Angulation Utility (3D)
This page contains a 3D positioning routine (by angulation).
Refer to the following diagram for the explanation that follows:
Assume this web page is a physical piece of paper that lies in the x-y plane of a 3D space. Two sensors are located a constant, fixed, distance, K1, apart. They lie along the x axis, and the left sensor (Sensor 1) is defined to be the origin. The z axis is perpendicular to the x and y axes, with positive z pointing straight out of the page toward you. Both sensors are able to detect objects located in the positive-y hemisphere only; they cannot detect objects that are located in the negative y hemisphere.
Parameters are defined as follows:
1) An object's position, P, is defined by its (x, y, z ) coordinates of a coordinate system whose origin is at the left sensor (see the diagram above);
2) Each sensor detects two angles.
(i) the angle between the x axis and the object's location projected into the x-y plane (the left sensor detects angle A1; the right sensor detects angle, A2), and
(ii) the angle between the x-y plane, measured from the sensor, and P. Sensor 1 detects angle B1; Sensor 2 detects angle B2.
In case the above explanation is not clear, here is a more detailed description of how the B angles are defined.
Consider the point, P, to be projected into the x-y plane. Call this point Pxy.
Now consider two vectors originating at the origin (Sensor 1), one vector is from Sensor 1 to the original point, P, and the second vector is from Sensor 1 to the projected point, Pxy. Angle B1 is the angle between these two vectors (the angle is measured positive in the positive z direction).
Angle B2 is formed in a similar way. Consider two vectors originating at Sensor 2, one vector is from Sensor 2 to the original point, P, and the second vector is from Sensor 2 to the projected point, Pxy. Angle B2 is the angle between these two vectors.
If this description still does not clarify the definitions of the angles, please see the worked example.
The sensors cannot detect the distance to the object--only the angles mentioned above.
ANGLES A1 AND A2 CANNOT BE LESS THAN OR EQUAL TO ZERO, OR GREATER THAN OR EQUAL TO 180 DEGREES.
ANGLES B1 AND B2 CANNOT BE LESS THAN NEGATIVE 90 DEGREES, OR GREATER THAN 90 DEGREES.
The utility posted on this page accepts five inputs, the values of K1, A1, A2, B1, and B2, and outputs the x, y, z coordinates of the object.
Keep in mind that units will be consistent. For example, if the value you enter for K1 represents meters, the values output for x, y, and z will also represent meters.
IMPORTANT: The values you enter for A1, A2, B1, and B2 must be degrees, NOT radians.