我有一条从a到B的直线和一个半径为R的圆。
用什么算法来检查直线是否与圆相交?它在圆边的哪个坐标上?
当前回答
我发现这个解决方案似乎比其他一些解决方案更容易遵循。
采取:
p1 and p2 as the points for the line, and
c as the center point for the circle and r for the radius
我可以用斜截式来解直线方程。但是，我不想处理以c为点的复杂方程，所以我只是平移了坐标系使圆在(0,0)处
p3 = p1 - c
p4 = p2 - c
顺便说一下，当我相互减分的时候，我是在减去x再减去y，然后把它们放到一个新的点里，以防有人不知道。
不管怎样，我现在解出p3和p4的直线方程
m = (p4_y - p3_y) / (p4_x - p3) (the underscore is an attempt at subscript)
y = mx + b
y - mx = b (just put in a point for x and y, and insert the m we found)
好的。现在我需要让这两个方程相等。首先我需要解圆的x方程
x^2 + y^2 = r^2
y^2 = r^2 - x^2
y = sqrt(r^2 - x^2)
然后我让它们相等:
mx + b = sqrt(r^2 - x^2)
求二次方程(0 = ax^2 + bx + c)
(mx + b)^2 = r^2 - x^2
(mx)^2 + 2mbx + b^2 = r^2 - x^2
0 = m^2 * x^2 + x^2 + 2mbx + b^2 - r^2
0 = (m^2 + 1) * x^2 + 2mbx + b^2 - r^2
现在我有了a b c。
a = m^2 + 1
b = 2mb
c = b^2 - r^2
我把这个代入二次公式
(-b ± sqrt(b^2 - 4ac)) / 2a
用值代入，然后尽可能简化:
(-2mb ± sqrt(b^2 - 4ac)) / 2a
(-2mb ± sqrt((-2mb)^2 - 4(m^2 + 1)(b^2 - r^2))) / 2(m^2 + 1)
(-2mb ± sqrt(4m^2 * b^2 - 4(m^2 * b^2 - m^2 * r^2 + b^2 - r^2))) / 2m^2 + 2
(-2mb ± sqrt(4 * (m^2 * b^2 - (m^2 * b^2 - m^2 * r^2 + b^2 - r^2))))/ 2m^2 + 2
(-2mb ± sqrt(4 * (m^2 * b^2 - m^2 * b^2 + m^2 * r^2 - b^2 + r^2)))/ 2m^2 + 2
(-2mb ± sqrt(4 * (m^2 * r^2 - b^2 + r^2)))/ 2m^2 + 2
(-2mb ± sqrt(4) * sqrt(m^2 * r^2 - b^2 + r^2))/ 2m^2 + 2
(-2mb ± 2 * sqrt(m^2 * r^2 - b^2 + r^2))/ 2m^2 + 2
(-2mb ± 2 * sqrt(m^2 * r^2 + r^2 - b^2))/ 2m^2 + 2
(-2mb ± 2 * sqrt(r^2 * (m^2 + 1) - b^2))/ 2m^2 + 2
这几乎是化简的极限了。最后，分离出带有±的方程:
(-2mb + 2 * sqrt(r^2 * (m^2 + 1) - b^2))/ 2m^2 + 2 or
(-2mb - 2 * sqrt(r^2 * (m^2 + 1) - b^2))/ 2m^2 + 2
然后简单地将这两个方程的结果代入mx + b中的x。为了清晰起见，我写了一些JavaScript代码来演示如何使用这个:
function interceptOnCircle(p1,p2,c,r){
//p1 is the first line point
//p2 is the second line point
//c is the circle's center
//r is the circle's radius
var p3 = {x:p1.x - c.x, y:p1.y - c.y} //shifted line points
var p4 = {x:p2.x - c.x, y:p2.y - c.y}
var m = (p4.y - p3.y) / (p4.x - p3.x); //slope of the line
var b = p3.y - m * p3.x; //y-intercept of line
var underRadical = Math.pow((Math.pow(r,2)*(Math.pow(m,2)+1)),2)-Math.pow(b,2)); //the value under the square root sign
if (underRadical < 0){
//line completely missed
return false;
} else {
var t1 = (-2*m*b+2*Math.sqrt(underRadical))/(2 * Math.pow(m,2) + 2); //one of the intercept x's
var t2 = (-2*m*b-2*Math.sqrt(underRadical))/(2 * Math.pow(m,2) + 2); //other intercept's x
var i1 = {x:t1,y:m*t1+b} //intercept point 1
var i2 = {x:t2,y:m*t2+b} //intercept point 2
return [i1,i2];
}
}
我希望这能有所帮助!
附注:如果任何人发现任何错误或有任何建议，请评论。我是新手，欢迎大家的帮助/建议。
其他回答
好吧，我不会给你代码，但既然你已经标记了这个算法，我认为这对你来说无关紧要。 首先，你要得到一个垂直于这条直线的向量。
y = ax + c是一个未知变量c是未知变量 为了解决这个问题，计算直线经过圆心时的值。
也就是说, 将圆心的位置代入直线方程，解出c。 然后计算原直线与其法线的交点。
这样就能得到直线上离圆最近的点。 计算该点到圆中心之间的距离(使用矢量的大小)。 如果这个小于圆的半径，看，我们有一个交点!
