In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.
In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.
Specifically in surveying, triangulation involves only angle measurements at known points, rather than measuring distances to the point directly as in trilateration; the use of both angles and distance measurements is referred to as triangulateration.
Computer stereo vision and optical 3D measuring systems use this principle to determine the spatial dimensions and the geometry of an item. [2] 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 base b and 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.
Triangulation today is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and, in the military, the gun direction, the trajectory and distribution of fire power of weapons.
The use of triangles to estimate distances dates to antiquity. In the 6th century BC, about 250 years prior to the establishment of the Ptolemaic dynasty, the Greek philosopher Thales is recorded as using similar triangles to estimate the height of the pyramids of ancient 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). [3] 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. [4] Such techniques would have been familiar to the ancient Egyptians. Problem 57 of the Rhind papyrus, a thousand years earlier, defines the seqt or seked as the ratio of the run to the rise of a slope, i.e. the reciprocal of gradients as measured today. The slopes and angles were measured using a sighting rod that the Greeks called a dioptra , the forerunner of the Arabic alidade. A detailed contemporary collection of constructions for the determination of lengths from a distance using this instrument is known, the Dioptra of Hero of Alexandria (c. 10–70 AD), which survived in Arabic translation; but the knowledge became lost in Europe until in 1615 Snellius, after the work of Eratosthenes, 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, [5] while Liu Hui (c. 263) gives a version of the calculation above, for measuring perpendicular distances to inaccessible places. [6] [7]
Geodesy or geodetics is the science of measuring and representing the geometry, gravity, and spatial orientation of the Earth in temporally varying 3D. It is called planetary geodesy when studying other astronomical bodies, such as planets or circumplanetary systems. Geodesy is an earth science and many consider the study of Earth's shape and gravity to be central to that science. It is also a discipline of applied mathematics.
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight and is measured by the angle or half-angle of inclination between those two lines. Due to foreshortening, nearby objects show a larger parallax than farther objects, so parallax can be used to determine distances.
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle. The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex; the shortest segment between base and apex is the height. The area of a triangle equals one-half the product of height and base length.
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. These points are usually on the surface of the Earth, and they are often used to establish maps and boundaries for ownership, locations, such as the designated positions of structural components for construction or the surface location of subsurface features, or other purposes required by government or civil law, such as property sales.
Liu Hui was a Chinese mathematician who published a commentary in 263 CE on Jiu Zhang Suan Shu. He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state of Cao Wei during the Three Kingdoms period of China.
Triangulation is the process of determining the location of a point by forming triangles to it from known points.
A geodetic control network is a network, often of triangles, that are measured precisely by techniques of control surveying, such as terrestrial surveying or satellite geodesy. It is also known as a geodetic network, reference network, control point network, or simply control network.
3D scanning is the process of analyzing a real-world object or environment to collect three dimensional data of its shape and possibly its appearance. The collected data can then be used to construct digital 3D models.
In the fields of computing and computer vision, pose represents the position and orientation of an object, usually in three dimensions. Poses are often stored internally as transformation matrices. The term “pose” is largely synonymous with the term “transform”, but a transform may often include scale, whereas pose does not.
Acoustic location is a method of determining the position of an object or sound source by using sound waves. Location can take place in gases, liquids, and in solids.
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.
In geometry, a base is a side of a polygon or a face of a polyhedron, particularly one oriented perpendicular to the direction in which height is measured, or on what is considered to be the "bottom" of the figure. This term is commonly applied in plane geometry to triangles, parallelograms, trapezoids, and in solid geometry to cylinders, cones, pyramids, parallelepipeds, prisms, and frustums.
The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two rays with a common starting point are intercepted by a pair of parallels. It is equivalent to the theorem about ratios in similar triangles. It is traditionally attributed to Greek mathematician Thales. It was known to the ancient Babylonians and Egyptians, although its first known proof appears in Euclid's Elements.
In computer vision, triangulation refers to the process of determining a point in 3D space given its projections onto two, or more, images. In order to solve this problem it is necessary to know the parameters of the camera projection function from 3D to 2D for the cameras involved, in the simplest case represented by the camera matrices. Triangulation is sometimes also referred to as reconstruction or intersection.
In computer vision and computer graphics, 3D reconstruction is the process of capturing the shape and appearance of real objects. This process can be accomplished either by active or passive methods. If the model is allowed to change its shape in time, this is referred to as non-rigid or spatio-temporal reconstruction.
Sensors for arc welding are devices which – as a part of a fully mechanised welding equipment – are capable to acquire information about position and, if possible, about the geometry of the intended weld at the workpiece and to provide respective data in a suitable form for the control of the weld torch position and, if possible, for the arc welding process parameters.
Tree height is the vertical distance between the base of the tree and the tip of the highest branch on the tree, and is difficult to measure accurately. It is not the same as the length of the trunk. If a tree is leaning, the trunk length may be greater than the height of the tree. The base of the tree is where the projection of the pith (center) of the tree intersects the existing supporting surface upon which the tree is growing or where the seed sprouted. If the tree is growing on the side of a cliff, the base of the tree is at the point where the pith would intersect the cliff side. Roots extending down from that point would not add to the height of the tree. On a slope this base point is considered as halfway between the ground level at the upper and lower sides of the tree. Tree height can be measured in a number of ways with varying degrees of accuracy.
In forestry, a tree crown measurement is one of the tree measurements taken at the crown of a tree, which consists of the mass of foliage and branches growing outward from the trunk of the tree. The average crown spread is the average horizontal width of the crown, taken from dripline to dripline as one moves around the crown. The dripline is the outer boundary to the area located directly under the outer circumference of the tree branches. When the tree canopy gets wet, any excess water is shed to the ground along this dripline. Some listings will also list the maximum crown spread which represents the greatest width from dripline to dripline across the crown. Other crown measurements that are commonly taken include limb length, crown volume, and foliage density. Canopy mapping surveys the position and size of all of the limbs down to a certain size in the crown of the tree and is commonly used when measuring the overall wood volume of a tree.
In surveying, triangulation is the process of determining the location of a point by measuring only angles to it from known points at either end of a fixed baseline by using trigonometry, rather than measuring distances to the point directly as in trilateration. The point can then be fixed as the third point of a triangle with one known side and two known angles.
Geopositioning is the process of determining or estimating the geographic position of an object or a person.
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