Triangulation

This article is about measurement by the use of triangles. For other uses, seeTriangulation (disambiguation).

Not to be confused withTrilaterationorTriangularization.

Intrigonometryandgeometry,triangulationis the process of determining the location of a point by formingtrianglesto the point from known points.

Estimating the height of a mountain using triangulation

Atriangulation stationsigned by iron rod

Applications

Finding the position of a distant object B with the angles observed from points A and C and the baseline b between them

In surveying

Main article:Triangulation (surveying)

Specifically insurveying,triangulation involves onlyanglemeasurements at known points, rather than measuring distances to the point directly as intrilateration;the use of both angles and distance measurements is referred to astriangulateration.

In computer vision

Main article:Triangulation (computer vision)

Computer stereo visionandoptical 3D measuringsystems use this principle to determine the spatial dimensions and the geometry of an item.Basically, the configuration consists of two sensors observing the item. One of the sensors is typically a digital camera device, and the other one can also be a camera or a light projector. The projection centers of the sensors and the considered point on the object's surface define a (spatial) triangle. Within this triangle, the distance between the sensors is the baseband must be known. By determining the angles between the projection rays of the sensors and the basis, the intersection point, and thus the 3D coordinate, is calculated from the triangular relations.

History

Measuring the height of a building with aninclinometer

Triangulation today is used for many purposes, includingsurveying,navigation,metrology,astrometry,binocular vision,model rocketryand, in the military, the gun direction, the trajectory and distribution of fire power ofweapons.

The use of triangles to estimate distances dates to antiquity. In the 6th century BC, about 250 years prior to the establishment of thePtolemaic dynasty,the Greek philosopherThalesis recorded as usingsimilar trianglesto estimate the height of thepyramidsofancient Egypt.He measured the length of the pyramids' shadows and that of his own at the same moment, and compared the ratios to his height (intercept theorem).Thales also estimated the distances to ships at sea as seen from a clifftop by measuring the horizontal distance traversed by the line-of-sight for a known fall, and scaling up to the height of the whole cliff.Such techniques would have been familiar to the ancient Egyptians. Problem 57 of theRhind papyrus,a thousand years earlier, defines theseqtorsekedas the ratio of the run to the rise of aslope,i.e.the reciprocal of gradients as measured today. The slopes and angles were measured using a sighting rod that the Greeks called adioptra,the forerunner of the Arabicalidade.A detailed contemporary collection of constructions for the determination of lengths from a distance using this instrument is known, theDioptraofHero of Alexandria(c. 10–70 AD), which survived in Arabic translation; but the knowledge became lost in Europe until in 1615Snellius,after the work ofEratosthenes,reworked the technique for an attempt to measure the circumference of the earth. In China,Pei Xiu(224–271) identified "measuring right angles and acute angles" as the fifth of his six principles for accurate map-making, necessary to accurately establish distances,whileLiu Hui(c. 263) gives a version of the calculation above, for measuring perpendicular distances to inaccessible places.

See also

• Direction finding
• GSM localization
• Multilateration,where a point is calculated using the time-difference-of-arrival between other known points
• Parallax
• Resection (orientation)
• Stereopsis
• Tessellation,covering a polygon with triangles
• Trig point
• Wireless triangulation