(14,1)-u(20,1))/d_mercury_sat^3;
udot(24,1) = m_sun*G*(u(9,1)-u(21,1))/d_mercury_sun^3 + m_earth*G*(u(3,1)-u(21,1))/d_mercury_earth^3 + m_sat*G*(u(15,1)-u(21,1))/d_mercury_sat^3;
end