从点和线获取矩形坐标

Question

I have the coordinates of 3 points [x1,y1] , [x2,y2] and [x3,y3] as shown below.

They define the line which is one side of a rectangle, and a point which lies on the parallel/opposite side of the rectangle. I want to get the coordinates of the other two corners.

How can I calculate points [xa, ya] and [xb, yb] as shown?

clc;
clear;

I = imread('peppers.png'); 
imshow(I);
h = imline;
lineEndPoints = wait(h);

x1 = round(lineEndPoints(1,1),2);
y1 = round(lineEndPoints(1,2),2);
x2 = round(lineEndPoints(2,1),2);
y2 = round(lineEndPoints(2,2),2);
hold on 

[x3, y3] = ginput(1);
plot(x3, y3,'b*');

slope = (y2 - y1)/ (x2 - x1);
slopePerp = -1/slope;

Answer 1

You have the slope between [x1, y1] and [x2, y2] ( slope ) and the slope for the perpendicular line to [x3, y3] ( slopPerp ).

So you have the y-intercept for the line [x1, y1] to [x2, y2] as

% From y=mx+c -> c=y-mx
c = y1 - slope*x1;

You can also get the y-intercept of the perpendicular line passing through [x3, y3]

cPerp = y3 - slopePerp*x3;

Then the point where your two black lines meet, let's call it [x4,y4] is

% Simultaneous equations  
% y = slope*x + c
% y = slopePerp*x + cPerp
% slope*x + c = slopePerp*x + cPerp
% x*(slope - slopePerp) = cPerp - c
x4 = (cPerp - c) / (slope - slopePerp);
y4 = slope*x4 + c;

Now all we need is the x and y differences

xdiff = x3 - x4; % So x4 + xdiff = x3
ydiff = y3 - y4; % So y4 + xdiff = y3

And to add these to our 1 and 2 points

xa = x1 + xdiff;
ya = y1 + ydiff;
xb = x2 + xdiff;
yb = y2 + ydiff;

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Note, with all these repeated actions it would probably be neater to store your x and y values in arrays rather than as separate variables.

Also, there is no reason to be using round , it will only make the result less accurate. If you are rounding because you want to display the values, use sprintf or round as you display, not before calculations.

Answer 2

Vector approach uses projection of point P3 onto line P1P2 and works for any rotation of rectangle (note that slope does not exist for axis-aligned rectangle)

  P4 = P1 + (P2 - P1) * DotProduct(P3 - P1, P2 - P1) / DotProduct(P2 - P1, P2 - P1)
  Pa = P1 + (P3 - P4)
  Pb = P2 + (P3 - P4)