the length of the arc L, the radius r and the starting point A are given

• A circles circumference is defined by 2 * r * pi . A whole circumference, calculated using r=5 : fullCircumference = 31.4159265359 .
• A partical circumference (the arc) is given by L , we know it's L/fullCircumference gives 0.21263598811 which is the fraction that L takes from the whole circumference.
• We can convert the arc fraction to radians, given that a whole circle has 2PI -> 0.21263598811 * 2 * PI = 1.33603131627 rad
• Now to the Starting Point A. I assume it's always ON the arc somewhere. Though, if we start with sin(0) and cos(0) we are at "12 o Clock":
  
  So we need to calculate the offset in radians around the circle.
• Though you didn't give us B and I can't assume that though the radius alone. So I will just ignore this for now and offset the forumla by 90° (or half pi) to get to A
• D's x & y would then be defined by: D(sin(x - pi/2) * r, cos(x - pi/2)) * r;
• And if we enter the 1.336 rad we calculated, it looks about right:
  

The c# code would look like this (didn't compile though):

Vector2 pointOnCircle(float L, float r){
    
    float fullCircumference = 2f * Mathf.PI * r;
    float partialCircumference = L / fullCircumference;
    float arcRadians = partialCircumference * 2f * Mathf.PI;

    float circularOffset = Mathf.PI / 2f;
    Vector2 D = new Vector2();
    D.x = Mathf.Sin(arcRadians - circularOffset) * r;
    D.y = Mathf.Cos(arcRadians - circularOffset) * r;

    return D;
}

• So we got C , now you want a random point D between A and C on that arc.
• We now need to sample a point on the arc using a fraction between 0 and 1 (0 being at A, 1 being at C):
• The arc's length is arcRadians in radians, so we can just mutiply that with 0..1 :

code:

Vector2 pointOnArc(float L, float r, float fraction){ // fraction is between 0 and 1
    
    float fullCircumference = 2f * Mathf.PI * r;
    float partialCircumference = L / fullCircumference;
    float arcRadians = partialCircumference * 2f * Mathf.PI;

    float circularOffset = Mathf.PI / 2f;
    Vector2 D = new Vector2();
    D.x = Mathf.Sin(fraction * arcRadians - circularOffset) * r;
    D.y = Mathf.Cos(fraction * arcRadians - circularOffset) * r;

    return D;
}

And there you go:

Percentage介于0和1之间，length是弧长。您可以将中心移到0，0的中心，也可以偏移中心，使点从所需的位置开始。
如果你想让它在另一边，让半径为负。如果你想让它在另一边，让弧长为负。
终点是从弧的起点到弧的终点的Angular 。Angular 是终点乘以百分比。然后将极坐标（Angular 和半径）转换为笛卡尔坐标（x，y）并将其偏移圆心。

Vector2 getPoint(Vector2 center, float length, float radius, float percentage)
{
    float end = length / radius;
    float angle = end * percentage;

    float x = center.x + (Mathf.Cos(angle) * radius);
    float y = center.y + (Mathf.Sin(angle) * radius);

    return new Vector2(x, y);
}