Thursday, January 21, 2016

Unizor - Geometry3D - Spheres - Latitude, Longitude

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Notes to a video lecture on

Coordinates on a Sphere

This lecture is about geographical coordinates - the system of coordinates on a surface of our spherical (for all intents and purposes) planet Earth.
Our purpose is to numerically identify a position of any point on Earth - a very useful thing in many practical cases.
There are more than one way to accomplish this, but we will talk only about one particular method, standard across many areas of activity, based on longitude and latitude.

Let's assume that we have a sphere that represents our planet Earth (we will use terms "sphere" and "Earth" interchangeably) and a point on it. We assume that our sphere is rotating around some axis that connects North Pole and South Pole. We also assume that direction from each point on this sphere that coincides with the direction of rotation is called East and the opposite direction is called West. The pole to the left from a point while looking in the direction of rotation is the North Pole, the other is the South Pole.

We can identify the position of any point on this sphere with two numbers - both angular measures - using the following definitions.

1. Let's define a concept of a meridian going through some point P on a sphere.

Meridian is a semi-circle on a surface of a sphere, going from North Pole to South Pole through our point, which is an intersection of a sphere with a half-plane going through an axis of rotation of the Earth from North Pole to South Pole and point P.

There is one and only one meridian that goes through any point on a sphere, except the two poles - just draw a one and only one half-plane through this point and an axis of rotation of the Earth, this half-plane's intersection with a surface of the Earth is a semi-circle going through our point from pole to pole, that is the meridian of this point.

To identify a specific meridian, we choose an angular measure of a dihedral angle between a half-plane generating this meridian and some fixed half-plane generating some base (fixed) meridian. For historical reasons the fixed half-plane serving as the base for this measurement is the one generating a meridian that goes through Royal Observatory in Greenwich - now a district of London. This Greenwich meridian is called Prime Meridian and is characterized by a dihedral angle of 0o.
From Prime Meridian all those towards East will have a measure of corresponding dihedral angle from 0o to +180o (or 180oE). Those towards West will have measures from 0o to −180o (or 180oW).
This angular measure of deviation of a meridian of some point from Prime Meridian is called longitude and is denoted by letter λ on the picture above. Knowing the longitude of a point on a sphere, we can construct its meridian - a curve on which this point is located.

2. To identify a position of a point on its meridian we will introduce a concept of a parallel and another measure - its latitude.

Parallel is a circle on our sphere formed by an intersection of any plane perpendicular to a sphere's axis with a sphere's surface. The parallel formed by a plane cutting the axis in the middle (that is, going through a center of a sphere) is called an equator.

The position of any parallel on a sphere is fully defined by an angle between an equatorial plane and a radius from a sphere's center to any point on this parallel (angle φ on the picture above). The measure of this angle is called latitude. It ranges from 0o for any point on the equator to 90o for both poles. Points closer to North Pole than equator will have a designation N with their latitude, points towards South will be designated with letter S.

As we see, for any point on a sphere we can determine its longitude (a specific meridian it is on) and latitude (a specific parallel it is on). Combination of these two curves, a meridian and a parallel, on a surface of a sphere fully defines the position of a point.

Every point on a sphere, except two poles, uniquely defines its coordinates, longitude and latitude. In case of North Pole, it is reasonable to attribute only latitude of 90oN to it. Similarly, for South Pole it's 90oS.

To more precisely identify a position, smaller than 1o units of angle measurement are used: angular minute (1') equals to 1/60 of 1o;
angular second (1") equals to 1/60 of 1'.
Alternatively, decimal fractions of 1o can be used as well.

In addition, it should noted that in reality our planet is not a perfect sphere and precise definitions of longitude and latitude are slightly different from those explained above.
Also, for those interested in coordinates above the surface of the Earth, they can add a height above the surface as a third coordinate. So, an object above the ground will have longitude λ and latitude φ of its projection on the ground plus height h above that point.

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