Distance circle
As a distance circle , a circle on the earth's surface - more specifically to their mathematical realization, an earth ellipsoid - denotes that all the points P i contains the same distance , measured in the reference surface from a given point Q have.
Properties and uses
The distance circle is a mathematically exact circle - a small circle or a great circle - only if the earth is considered to be spherical . In every other case it is an algebraic curve of a higher order, which is also called a geodetic circle with regard to the ellipsoid or a mathematically uniquely describable surface . This name means that the constant distances P i -Q are each measured along a geodetic line .
The distance circles are of practical importance, especially when determining coordinates in geodesy and navigation . A point P on the earth's surface can be prepared by two measured distances from two positionally known fixed points Q 1 and Q 2 are determined by the distance of two circles are placed around the fixed points and brought to the interface. The task is also called a bow cut . Their fundamental ambiguity (the question of whether P is on the right or left side of the line Q 1 Q 2 ) is usually irrelevant in practice. However, the position of the new point becomes imprecise if the two circles grindly intersect in a dangerous location .
In three-dimensional space one needs three distance measurements to three given points; The task is called a spatial arc cut (trispheration) - because instead of two distance circles, three “distance spheres” have to be cut in 3D space. The most common application of this method is to determine the location with GPS - although, strictly speaking, there are even four spheres here.
Related topics
Other cutting methods also have circles as geometric locations . The most important are the procedures in which a peripheral circle occurs - such as when measuring a new point by measuring the direction according to 3 known points, the so-called backward cut . It can be traced back to the measurement of two angles , with each square having a circle through the two fixed points involved , the radius of which depends on the angle.
With dead reckoning , the navigator sometimes uses additional circles to verify the result by including radar or height measurements on landmarks . For the emerging systems of pedestrian navigation , distance and visual circles are of certain importance. Their conception is based as much as possible on people's natural sense of orientation and space , which is closely linked to the constantly "updated" mental model of the environment and the sense of balance .
The term “distance circle ” is often confused with the similar word distance circle . This term is mostly used in perspective , namely to define vanishing points . The vanishing points of straight lines which a given straight line at an angle of z. B. cut 45 °, lie on a circle around the primary vanishing point, which is called "distance circle" in the geometry .