Cosine Error
Microwave and Laser Radars
Geometry
Cosine Error
Microwave and Laser Radars
Geometry
Figure 4.1-1 -- Cosine Effect Setup
The cosine effect angle (alpha) is the angle between the radar and the target direction of travel. Target range from radar and radar distance off the road (really the distance between radar and the point the target would be closest to radar if target continues in same direction) determine the cosine effect angle. Note that the road direction and antenna direction (direction antenna pointed) are completely irrelevant, only the angle (alpha) matters (radar stationary).
Antenna direction (alignment to patrol car direction) is important in moving mode radar. A mis-aligned antenna measures target speed high if the misalignment is great enough, the target is approaching the moving radar, and the target is traveling slower than the radar (chapter 4.3 -- Cosine Effect on Moving Radar).
As long as the angle (alpha) remains relatively small, the error (cosine of alpha) is tolerable. The larger the angle, the larger the error and the lower the displayed (relative) speed. On a straight section of road, radar distance from the road and the range of the target determine the angle. The greater the distance the radar is off the road and/or the closer the target, the larger the angle (and error). When the target is even with the radar (alpha equals 90 degrees) the target speed, with respect (relative) to the radar, is zero.
The Cosine Effect applies to both microwave radars and laser radars (ladars) as well as to targets traveling in any direction (on-coming or going traffic at any angle). Most traffic radars do not account for the Cosine Effect; across the road microwave radars (such as photo radars) are an exception. These systems point the beam at a known fixed angle across the road and compensate the measured target speed for the Cosine Effect.
Overpass
Figure 4.1-2 --
Cosine Effect from an Overpass
The radar distance from (off) the road is the line-of-sight distance from the radar to the road (target path). If the radar is on an overpass (shooting cars running under the overpass) or hill for example, the radar distance from the road is the distance from the radar position to the road (target path) as in the side figure. In the figure the traffic is traveling directly away or into the page.
In the side figure, x represents the horizontal distance and y represents the vertical distance from the road to the radar. The line-of-sight distance is d. If either the horizontal or vertical component is zero, the equations reduce to that shown in figure 4.1-1 -- where d = y (if x = 0), or d = x (if y = 0). When applying the equations, all distances must use the same unit dimensions (feet, meters, etc.).
When calculating the angle (alpha) using the inverse tangent function (arctan), the unit dimension of the calculated angle is radians (rad), not degrees. Pi (3.14159...) radians equals 180 degrees; one radian approximately equals 57.3 degrees. Degrees = radians x (180/pi).
Hills or Curves
On hills or curves target direction (with respect to radar) is changing, this causes the Cosine Effect angle (alpha) to change. A changing Cosine Effect angle results in measured target speed changing, the faster the angle changes the faster measured target speed changes (acceleration or deceleration component). If measured speed changes too fast the radar misses (does not display) target speed.
Note the above figures illustrates targets on a hill (side view), or a curved road (top view). Alpha is the Cosine Effect angle, d is radar (closest) distance from target path. The steeper the hill or the tighter the curve the greater the angle alpha, and the greater the measured speed error and the greater the acceleration component. Moving radar introduces another component that generally increases the target acceleration component for approaching or receding targets, and decreases for same-lane targets.
Also see;
chapter 4.5 -- Targets on a Curve
chapter 5.4 -- Operational problems / Target Acceleration
Cosine Error
The below figure is a graphical representation of the Cosine Effect for measured speed, as a percentage of true speed versus angle (alpha) between radar and target -- the larger the angle the larger the error and the lower the measured target speed. For example at angles of only a few degrees the measured speed is 99 to 100 percent of actual; at an angle of 60 degrees the measured speed is half (50 percent) the actual target speed.
Just because the Cosine Effect works in favor of the motorist (moving mode radar has an exception) does not mean one cannot become a victim of the Cosine Effect. See chapter 7.2 -- The Courtroom--Cosine Effect Defense.
