Finding the coordinates of a point given the gradient, distance and one set of coordinates

Hi im trying to find the coordinates of a point given the gradient (m), distance (s) and one set of coordinates (x1,y1). I have a simultaneous equation that needs solving but cannot seem to find the numerical answers i have calulated by hand in MATLAB.

m = y2-y1/x2-x1 and s = sqrt((x2-x1)^2 + (y2-y1)^2)

given that:

m = 0.4

x1 = 10

y1 = 4

s = 15

and solving to find x2 and y2.

Answer 1

[Torsten]

m = 0.4;
x1 = 10;
y1 = 4;
s = 15;
z = [x1 y1] + s/sqrt(1+m^2)*[1 m];
x2 = z(1)

x2 = 23.9272

y2 = z(2)

y2 = 9.5709

[Adil Saeed]

thank you

[Adil Saeed]

one thing, could you explain how you did that?

[Torsten]

The direction from [x1 y1] to [x2 y2] is given by +/-[1 m] because m is the slope of the line connecting the two points. If you normalize the direction to be a vector with norm 1, you can directly multiply it by s to get the connecting line between [x1 y1] and [x2 y2]. And the vector +/-[1 m] normalized to be of norm 1 is +/-[1 m]/sqrt(m^2+1).

A second point that might have to be considered is thus

z = [x1 y1] - s/sqrt(1+m^2)*[1 m]:

m = 0.4;
x1 = 10;
y1 = 4;
s = 15;
z1 = [x1 y1] + s/sqrt(1+m^2)*[1 m];
z2 = [x1 y1] - s/sqrt(1+m^2)*[1 m];
m1 = (z1(2)-y1)/(z1(1)-x1)

m1 = 0.4000

s1 = sqrt((z1(1)-x1)^2+(z1(2)-y1)^2)

s1 = 15.0000

m2 = (z2(2)-y1)/(z2(1)-x1)

m2 = 0.4000

s1 = sqrt((z2(1)-x1)^2+(z2(2)-y1)^2)

s1 = 15.0000

[Adil Saeed]

Following on from this, I have furthered my code to encorpate the coordinates as a matrix array to cater for more coordinates working together as one, rather than doing it all individually (hope this made sense) i.e.

x0 = 0;
y0 = 0;
x1 = [1;2;3;4;5];
y1 = [6;7;8;9;10];

so the coordinate pairs are: (1,6) (2,7) (3,8) (4,9) (5,10)

m = [y1(1)/x1(1); y1(2)/x1(2); y1(3)/x1(3); y1(4)/x1(4); y1(5)/x1(5)];

I know this maths is not technically correct as you cannot divide matrices however i found this way useful in order to find the gradients as a 5x1.

d = sqrt((x1-x0).^2 + (y1-y0).^2);

This was used to find the distance between the points.

Now the confusions sets in when finding the new coordinates as a 5x2 matrix if this is possible to be broken into 2 5x1 matrices for x2 and y2. Hope this all made sense.

Thanks

[Torsten]

What is (x0,y0) ? Are m and s the same for all pairs (x1(i),y1(i)) ?

[Adil Saeed]

so (x0,y0) is the origin and the source for this code. m and s should change for each pair depending on the coordinate pair. so if we analyse the first pair of coordinates (1,6) this should give m to be 6 and the distance s as 6.0828. for the coordinates (2,7) this should give m to be 3.5 and s to be 7.28 as so on...

i represented this as a matrix and m and s (or d) should equal:

m = [6;3.5;2.666;2.25;2];
d = [6.0828;7.28;8.5440;9.8489;11.1803];

now based on this i want the new coordinates for each of the pairs all represented in a matrix (5x2) of x2 and y2.

