The four little stars on the five-star red flag are identical, and their difference is "position" and "angle". To get the others from one, you need to "translate" and "rotate".
Every regular five-pointed star has a circumscribed circle. Then, as long as the sizes of these five circumscribed circles and their positions in the rectangular area are determined, the sizes and positions of these five stars can be determined. As for the angles of these five stars, we know that the general situation is: the big stars are in full bloom; Each of the four small stars has an angle pointing to the center of the big star. As for the exact angle, it needs to be calculated.
Size: What is the radius of the circumscribed circle of the big star? 3? (unit length), what are the radii of the circumscribed circles of the four little stars? 1? (unit length).
Location: we will put the rectangular area above (the upper left corner of the flag? 1/4? ) put it right, take the vertex in the lower left corner of the rectangular area as the center o, and the bottom edge as? x？ Axis (right), and left? y？ Axis (forward), with the unit length as the coordinate length, determine a coordinate system, so the rectangular area is just in the first quadrant of this coordinate system. Now, the positions of the five circumscribed circles can be determined by the coordinates of their centers:
Big star: s (? 5,? 5)
Xiaoxing: A( 10,? 8)
Xiaoxing: B( 12,? 6)
Xiaoxing: C( 12,? 3)
Xiaoxing: D( 10,? 1)
Angle: It is known that the relationship between each small star and the big star is that there is an angle (vertex) pointing to the center of the big star. Connect the big star center S and each small star center respectively, and get four straight lines: SA, SB, SC, SD. Let the included angle between each straight line and the X axis (the angle below the X axis is negative) be ∠A, ∠B, ∠C and ∠D respectively. The angles between adjacent straight lines ∠ASB, ∠BSC and ∠CSD are the angles that each small star needs to rotate when it is transformed into an adjacent small star, which is also the result we finally require. According to the coordinates of points S, A, B, C and D, these angles can be found by tangent and arc tangent functions:
tanA=？ 3/5= >？ ∠A？ =? arctan(？ 3/5)? ≈30.963757
tanB=？ 1/7= >？ ∠B？ =? arctan(？ 1/7)? ≈8. 130 102
tanC=-2/7= >？ ∠C？ =? arctan(-2/7)？ ≈- 15.945396
tanD=-4/5= >？ ∠D？ =? arctan(-4/5)？ ≈-38.659808
∠ASB？ =? ∠A？ -? ∠B？ =? arctan(？ 3/5)? -? arctan( 1/7)？ =? Arctangent (8/19) ≈ 19989.19989888887
∠BSC？ =? ∠B？ -? ∠C？ =? arctan(？ 1/7)? -? arctan(？ -2/7)? =? arctan(2 1/47)？ ≈24.075498
∠CSD？ =? ∠C？ -? ∠D？ =? arctan(-2/7)？ -? arctan(？ -4/5)? =? arctan( 18/43)？ ≈22.7 144 12
Now that the size and position of each star and the angle difference between adjacent small stars have been determined, we can know how the small stars overlap by moving. For example: from Little Star A? Get "Little Star B":
Step 1: pan to the right? 2 (unit length), and then translate down by 2; (This is the way to move point A to point B)
Step 2: Turn clockwise about? 22.833654 。 (This is the method of moving (rotating) the straight line SA to the straight line SB)
The same is true of the movement between other small stars.