E3.1 Shortest paths and distances on the circle (20 points) Given two points on the circle 1 and 2 (represented as values in the interval-, the counter-clockwise distance from to 2, denoted by distcc(, 2), is the length of the counter-clockwise arc starting at and ending at 2 Similarly, the clockwise distance, denoted by diste 2 is the length of the clockwise arc from 1 to 2. Finally, the distance between 1 to 2 is the smallest of the two counter-clockwise and clockwise distances. Hint: On the unit circle, the length of an arc is equal to the angle subtended by the arc and is measured in radians. (i) (2 points) Compute counter-clockwise and the clockwise distances between -/3 and 2--3/4. (ii) (5 points) Provide a formula for the counter-clockwise distance from arbitrary 01 to arbitrary 02 Hint: Recall the "modulo" operator: 1n0d(,2) is the remainder of the division of by 2. (iii) (5 points) Provide a formula for the clockwise distance from to 2, denoted by distc(91,02) (iv) (3 points) Define the distance between two angles to be the smallest between their counterclockwise and clockwise distance. Provide a formula for the distance between 01 and 02, denoted by distcircle (01,02) For further reference, note the following general properties distcc(01,02) distcc (02,01), distcc(4 M)-distc(02,01 ), lack of symmetry of distcc relationship between distcc and distc, and symmetry of distcircle