Cleo Reihl, winter semester 2018/19

Triangulation is done in several fields of science. In measurement engineering it means the distance measurement via angular measurements in triangles. The distance between the laser and the test object is measured via the angle under which the light that is scattered on the object surface hits a detector next to the laser. If the distances are different, and therefore also the angles are different, the detector is hit at a different position.

Physical Basics [1]

In geometrical optics light is seen as a form of energy that spreads in space along light rays. These light rays are refracted when they meet the boundary between two transparent materials, e.g. air and glass. This is used by lenses which can either scatter or bundle incoming light rays.

Figure 1 shows a thin convex lens (thickness is idealized to zero) that maps the point P in front of the lens to the point P’ behind the lens. The construction of the position of the mapped point P’ is done with the parallel ray (1), the central ray (2) and the focal ray (3). The parallel ray is parallel to the optical axis of the lens und is refracted by the lens in a way that it passes the focal point behind the lens. The central ray is not refracted by the lens and passes directly. The focal ray which passes the focal point in front of the lens is refracted in a way that behind the lens it is parallel to the optical axis. In the point where all three rays intersect, the mapped point P’ is positioned.

When the lens is symmetrical, both focal lengths in front of and behind the lens are equal.

Figure 1: Point and mapped point in the case of a thin convex lens

Principle of Measurement [2]

The measurement consists of four main components: The laser, the test object, the lens and the detector. The laser emits a beam which hits the test object and produces a light point on the test object. This point is mapped on the detector by a convex lens. This is only possible when the light beam is scattered on the object surface, because otherwise no light might hit the lens. Figure 2 shows the principle of the measurement with the scattered light being focused by the lens. In figure 3 the same concept is shown, but this time with a single idealized light ray that passes the centre of the lens.

Figure 2: Principle of measurementFigure 3: Simplified principle of measurement

Depending on the distance between the laser and the test object, the angle, under which the scattered light hits the lens, is different and therefore the position of the mapped point on the detector is different. This can be seen in figure 4, where the two light rays resulting from interactions with two test objects at different distances hit the detector at different points. When a detector is hit by a light ray, a signal is recorded depending on the location where the light hits the detector.

As the image points of points with different distances to the lens won’t be mapped at the same distance to the lens either, the detector must be inclined relative to the optical axis of the lens to make sure the mapped points lie directly on the detector. Figure 5 shows the principle for two randomly chosen points.

Figure 4: Principle of measurement for two different distancesFigure 5: Image points lie on a line inclined relative to the optical axis (here the inclination to optical axis is exaggerated)

The relation between the distance z between the laser and the test object and the distance z’ between the laser and the mapped point on the detector is nonlinear:

z'(z)\;=\; m_{L} \cdot f \sqrt{(1+(m_{L} - \frac {u_{0}}{f})²} \cdot \frac {\frac {z} {\sqrt{1+m_{L}²}}} {(\frac {z} {\sqrt{1+ m_{L}^2}} + \frac {u_{0}} {m_{L}} - f)(m_{L} - \frac{u_{0}} {f})} [2]

With f being the focal length of the lens, u_{0} being the distance between the centre of the lens and the laser and m_{L} being the gradient of the scattered light ray referring to the optical axis of the lens (for the denomination see figure 3 and 4). In figure 3 the ray is identical to the axis; the gradient is therefore zero.

Implementation of measurement principle [2] [3] [4] [5]

Typical wavelengths for the laser are in the red and near-infrared spectral range, e.g. 660, 670, 685 and 780 nm. The optical power varies between 1 and 100 mW. Possible detectors are lateral effect diodes, CCD (Charge Coupled Device) or CMOS (Complementary Metal Oxide Semiconductor) line arrays. For more information see[2].