我发现这个解决方案似乎比其他一些解决方案更容易遵循。
采取:
p1 and p2 as the points for the line, and
c as the center point for the circle and r for the radius
我可以用斜截式来解直线方程。但是，我不想处理以c为点的复杂方程，所以我只是平移了坐标系使圆在(0,0)处
p3 = p1 - c
p4 = p2 - c
顺便说一下，当我相互减分的时候，我是在减去x再减去y，然后把它们放到一个新的点里，以防有人不知道。
不管怎样，我现在解出p3和p4的直线方程
m = (p4_y - p3_y) / (p4_x - p3) (the underscore is an attempt at subscript)
y = mx + b
y - mx = b (just put in a point for x and y, and insert the m we found)
好的。现在我需要让这两个方程相等。首先我需要解圆的x方程
x^2 + y^2 = r^2
y^2 = r^2 - x^2
y = sqrt(r^2 - x^2)
然后我让它们相等:
mx + b = sqrt(r^2 - x^2)
求二次方程(0 = ax^2 + bx + c)
(mx + b)^2 = r^2 - x^2
(mx)^2 + 2mbx + b^2 = r^2 - x^2
0 = m^2 * x^2 + x^2 + 2mbx + b^2 - r^2
0 = (m^2 + 1) * x^2 + 2mbx + b^2 - r^2
现在我有了a b c。
a = m^2 + 1
b = 2mb
c = b^2 - r^2
我把这个代入二次公式
(-b ± sqrt(b^2 - 4ac)) / 2a
用值代入，然后尽可能简化:
(-2mb ± sqrt(b^2 - 4ac)) / 2a
(-2mb ± sqrt((-2mb)^2 - 4(m^2 + 1)(b^2 - r^2))) / 2(m^2 + 1)
(-2mb ± sqrt(4m^2 * b^2 - 4(m^2 * b^2 - m^2 * r^2 + b^2 - r^2))) / 2m^2 + 2
(-2mb ± sqrt(4 * (m^2 * b^2 - (m^2 * b^2 - m^2 * r^2 + b^2 - r^2))))/ 2m^2 + 2
(-2mb ± sqrt(4 * (m^2 * b^2 - m^2 * b^2 + m^2 * r^2 - b^2 + r^2)))/ 2m^2 + 2
(-2mb ± sqrt(4 * (m^2 * r^2 - b^2 + r^2)))/ 2m^2 + 2
(-2mb ± sqrt(4) * sqrt(m^2 * r^2 - b^2 + r^2))/ 2m^2 + 2
(-2mb ± 2 * sqrt(m^2 * r^2 - b^2 + r^2))/ 2m^2 + 2
(-2mb ± 2 * sqrt(m^2 * r^2 + r^2 - b^2))/ 2m^2 + 2
(-2mb ± 2 * sqrt(r^2 * (m^2 + 1) - b^2))/ 2m^2 + 2
这几乎是化简的极限了。最后，分离出带有±的方程:
(-2mb + 2 * sqrt(r^2 * (m^2 + 1) - b^2))/ 2m^2 + 2 or
(-2mb - 2 * sqrt(r^2 * (m^2 + 1) - b^2))/ 2m^2 + 2
然后简单地将这两个方程的结果代入mx + b中的x。为了清晰起见，我写了一些JavaScript代码来演示如何使用这个:
function interceptOnCircle(p1,p2,c,r){
//p1 is the first line point
//p2 is the second line point
//c is the circle's center
//r is the circle's radius
var p3 = {x:p1.x - c.x, y:p1.y - c.y} //shifted line points
var p4 = {x:p2.x - c.x, y:p2.y - c.y}
var m = (p4.y - p3.y) / (p4.x - p3.x); //slope of the line
var b = p3.y - m * p3.x; //y-intercept of line
var underRadical = Math.pow((Math.pow(r,2)*(Math.pow(m,2)+1)),2)-Math.pow(b,2)); //the value under the square root sign
if (underRadical < 0){
//line completely missed
return false;
} else {
var t1 = (-2*m*b+2*Math.sqrt(underRadical))/(2 * Math.pow(m,2) + 2); //one of the intercept x's
var t2 = (-2*m*b-2*Math.sqrt(underRadical))/(2 * Math.pow(m,2) + 2); //other intercept's x
var i1 = {x:t1,y:m*t1+b} //intercept point 1
var i2 = {x:t2,y:m*t2+b} //intercept point 2
return [i1,i2];
}
}
我希望这能有所帮助!
