To capture three stars together in a photo, how wide must the field of view be?
Try tightening a circular frame and see who hits the edge first.
Sky coordinates and observations
You want to take a photo of three celestial bodies that are not far apart. If the field of view of the lens is too small, one lens will be missed; if it is too large, the frame will be wasted. The minimum containing circle tells you: at least how wide the sky should be.
Guess first
Do all three stars have to be on the edge of the smallest circle?
Show answer and explanation
uncertain — If a triangle has obtuse angles, the two furthest points determine the circle, and the third point will be inside the circle.
Then connect experience and science
Decide first who you want to include in the photo
The directions of the three stars are three points. After selecting the center of the picture, the circular field of view must be large enough to accommodate them all.
Use three diagonal distances to form a small field of view triangle
Sometimes the two furthest points are enough
If the third point is already within the circle determined by the two furthest points, there is no need to enlarge the circle for it.
If a² ≥ b²+c²: diameter Δ = a
Sometimes all three points are supported on the edge of the circle
The three stars of an acute triangle all fall on the sides of the smallest circle. Their circumscribed circles are then needed.
Δ = 2abc / √[(a+b+c)(a+b−c)(b+c−a)(a+c−b)]
When moving a celestial body, recalculate each time
The three-point relationship will switch from one branch to another; do not directly use the final circle diameter at several moments as a smooth polynomial interpolation. The celestial coordinates should be obtained first, and then the circle should be recalculated.
Interpolate the coordinates first, then find the angular distance and minimum circle
a takes the maximum value among the three pairs of angular distances, b and c are the other two sides, all using the same angular unit. Make an obtuse/right angle judgment first, and then select the diameter of the longest side or the diameter of the circumscribed circle; this is a plane approximation for a small field of view.
give it a try
Drag the field of view to see which stars are missed; compare interior, three-point welts and right-angled boundaries. The moving example only changes the number of days and then recalculates the positions of the three celestial bodies.
No need to memorize symbols
- a,b,c
- Three diagonal distances, a is the longest side
- Δ
- angular diameter of smallest circle
- I / II
- The longest side determines the circle/three-point circumcircle
- field of view
- The directional range covered by the screen, not the physical distance
Algorithm, accuracy and scope of application
Printed pages 127–130 of Chapter 20 of the original book, Equation (20.1), Examples 20.a and 20.c. The original method is suitable for aggregations within about 6° of mutual distance, and uses the local plane approximation of angular distance triangles; this lesson is not considered an accurate minimum spherical hat algorithm for the entire sky. The equatorial coordinates in Example 20.a of the original book are checked item by item with the scanned original page, and 4.26364° is obtained; the moving example uses the assumption of constant latitude and constant celestial longitude in Example 20.c of the original book, which does not represent the true planetary ephemeris. In order to accurately present the calculations in the book, the picture is reconstructed from three diagonal distances to a planar triangle, and then translated to the optimal center of the circle, which is not the true southeast, northwest, and northwest directions. The user's field of view remains at the center of this optimal circle; camera projection, distortion, or actual imaging is not simulated. Additional numerical treatment of coincident, collinear and right-angled boundaries. Comparison example 20.a with locally compiled AA+ SmallestCircle; 20.b complete daily planet table and minimum time search has not been implemented.
my understanding check
The third point is already within the circle determined by the two farthest points. Does it still need to enlarge the circle?
unnecessary — The smallest containing circle allows points to fall within it.
Does the "4° field of view" in this lesson mean that the lens diameter is 4 cm?
No, it's the angle of the sky — The angular diameter describes how wide the directional range can fit in the sky, not the physical size of the lens.
To find the smallest circle when moving celestial bodies gather, what should be the priority interpolation?
Celestial coordinates — The circle diameter will change the shape of the curve due to the switching of geometric branches, and direct interpolation may cause distortion.
First distinguish what kind of triangle it is, and then tighten the circular field of view; some points can be within the circle, not all of them need to be close to the edge.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 20 chapter