圆内矢量圆上的交集

Elementary vector algebra .

O — center of circle (vector)
r — its radius       (scalar)
A — origin of ray    (vector)
k — direction of ray (vector)

解决 (A + kt - O)² = r² 标量 t ,选择正根,然后 A + kt 这就是你的观点。

进一步解释:

. 是 dot product ,对于向量,是向量与其自身的点积。扩展LHS

(A + kt - O)² = (A - O)² + 2(k.(A - O))t + k²t².

二次方是 k²t² + 2(k.(A - O))t + (A - O)² - r² = 0 . 就你的变量而言,这就变成了 (rayVX² + rayVY²)t² + 2(rayVX(rayX - circleX) + rayVY(rayY - circleY))t + (rayX - circleX)² + (rayY - circleY)² - r² = 0 .

非常感谢Anton Tykhyy的详细回答。这就是由此产生的Java代码:

float xDiff = rayX - circleX;
float yDiff = rayY - circleY;
float a = rayVX * rayVX + rayVY * rayVY;
float b = 2 * (rayVX * (rayX - circleX) + rayVY * (rayY - circleY));
float c = xDiff * xDiff + yDiff * yDiff - r * r;
float disc = b * b - 4 * a * c;
if (disc >= 0) {
    float t = (-b + (float)Math.sqrt(disc)) / (2 * a);
    float x = rayX + rayVX * t;
    float y = rayY + rayVY * t;
    // Do something with point.
}