已知两点坐标、边长及夹角，求第三点坐标的方法

Alright, let's break this down step by step—this is a classic coordinate geometry problem that's totally solvable with vector rotation and basic coordinate transformations. Here's how to do it clearly:

Core Idea #

We'll translate the problem to a temporary coordinate system centered at point B (to simplify calculations), rotate the vector BA by the given angle to get the direction of BC, scale it to the length of BC, then translate back to the original coordinate system. This works because ∠ABC is the angle between vectors BA and BC at point B.

Step-by-Step Solution #

Let's define our known values first:

• Point A: (x_A, y_A)

• Point B: (x_B, y_B)

• Length AB: L_AB (we can also calculate this from A and B's coordinates, but we'll use the given value if provided)

• Length BC: L_BC

• ∠ABC: θ (note: this angle can be clockwise or counterclockwise, which will give two possible positions for C unless θ is 0° or 180°)

• Calculate Vector BA
  First, shift our perspective to make B the origin. The vector from B to A is:

  BA_x = x_A - x_B
  BA_y = y_A - y_B
  

• Get the Unit Vector of BA
  We need the direction of BA without its length, so we normalize it to a unit vector (length = 1):

  unit_BA_x = BA_x / L_AB
  unit_BA_y = BA_y / L_AB
  

  (If you didn't have L_AB given, you could calculate it as sqrt(BA_x² + BA_y²) instead.)

• Rotate the Unit Vector to Get BC's Direction
  The angle θ is between BA and BC, so we rotate the unit BA vector by θ to get the direction of BC. There are two possible rotations, which give the two possible positions of C:

  • Counterclockwise rotation (one possible C):

    unit_BC_x = unit_BA_x * cosθ - unit_BA_y * sinθ
    unit_BC_y = unit_BA_x * sinθ + unit_BA_y * cosθ
    

  • Clockwise rotation (the other possible C):

    unit_BC_x = unit_BA_x * cosθ + unit_BA_y * sinθ
    unit_BC_y = -unit_BA_x * sinθ + unit_BA_y * cosθ
    

  ⚠️ Important: Make sure θ is in radians if you're using calculator or programming functions (most cos()/sin() functions use radians). Convert degrees to radians with: θ_rad = θ_deg * π / 180

• Scale to Get Vector BC
  Multiply the unit BC vector by the length of BC to get the full vector from B to C:

  BC_x = unit_BC_x * L_BC
  BC_y = unit_BC_y * L_BC
  

• Convert Back to Original Coordinates
  Add the BC vector to point B's coordinates to get point C:

  x_C = x_B + BC_x
  y_C = y_B + BC_y
  

Example Walkthrough #

Let's say:

• A = (0, 0), B = (2, 0)
• L_AB = 2, L_BC = 2
• ∠ABC = 90° (π/2 radians)

1. Vector BA: BA_x = 0-2 = -2, BA_y = 0-0 = 0
2. Unit BA: unit_BA_x = -2/2 = -1, unit_BA_y = 0/2 = 0
3. Counterclockwise rotation (90°):
   unit_BC_x = (-1)*cos(π/2) - 0*sin(π/2) = 0
unit_BC_y = (-1)*sin(π/2) + 0*cos(π/2) = -1
4. Vector BC: BC_x = 0*2 = 0, BC_y = -1*2 = -2
5. Point C: x_C = 2+0=2, y_C=0+(-2)=-2 → C=(2,-2)

For the clockwise rotation (90°):
3. unit_BC_x = (-1)*cos(π/2) + 0*sin(π/2) = 0
unit_BC_y = -(-1)*sin(π/2) + 0*cos(π/2) = 1
4. BC_x=0*2=0, BC_y=1*2=2
5. Point C: (2+0, 0+2) → C=(2,2)

Which matches our intuition—two points forming a right angle with AB at B.

内容的提问来源于stack exchange，提问作者Alex Dave