Finding several polar coordinates of a single point

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Finding several polar coordinates of a single point


Use a value of 1mm of uncertainty on non-USGS maps. By this rule, the uncertainty for a map of scale 1: Distance uncertainties in any given direction are linear and additive. Following is an example of a simple locality description and an explanation of the manner in which multiple sources of uncertainty interact to result in an overall maximum error distance.

The possible sources of uncertainty for this example are 1 the extent of Bakersfield, 2 unknown datum, 3 distance imprecision, and 4 map scale. Suppose the center of Bakersfield is 3 km from the eastern city limit and the distance is being measured on a USGS map at 1: The uncertainty due to the extent of Bakersfield is 3 km, there is no uncertainty due to an unknown datum assuming the datum is recorded with the coordinatesthe distance imprecision is 1 km, and the uncertainty due to map scale is 51 meters.

The maximum error distance for this locality is the sum of these, or 4. Ignore for the moment all sources of uncertainty except those arising from distance imprecision. Under this simplification, a proper description of the uncertainty is a bounding box centered on the point 6 km E and 8 km N of Bakersfield.

Each side of the box is 2 km in length 1 km uncertainty in each cardinal direction from the center. Since we are characterizing the maximum error as a single distance measurement, we need the circle that circumscribes the above-mentioned bounding box. The radius of this circle is the distance from the center of the box to any corner.

The radius could either be measured on a map or calculated using a right triangle, the hypotenuse of which is the line between the center of the bounding box and a corner.

Given the rule that the distance precision is the same in both cardinal directions, the triangle will always be a right isosceles triangle and the hypotenuse will always be the the square root of 2 times the distance precision.

So, for the above example the error distance associated with only the distance precision would be 1.

Dennis Rawlins: Contributions

So far we have accounted only for distance imprecision for this example. How do we take into account the uncertainty due to the shape of the named place?

There are many methods that could be used to determine the coordinates and error for this situation.

Finding several polar coordinates of a single point

The method presented here is quite conservative, resulting in errors larger than they need to be. A better alternative would be to multiply only the distance precision error by the square root of 2 contributions in both dimensionsand then sum that with all other sources, which already account for the two dimensions.

Translation from Polar to Cartesian Coordinates

This second method is the one used in the Georeferencing Calculator since version Determine the furthest distance within the named place from the geographic center of the named place in either of the two cardinal directions mentioned in the locality description.

Add this distance to the distance precision and take the square root of 2 times the sum to get the maximum error distance associated with the combination of distance precision and the extent of the named place.

For the example above, suppose the furthest extent of the city limits of Bakersfield either E or N from the geographic center is 3 km. There would be a total of 4 km of uncertainty in each of the two directions and the radius of the circumscribing circle would be 4 km times the square root of 2, or 5.

What other sources of uncertainty need to be accounted for in this example? Suppose the coordinates for Bakersfield were taken from the GNIS database, in which there is no reference to datum and the coordinates are given to the nearest second.

At this location the uncertainty due to an unknown datum is 79 meters. The datum uncertainty contributes in each of the orthogonal directions. Thus, the uncertainty in each direction would be 4.

The coordinates in the GNIS database are given to the nearest second. Based on the Uncertainty associated with coordinate precision section, above, the uncertainty due to coordinate precision alone is about 39 meters at the latitude of Bakersfield.

Finding several polar coordinates of a single point

This number already accounts for the contributions in both cardinal directions, so it must not be multiplied by the square root of 2. Instead, simply add the coordinate precision uncertainty to the calculated sum of uncertainties from the other sources.The CF conventions generalize and extend the COARDS extensions include metadata that provides a precise definition of each variable via specification of a standard name, describes the vertical locations corresponding to dimensionless vertical coordinate values, and provides the spatial coordinates of non-rectilinear gridded data.

Mentor: The polar coordinate system is a circular rather than a rectangular coordinate system.

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You plot points given in terms of a value, r which is the distance from the center; and an angle measure which is given in degrees, or more usually radians. Oct 29,  · Finding multiple sets of polar coordinates for a given point Polar Coordinates Naming All Polar Coordinates for a Point - Duration: Brendon Ferullo 7, views.

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In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate contrasts with synthetic geometry.. Analytic geometry is widely used in physics and engineering, and also in aviation, rocketry, space science, and is the foundation of most modern fields of geometry, including algebraic.

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Polar coordinates - Math Insight