Geodesic and Planar methods are two primary methods used for calculating the distance between two locations in GIS. Geodesic method is used to get the exact estimation of direction and distance between two points on the Earth's surface when a 2D representation its surface is used. This method takes the globes curvature into consideration and determines the accurate pathway. This is very useful while calculating the distance, area and direction across continents.
Planar method does not consider the curvature of out planet while calculating distances. Mostly planar methods produce straight lines and are generally used to measure smaller distance in the 2D maps, but if the map represented as a globe than planar methods can be used to calculate the distance, area and direction across large areas for better aesthetics.
The map depicts the 5000 Km buffer from the center of India in GCS (Top), Mercator projection (Bottom Left) and Earth from Space projection (Bottom Right) with WGS_84 datum. From the image, the buffer shape varies with different projection systems, but the area encompassed inside the buffer remained consistent as the Geodesic and Planar systems were used were ever these methods were appropriate.