How Far Did the Wheel Roll?

A small delivery robot counts how far each of its wheels has turned. Every control tick, the wheel's encoder reports the turn since the last tick in degrees: positive when the wheel turned forwards, negative when it turned backwards.

A wheel of radius r that turns by an angle \theta in radians and does not slip rolls the distance

(this is the length of the arc the rim lays on the floor). One full turn is 360 degrees or 2\pi radians, so \theta_{\text{rad}} = \theta_{\text{deg}} \cdot \pi / 180; math.radians does that conversion.

Write wheel_distances(radius, turns) that returns a tuple (distances, net):

• distances: the list of distances rolled in each tick, in the order of turns, in the units of radius,
• net: the sum of those distances (how far the robot is from where it started along its straight path; 0.0 for no ticks).

Examples

Input:  radius = 5, turns = [90, 180, -45]
Output: ([7.853981633974483, 15.707963267948966, -3.9269908169872414], 19.634954084936208)
Explanation: 90 degrees is pi/2 radians, so the wheel rolls 5 * pi/2 = 7.854 cm.
The last tick turned backwards and rolled 3.927 cm back.

Input:  radius = 3, turns = [360]
Output: ([18.84955592153876], 18.84955592153876)
Explanation: one full turn rolls the whole rim, 2 * pi * 3.

Constraints

• answers are compared with a tolerance of 1e-6; do not round

Goals

• Convert an angle from degrees to radians
• Use the arc length $s = r\,\theta$ to turn a wheel rotation into a distance
• Keep the sign: a backwards turn rolls a negative distance