Rotation angles

The following image depicts a typical example of how the Radial/Tangential axis pair might be configured. During configuration the user specifies a rotation angle (Ø), which is the angle from the Northing axis to the Radial axis, measured in a clockwise direction.

In this case the Rotation Angle is configured so that the bearing of the positive Radial axis corresponds with the downstream direction of the dam structure.

The diagrams below show the mathematical relationship between the displacement components in the Northing and Easting directions and the displacement components in the Radial and Tangential components.

Let us assume a hypothetical example where the Rotation Angle (Ø) between the Northing axis and the Radial axis is 120° and where a prism is displaced by 3 mm in the Northing direction and 4 mm in the Easting direction.

Given the above, the following will hold:

Value	Symbol	Calculation
Axis Rotation	Ø	User Defined = 120°
Planar Displacement Magnitude	d2D	d2D = √(dN² + dE²) = √(3² + 4²) = 5 mm
Azimuth of the Displacement	θ	θ = ArcTan(dE/dN) = ArcTan(4/3) = 53.13°
Northing Displacement	dN	dN = d2D.Cos(θ) = 5.Cos(53.13°) = 3 mm
Easting Displacement	dE	dE = d2D.Sin(θ) = 5.Sin(53.13°) = 4 mm
Radial Displacement	dR	dR = d2D.Cos(Ø - θ) = 5.Cos(120°- 53.13°) = 1.964 mm
Tangential Displacement	dT	dT = d2D.Cos(Ø - θ + 90°) dT = d2D.Sin(θ - Ø) dT = 5.Sin(53.13°- 120°) dT = - 4.598 mm

Where applicable, the Add Sensor page includes an Axis Rotation tab. The input fields on this tab are used to configure the bearing of the Radial axis (and by implication the bearing of the Tangential axis) as described in the example above.

There are four different modes that can be used to define the Rotation Angle.

No Rotation

The No Rotation option applies a rotation angle of zero degrees. In this case the direction of the Radial axis coincides with the Northing axis and the direction of the Tangential axis coincides with the direction of the Easting axis.

Manual Input

The Manual Input mode allows for manual entry of the rotation angle.

The Angular Unit Preference as defined in the Project Unit Preferences applies. If the Angular Unit Preference is set to degrees, then toggle switch denoted with D°M’S” appears above this Rotation Angle field. Use this toggle switch to switch between Decimal Degrees and Degrees, Minutes and Seconds.

The arrow icon next to the Rotation Angle field depicts the direction of the Radial axis. The reference orientation for this depiction is such that the Northing axis points up and the Easting axis points right. This holds regardless of the true orientation of the Coordinate System.

There are two utility buttons for rotating the current rotation angle by 90 degrees in the CCW and CW directions:

Bearing to Point

The Bearing to Point entry mode can be used to calculate the Rotation Angle so that the Radial axis will coincide with a line drawn from the Coordinate of the sensor that is being configured to the specified Point.

The To Point drop-down list contains all total stations, prisms and GNSS receivers configured in T4D Server.

Bearing between Points

The Bearing between Points entry mode can be used to calculate a Rotation Angle so that the bearing of the Radial axis will correspond with the bearing of a line drawn between to the specified Points.

The From Point and To Point drop-down lists contain all the total stations, prisms and GNSS receivers configured in T4D Server.