The measurement can be performed by a single laser beam, but also by a laser line to measure contours or by several laser lines to examine the shape of a test object.[2]

Influences on the measurement quality:[2] [3]

  • Laser beam propagation:
    Type of laser source and the beam shaping optics, e.g. diameter of laser beam
    speckle (interference of the scattered laser light causes an intensity pattern on the detector)
  • Properties of the object surface:
    e.g. the scattering properties of the surface. An ideal specular object surface would reflect the incoming beam in only one direction, but no diffuse light scattering would appear. This on the other hand is needed to detect the laser beam on the detector. Also varying reflectivity (e.g. by surface scratches) leads to a varying intensity distribution on the detector and therefore to varying results.
  • Imaging aberrations:
    Errors that are caused by the lenses (e.g. coma, astigmatism, distortion, spherical aberrations)
  • Detector and signal evaluation:
    How the position of the mapped light spot on the detector is evaluated, e.g. by integration of the intensity distribution in the case of a Lateral Effect Diode
    Detector noise
  • Atmospheric conditions:
    Variations in the refractive index of the surrounding atmosphere, especially near hot test objects.
  • Mechanical influences like vibrations

The sensitivity of a laser-based triangulation measurement varies depending on the measured distance: The sensitivity is lower when the test object is further away. When the test object is further away, a certain change in distance perpendicular to the ray results in a relatively small change in angle for the light ray that hits the detector. When on the other hand the test object is closer to the laser, the same change in distance results in a larger change in angle for the light ray.

Typical measured distances range from 1 – 400 mm [2] [4], the measurement error can be, depending on the used system configuration, around ±0,2mm[4].

Advantages of laser-based triangulation measurements are: [3] [5]

  • Fast measurement rates
  • Small size of device
  • Automation
  • Non-contact measurement
  • High precision
  • Good scalability for varying specimen dimensions

Applications [2] [4] [5] [6]

Laser triangulation is often used in process control and quality assurance by measuring the geometric quantities of moving objects.[2]

The examination of welded joints, where laser triangulation can be used to measure object dimensions or find defects, is an example for this application. Here, the laser is moved along the weld seam and projects a laser line onto the surface of the welded test object. At each time step the laser stripe that hits the detector is analysed by an image processing module that extracts the geometric information on the weld seam. The result is a profile of the welded joint where the dimensions as well as defects like mismatch between the two welded components can be examined. Moreover, the weld surface can be reconstructed by merging all generated profiles and thus generating a 3D image.[4]

Another example for the operation of laser triangulation is the measurement of pavement roughness of roads after the completion of the construction work. The roughness is measured via the variation of the measured distance.[5]

Typical functions of systems using laser triangulation are: [6]

  • Noncontact determination of the presence and (or) absence of an object
  • Determination of the distance to the surface of an object
  • Measurement and inspection of the profile of the surface of articles with complex configuration
  • Measurement of geometric parameters (thickness, length, width, non-rectilinearity, inner and outer diameters, planarity, rippling)
  • Determination of the position of an object
  • Recognition of manufactured objects
  • Measurement and inspection of the levels of liquids and friable bodies
  • Classification of objects
  • Measurement and inspection of positioning precision


  1. Velzel, Chris: A Course in Lens Design. Springer Series in Optical Sciences, Volume 183. Springer, Dordrecht (2014)
  2. Donges, Axel. Noll, Reinhard: Laser Measurement Technology. Fundamental and Applications. Springer Series in Optical Sciences, Volume 188. Springer, Heidelberg (2015)
  3. Mueller, Thomas. Poesch, Andreas. Reithmeier, Eduard: Measurement Uncertainty of Microscopic Laser Triangulation on Technical Surfaces. in Microscopy and Microanalysis (2015) 21:6, p. 1443-1454.
  4. Nguyen, Huu-Cuong. Lee, Byung-Ryong: Laser-vision-based quality inspection system for small-bead laser welding. in International Journal of Precision Engineering and Manufacturing (2014) 15:3, p. 415-423.
  5. Chen, Wenxue. Ni, Zhibin. Hu, Xinhan. Lu, Xiaofeng: Research on pavement roughness based on the laser triangulation. in Photonic Sensors (2016) 6:2, p. 177-180.
  6. Klimanov, M.: Triangulating laser system for measurements and inspection of turbine blades. in Measurement Techniques (2009) 52:7, p. 725-731.