Concept of Computational Geometry, Geometrical objects and its properties

Computational Geometry:

Computational Geometry is a significant subject. Mastering this subject can really help you in programming contests as every contest generally comprise 1-2 geometrical problems.

Geometrical objects and its properties:

Earth coordinate system:

People employ latitudes (or horizontal lines) and longitudes (or vertical lines) in Earth coordinate system.

Longitude spans from 0 degrees (that is, Greenwich) to +180* East and -180* West. Latitude spans from 0 degrees (that is, Equator) to +90* (or North Pole) and -90* (or South Pole).

The most fascinating question is what the spherical or geographical distance between two cities p and q on earth with radius r, symbolized by (p_lat,p_long) to (q_lat,q_long). All the coordinates are in radians. (that is, convert [-180..180] range of longitude and [-90..90] range of latitudes to [-pi..pi] correspondingly.

On deriving the mathematical equations. The answer is as shown below:

spherical_distance(p_lat,p_long,q_lat,q_long) = 
acos( sin(p_lat) * sin(q_lat) + cos(p_lat) * cos(q_lat) * cos(p_long -
q_long) ) * r
since cos(a-b) = cos(a)*cos(b) + sin(a)*sin(b), we can simplify the above
formula to:
spherical_distance(p_lat,p_long,q_lat,q_long) = 
acos( sin(p_lat) * sin(q_lat) + 
        cos(p_lat) * cos(q_lat) * cos(p_long) * cos(q_long) +
        cos(p_lat) * cos(q_lat) * sin(p_long) * sin(q_long)
       ) * r


Convex Hull:

Fundamentally, Convex Hull is the most fundamental and most popular computational geometry problem. Most of the algorithms are available to resolve this capably, with the best lower bound O(n log n). This lower bound is already confirmed.

Convex Hull problem (2-D version):

Input: A set of points in the Euclidian plane
Output: Determine the minimum set of points which enclosed all other points.

Convex Hull algorithms:

i) Jarvis March or Gift Wrapping
ii) Graham Scan
iii) Quick Hull
iv) Divide and Conquer

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