附注:如果任何人发现任何错误或有任何建议，请评论。我是新手，欢迎大家的帮助/建议。
我根据chmike给出的答案为iOS创建了这个函数
+ (NSArray *)intersectionPointsOfCircleWithCenter:(CGPoint)center withRadius:(float)radius toLinePoint1:(CGPoint)p1 andLinePoint2:(CGPoint)p2
{
NSMutableArray *intersectionPoints = [NSMutableArray array];
float Ax = p1.x;
float Ay = p1.y;
float Bx = p2.x;
float By = p2.y;
float Cx = center.x;
float Cy = center.y;
float R = radius;
// compute the euclidean distance between A and B
float LAB = sqrt( pow(Bx-Ax, 2)+pow(By-Ay, 2) );
// compute the direction vector D from A to B
float Dx = (Bx-Ax)/LAB;
float Dy = (By-Ay)/LAB;
// Now the line equation is x = Dx*t + Ax, y = Dy*t + Ay with 0 <= t <= 1.
// compute the value t of the closest point to the circle center (Cx, Cy)
float t = Dx*(Cx-Ax) + Dy*(Cy-Ay);
// This is the projection of C on the line from A to B.
// compute the coordinates of the point E on line and closest to C
float Ex = t*Dx+Ax;
float Ey = t*Dy+Ay;
// compute the euclidean distance from E to C
float LEC = sqrt( pow(Ex-Cx, 2)+ pow(Ey-Cy, 2) );
// test if the line intersects the circle
if( LEC < R )
{
// compute distance from t to circle intersection point
float dt = sqrt( pow(R, 2) - pow(LEC,2) );
// compute first intersection point
float Fx = (t-dt)*Dx + Ax;
float Fy = (t-dt)*Dy + Ay;
// compute second intersection point
float Gx = (t+dt)*Dx + Ax;
float Gy = (t+dt)*Dy + Ay;
[intersectionPoints addObject:[NSValue valueWithCGPoint:CGPointMake(Fx, Fy)]];
[intersectionPoints addObject:[NSValue valueWithCGPoint:CGPointMake(Gx, Gy)]];
}
// else test if the line is tangent to circle
else if( LEC == R ) {
// tangent point to circle is E
[intersectionPoints addObject:[NSValue valueWithCGPoint:CGPointMake(Ex, Ey)]];
}
else {
// line doesn't touch circle
}
return intersectionPoints;
}
You can find a point on a infinite line that is nearest to circle center by projecting vector AC onto vector AB. Calculate the distance between that point and circle center. If it is greater that R, there is no intersection. If the distance is equal to R, line is a tangent of the circle and the point nearest to circle center is actually the intersection point. If distance less that R, then there are 2 intersection points. They lie at the same distance from the point nearest to circle center. That distance can easily be calculated using Pythagorean theorem. Here's algorithm in pseudocode:
{
dX = bX - aX;
dY = bY - aY;
if ((dX == 0) && (dY == 0))
{
// A and B are the same points, no way to calculate intersection
return;
}
dl = (dX * dX + dY * dY);
t = ((cX - aX) * dX + (cY - aY) * dY) / dl;
// point on a line nearest to circle center
nearestX = aX + t * dX;
nearestY = aY + t * dY;
dist = point_dist(nearestX, nearestY, cX, cY);
if (dist == R)
{
// line segment touches circle; one intersection point
iX = nearestX;
iY = nearestY;
if (t < 0 || t > 1)
{
// intersection point is not actually within line segment
}
}
else if (dist < R)
{
// two possible intersection points
dt = sqrt(R * R - dist * dist) / sqrt(dl);
// intersection point nearest to A
t1 = t - dt;
i1X = aX + t1 * dX;
i1Y = aY + t1 * dY;
if (t1 < 0 || t1 > 1)
{
// intersection point is not actually within line segment
}
// intersection point farthest from A
t2 = t + dt;
i2X = aX + t2 * dX;
i2Y = aY + t2 * dY;
if (t2 < 0 || t2 > 1)
{
// intersection point is not actually within line segment
}
}
else
{
// no intersection
}
}
编辑:增加了代码来检查所找到的交点是否实际上在线段内。
基于@Joe Skeen的python解决方案
def check_line_segment_circle_intersection(line, point, radious):
""" Checks whether a point intersects with a line defined by two points.
A `point` is list with two values: [2, 3]
A `line` is list with two points: [point1, point2]
"""
line_distance = distance(line[0], line[1])
distance_start_to_point = distance(line[0], point)
distance_end_to_point = distance(line[1], point)
if (distance_start_to_point <= radious or distance_end_to_point <= radious):
return True
# angle between line and point with law of cosines
numerator = (math.pow(distance_start_to_point, 2)
+ math.pow(line_distance, 2)
- math.pow(distance_end_to_point, 2))
denominator = 2 * distance_start_to_point * line_distance
ratio = numerator / denominator
ratio = ratio if ratio <= 1 else 1 # To account for float errors
ratio = ratio if ratio >= -1 else -1 # To account for float errors
angle = math.acos(ratio)
# distance from the point to the line with sin projection
distance_line_to_point = math.sin(angle) * distance_start_to_point
if distance_line_to_point <= radious:
point_projection_in_line = math.cos(angle) * distance_start_to_point
# Intersection occurs whent the point projection in the line is less
# than the line distance and positive
return point_projection_in_line <= line_distance and point_projection_in_line >= 0
return False
def distance(point1, point2):
return math.sqrt(
math.pow(point1[1] - point2[1], 2) +
math.pow(point1[0] - point2[0], 2)
)
