Transforming Cartesian Coordinates Using Perpendicular Lines

[lolito]

Hi, I think this problem isn't too complicated, but I am not good at geometry. Can you help me?

I want to transform the origin of a cartesian system. I have a plane (Ax +By +Zc +D = 0) and two perpendicular lines, contained in the plane. I have the equations (parametric) of the two perpendicular lines too. I want to consider these two perpendicular lines as the new coordinates system (the intersection of two lines would be the new coordinates origin: 0.0.0).

I have several points in the planes (defined by using the XYZ coordinates). I need to change the coordinates of these points to the new reference system (X'Y'Z'). In this way I could know in which quadrant of the cartesian system defined by the above lines the points are (this is my final target).

Could anyone help me? Many many thanks.

 

[jeroen]

If you already know the points that you are interested in then it is a matter of subtracting the position of the new origin within the old system from these points:
point_new = point_old - origin_old

The equation for the plane would be modified like:
A*(point_x_new+origin_x_old) + B*(point_y_new+origin_y_old) + C*(point_z_new+origin_z_old) + D = 0
Or:
A*point_x_new + B*point_y_new + C*point_z_new + D + A*origin_x_old + B*origin_y_old + C*origin_z_old = 0
Or:
E = A*origin_x_old + B*origin_y_old + C*origin_z_old
A*point_x_new + B*point_y_new + C*point_z_new + D + E = 0

 

[lolito]

Thanks, jeroen. Your suggestion works. Now I can know the quadrant points belong to

 

[lolito]

Hi,

I still have another problem: I want the two lines (perpendicular and contained in the same plane) to be a new reference system. As above jeroen has said, the changes of coordinates is carried out, and considering the intersection of the two lines as the new origin, this point will be 0, 0, 0. But the two lines must be the new X,Y reference system (we have the XY plane) because I am seeking to locate which quadrant of the new XY system the points (with new X'Y'Z') belong to.

For example, a point with new coordinates (-X.XX, +Y.YYY, 0) must be located in the first quadrant. I need to know in which quadrant (of the system composed by the two perpendicular lines) the points are.

I need help. Can anyone help me? THANKS

 

[jeroen]

So you are not just translating the origin but also rotating the axes?

What you could do is take a point in the old system and then determine how far it is from both lines.
The distance to the x line is the y coordinate, the distance to the y line is the x coordinate.

 

[lolito]

Ok, I have a series of points in a plane, and two perpendicular lines contained in this plane (I know the intersection point, which I want it to be the 0,0,0). I need to determine the quadrant (regarding the reference system composed by the two perpendicular lines) the points belong to.

The easiest way would be if the coordinates of the points would be given in a new system composed by the two lines whose intersection was 0, 0, 0. Now if a point of the plane is -1,+1,0, I would know exactly its quadrant. The problem is that now I have original XYZ coordinates and I need the new coordinates considering the intersection point as the origin and the two lines as the X and Y axis.

How can I do it, please?

 

[lolito]

Ok jeroen, now I understand your previous answer: to compute the distance of the point to both lines (x axis and y axis), and I will have the coordinates, but how can I know the negative or plus sign, please?? Sorry, but I have not idea of geometry.

 

[lolito]

Thanks jeroen, now I have solved the problem and your help has been very useful!