After turning around in a full circle, why are you still facing the same house?
Use turning to find the direction, and distinguish "how much you turned" and "where you ended up heading."
General Algorithms and Numerical Computing
Stand at the intersection and start heading toward the red-roofed house on the right. Half a turn to the left will turn you away from it, and half a turn to the left will bring you face to face. You do go full circle, but end up back where you started.
Guess first
Turned 360° in place, what direction did you face at the end compared to where you started?
Show answer and explanation
same — After a full circle, the direction returns to its original position; the total angle turned is not zero.
Then connect experience and science
The negative sign controls the entire string of angles
Degrees, arc minutes, and arc seconds are sexagesimal. −0°30′ is legal and meaningful: the negative sign cannot be thrown away because the integer part is zero.
Angle = sign × (degrees + minutes/60 + seconds/3600)
You can turn many times in the same direction
Increasing or decreasing the direction angle by 360° will not change the direction. The blue pointer in the picture only retains the position after the circle.
Orientation angle normalized to 0° ≤ θ < 360°
Divide a circle into twenty-four hours
Right ascension and sidereal angles are often expressed in hours. A circle is also 360°, so the angle in 1 hour is equal to 15°.
Angle hours = θ / 15
The units are different and the inputs are also different.
The program's sin usually takes radians. Change the units before entering; do not rewrite the latitude −30° as 330° without authorization.
Radians = degrees × π / 180
Expand the calculation
θ = sign × (d + m/60 + s/3600) θ圈内 = ((θ mod 360) + 360) mod 360 αHour = θSpend/15；θ弧Spend = θSpend × π/180
First convert minutes and seconds into degrees, and then deal with the symbols. Normalization only changes the way the angle is written, not the direction.
give it a try
First set to −0.5° and then changed to 359.5°. The pointers are the same, but the signed raw input is different. Let’s look at how many hours 90° corresponds to.
No need to memorize symbols
- ° / ′ / ″
- Degrees, arc minutes, arc seconds; 1° = 60′ = 3600″
- h / m / s
- This is used here for equal angular hours in right ascension, not elapsed time.
- θ
- Normalize the direction angle, retaining only the direction within a circle
Algorithm, accuracy and scope of application
Corresponds to angle conversion, normalization and programming language tips in Chapter 1. DMS output is rounded to arcseconds and carries are handled; the minimum slider step for this lesson is 0.25°. AA+ DMSToDegrees uses independent positive and negative flags to avoid negative zero problems. The two sets of scales in the picture both represent the direction angle, and the right ascension hour cannot be directly regarded as a civilian clock.
my understanding check
With your back to the starting direction, how many degrees do you need to turn?
180° — Back-to-back means half a turn, which is 180°.
Are 0° and 360° pointing in the same direction and turning the same total angle?
different — One didn't turn, the other turned a full circle; in the end the direction was the same.
It says 6h in right ascension, does it mean that the action took six hours?
No, it’s an angle unit — A circle of right ascension is divided into 24h, with 6h corresponding to 90°.
The same orientation does not mean the same angle of rotation; astronomy often only requires the direction within a circle.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 1 chapter
General Algorithms and Numerical Computing · All interactive courses