Angles: degrees, radians and Atan2

Drawings use degrees: a 90° corner, a 45° chamfer, a wall at 30°. C#’s Math.Sin, Math.Cos and Math.Tan use radians. Mixing them up is one of the most common geometry bugs, so it is worth understanding what a radian is.

What is a radian?

Draw a circle of radius 1. Walk around its edge a distance of 1: the angle you have turned is one radian (about 57.3°). A full circle is 2π ≈ 6.283 radians, because the circumference of a unit circle is 2π. So 360° = 2π, 180° = π and 90° = π/2.

1static double ToRadians(double deg) => deg * Math.PI / 180;2static double ToDegrees(double rad) => rad * 180 / Math.PI;3 4Console.WriteLine(Math.Sin(90));                          // the bug: 90 radians5Console.WriteLine(Math.Sin(ToRadians(90)));               // what was meant6Console.WriteLine($"{ToDegrees(1):F4}");                  // one radian in degrees

0.8939966636005579
1
57.2958

The angle of a direction: use Atan2

A wall runs from (x₁, y₁) to (x₂, y₂). What angle does it make with the X axis? The slope is dy / dx, and the tangent of the angle is that slope, so it is tempting to write Math.Atan(dy / dx). That is wrong in two ways:

• dy / dx loses the signs: a wall going up-right (2, 2) and one going down-left (−2, −2) have the same slope, 1, so Atan gives 45° for both, though they point in opposite directions.
• A vertical wall has dx = 0. With double coordinates the division gives ±∞, which Atan happens to survive; with int coordinates it throws a DivideByZeroException, and a zero-length wall gives NaN.

Math.Atan2(dy, dx) takes the two values separately, so it knows the quadrant and handles dx = 0. It returns an angle between −π and π. Drag the point around the dial to compare the two.

1// Angle of a direction, counter-clockwise from +X, in [0, 360).2static double DirectionDeg(double dx, double dy)3{4    double deg = Math.Atan2(dy, dx) * 180 / Math.PI;5    return deg < 0 ? deg + 360 : deg;6}7 8Console.WriteLine($"{Math.Atan(-2.0 / -2.0) * 180 / Math.PI:0.##}");  // Atan: wrong quadrant9Console.WriteLine(DirectionDeg(-2, -2));10Console.WriteLine(DirectionDeg(0, 5));11Console.WriteLine(DirectionDeg(3, -3));

45
225
90
315

Normalising angles

Angles wrap: 370°, 10° and −350° are the same direction. Before comparing angles or showing them, bring them into one standard range. Two ranges are common: [0, 360) for directions, and (−180, 180] for turns, where the sign tells you left (+) or right (−).

1static double To360(double deg)2{3    deg %= 360;                      // C#'s % keeps the sign: -350 % 360 = -3504    return deg < 0 ? deg + 360 : deg;5}6 7static double ToPlusMinus180(double deg)8{9    deg = To360(deg);10    return deg > 180 ? deg - 360 : deg;11}12 13Console.WriteLine(To360(370));14Console.WriteLine(To360(-350));15Console.WriteLine(ToPlusMinus180(270));        // a 270 left turn is a 90 right turn16Console.WriteLine(ToPlusMinus180(350 - 10));   // turn from 10 to 350

10
10
-90
-20

The last line is the important one: the turn from heading 10° to heading 350° is not +340°, it is 20° to the right. Taking the difference and normalising it to (−180, 180] always gives the short way round.

Maths angles vs compass bearings

Maths measures angles counter-clockwise from east (+X). Surveyors and site plans use bearings: clockwise from north (+Y). Both describe the same direction, so converting is one line: bearing = 90 − angle, normalised to [0, 360).

1using Pt = (double X, double Y);2 3static double Dir(Pt a, Pt b) => Math.Atan2(b.Y - a.Y, b.X - a.X) * 180 / Math.PI;4 5static double ToPlusMinus180(double deg)6{7    deg %= 360;8    if (deg < 0) deg += 360;9    return deg > 180 ? deg - 360 : deg;10}11 12Pt[] run = { (0, 0), (6, 0), (6, 4), (2, 7) };13 14for (int i = 1; i < run.Length - 1; i++)15{16    double turn = ToPlusMinus180(Dir(run[i], run[i + 1]) - Dir(run[i - 1], run[i]));17    string side = turn >= 0 ? "left" : "right";18    Console.WriteLine($"At {run[i]}: {side} {Math.Abs(turn):F1} deg");19}

At (6, 0): left 90.0 deg
At (6, 4): left 53.1 deg