[Torsten]

So you want the point (x2, y2) which is d = sqrt((x0-x1)^2+(y0-y1)^2) apart from (x1, y1) and lies on the line connecting (x0,y0) and (x1,y1) ?

x0 = 0;
y0 = 0;
x1 = [1;2;3;4;5];
y1 = [6;7;8;9;10];
m = (y1-y0)./(x1-x0)

m = 5×1

6.0000 3.5000 2.6667 2.2500 2.0000

d = sqrt((x1-x0).^2+(y1-y0).^2)

d = 5×1

6.0828 7.2801 8.5440 9.8489 11.1803

z = [-x0+2*x1,-y0+2*y1] 

z = 5×2

2 12 4 14 6 16 8 18 10 20

[Adil Saeed]

Apologies for the confusion on my part. So the distance between (x0,y0) to (x1,y1) is determined by d = sqrt((x1-x0).^2+(y1-y0).^2). However the distance between (x1,y1) to (x2,y2) is already pre determined by s = vt (conditions i have set in place), where s = 10 m. This is the distance between (x1,y1) and (x2,y2). Again its like the orginal message where i have the simultaneous equation of:

m = y2-y1/x2-x1 and s = sqrt((x2-x1)^2 + (y2-y1)^2) solving for (x2,y2).

But this time need to consider the equation as a matrix rather than one coordinate.

I hope this makes it more clear.

[Torsten]

So you don't need (x0,y0) and you are given (x1,y1) in matrix form and s and m in vector form and you want to determine the two points (x21,y21) and (x22,y22) that have distance s from (x1,y1) with slope m ?

[Adil Saeed]

pretty much yes, but want all 5 points within a matrix:

(x2(1),y2(1)), (x2(2),y2(2)), (x2(3),y2(3)), (x2(4),y2(4)), (x2(5),y2(5))

and to be a 5x2 matrix, based on their own gradient m in a 5x1 matrix. the gradient is determined by

m = y1(i)-y0/x1(i)-x0

from (x0,y0) and (x1(i),y1(i)), as each new coordinate will be on the same coresponding gradient. The distance between (for the new coordinate (x2(i),y2(i)) will be of a distance 10 m away.

[Adil Saeed]

i have just calculated the new coordinates individually if they were not in a matrix, therefore the matrix should be:

[2.6,15.86; 4.74,16.61; 6.51,17.36; 8.06,18.13; 9.47, 18.94]

I have also attached my full code

x0 = 0;
y0 = 0;
Sr = [x0 y0];            %Source coordinates
SL = 220;                %Source Level dB re 1uPa @ 1m (made up  need to find example)
a = 0.25;                %Attenuation (made up need to find example)
IL = 180;                %Impact Level (made up need to find example)


%z1 = [randi([-10 10],1,5); randi([-10 10],1,5)];
%z1 = z1'
%x1 = z1(1:5,1);
%y1 = z1(1:5,2);

x1 = [1;2;3;4;5];
y1 = [6;7;8;9;10];

d = sqrt((x1-x0).^2+(y1-y0).^2);          %Pythagoras Theorem for distance between the 2 (Source and Reciever) points
m = (y1-y0)./(x1-x0);                     %Gradient between the points

r1 = d;

v = 2;      %harbor porpoises swim between 0.5 - 4.2 m/s
t =5;      %seconds
s = v*t;    %metres

z2 = [x1 y1] + s/sqrt(1+m.^2).*[1 m];     %New coordinates for new positions 5s later at a speed of 1.5m/s
x2 = z2(1);
y2 = z2(2);

r2 = d + s;                              %distance from source to new receive position

[Torsten]

v = 2;      %harbor porpoises swim between 0.5 - 4.2 m/s
t = 5;      %seconds
s = v*t;    %metres
x0 = 0;
y0 = 0;
x1 = [1;2;3;4;5];
y1 = [6;7;8;9;10];
d = sqrt((x1-x0).^2+(y1-y0).^2);          %Pythagoras Theorem for distance between the 2 (Source and Reciever) points
m = (y1-y0)./(x1-x0);                     %Gradient between the points
z1 = [x1,y1] + diag(s./sqrt(1+m.^2))*[ones(size(x1)),m]

z1 = 5×2

2.6440 15.8639 4.7472 16.6152 6.5112 17.3633 8.0614 18.1381 9.4721 18.9443

z2 = [x1,y1] - diag(s./sqrt(1+m.^2))*[ones(size(x1)),m]

z2 = 5×2

-0.6440 -3.8639 -0.7472 -2.6152 -0.5112 -1.3633 -0.0614 -0.1381 0.5279 1.0557

[Adil Saeed]

thank you so much, much appreciated