我需要一个基本函数来求点到线段的最短距离。你可以随意用任何你想要的语言来编写解决方案;我可以把它翻译成什么我正在使用(Javascript)。
编辑:线段由两个端点定义。线段AB由两点A (x1,y1)和B (x2,y2)定义。我要求的是这条线段到点C (x3,y3)的距离。我的几何技能生疏了,所以我看到的例子让我很困惑,我很遗憾地承认。
我需要一个基本函数来求点到线段的最短距离。你可以随意用任何你想要的语言来编写解决方案;我可以把它翻译成什么我正在使用(Javascript)。
编辑:线段由两个端点定义。线段AB由两点A (x1,y1)和B (x2,y2)定义。我要求的是这条线段到点C (x3,y3)的距离。我的几何技能生疏了,所以我看到的例子让我很困惑,我很遗憾地承认。
当前回答
下面是HSQLDB的SQL实现:
CREATE FUNCTION dist_to_segment(px double, py double, vx double, vy double, wx double, wy double)
RETURNS double
BEGIN atomic
declare l2 double;
declare t double;
declare nx double;
declare ny double;
set l2 =(vx - wx)*(vx - wx) + (vy - wy)*(vy - wy);
IF l2 = 0 THEN
RETURN sqrt((vx - px)*(vx - px) + (vy - py)*(vy - py));
ELSE
set t = ((px - vx) * (wx - vx) + (py - vy) * (wy - vy)) / l2;
set t = GREATEST(0, LEAST(1, t));
set nx=vx + t * (wx - vx);
set ny=vy + t * (wy - vy);
RETURN sqrt((nx - px)*(nx - px) + (ny - py)*(ny - py));
END IF;
END;
Postgres的实现:
CREATE FUNCTION dist_to_segment(px numeric, py numeric, vx numeric, vy numeric, wx numeric, wy numeric)
RETURNS numeric
AS $$
declare l2 numeric;
declare t numeric;
declare nx numeric;
declare ny numeric;
BEGIN
l2 := (vx - wx)*(vx - wx) + (vy - wy)*(vy - wy);
IF l2 = 0 THEN
RETURN sqrt((vx - px)*(vx - px) + (vy - py)*(vy - py));
ELSE
t := ((px - vx) * (wx - vx) + (py - vy) * (wy - vy)) / l2;
t := GREATEST(0, LEAST(1, t));
nx := vx + t * (wx - vx);
ny := vy + t * (wy - vy);
RETURN sqrt((nx - px)*(nx - px) + (ny - py)*(ny - py));
END IF;
END;
$$ LANGUAGE plpgsql;
其他回答
在f#中,点c到a和b之间的线段的距离为:
let pointToLineSegmentDistance (a: Vector, b: Vector) (c: Vector) =
let d = b - a
let s = d.Length
let lambda = (c - a) * d / s
let p = (lambda |> max 0.0 |> min s) * d / s
(a + p - c).Length
向量d沿着线段从a指向b。d/s与c-a的点积给出了无限直线与点c之间最接近点的参数。使用min和max函数将该参数钳制到范围0..s,使该点位于a和b之间。最后,a+p-c的长度是c到线段上最近点的距离。
使用示例:
pointToLineSegmentDistance (Vector(0.0, 0.0), Vector(1.0, 0.0)) (Vector(-1.0, 1.0))
忍不住用python来编码:)
from math import sqrt, fabs
def pdis(a, b, c):
t = b[0]-a[0], b[1]-a[1] # Vector ab
dd = sqrt(t[0]**2+t[1]**2) # Length of ab
t = t[0]/dd, t[1]/dd # unit vector of ab
n = -t[1], t[0] # normal unit vector to ab
ac = c[0]-a[0], c[1]-a[1] # vector ac
return fabs(ac[0]*n[0]+ac[1]*n[1]) # Projection of ac to n (the minimum distance)
print pdis((1,1), (2,2), (2,0)) # Example (answer is 1.414)
fortran也是一样:)
real function pdis(a, b, c)
real, dimension(0:1), intent(in) :: a, b, c
real, dimension(0:1) :: t, n, ac
real :: dd
t = b - a ! Vector ab
dd = sqrt(t(0)**2+t(1)**2) ! Length of ab
t = t/dd ! unit vector of ab
n = (/-t(1), t(0)/) ! normal unit vector to ab
ac = c - a ! vector ac
pdis = abs(ac(0)*n(0)+ac(1)*n(1)) ! Projection of ac to n (the minimum distance)
end function pdis
program test
print *, pdis((/1.0,1.0/), (/2.0,2.0/), (/2.0,0.0/)) ! Example (answer is 1.414)
end program test
这里它使用Swift
/* Distance from a point (p1) to line l1 l2 */
func distanceFromPoint(p: CGPoint, toLineSegment l1: CGPoint, and l2: CGPoint) -> CGFloat {
let A = p.x - l1.x
let B = p.y - l1.y
let C = l2.x - l1.x
let D = l2.y - l1.y
let dot = A * C + B * D
let len_sq = C * C + D * D
let param = dot / len_sq
var xx, yy: CGFloat
if param < 0 || (l1.x == l2.x && l1.y == l2.y) {
xx = l1.x
yy = l1.y
} else if param > 1 {
xx = l2.x
yy = l2.y
} else {
xx = l1.x + param * C
yy = l1.y + param * D
}
let dx = p.x - xx
let dy = p.y - yy
return sqrt(dx * dx + dy * dy)
}
在数学
它使用线段的参数描述,并将点投影到线段定义的直线中。当参数在线段内从0到1时,如果投影在这个范围之外,我们计算到相应端点的距离,而不是法线到线段的直线。
Clear["Global`*"];
distance[{start_, end_}, pt_] :=
Module[{param},
param = ((pt - start).(end - start))/Norm[end - start]^2; (*parameter. the "."
here means vector product*)
Which[
param < 0, EuclideanDistance[start, pt], (*If outside bounds*)
param > 1, EuclideanDistance[end, pt],
True, EuclideanDistance[pt, start + param (end - start)] (*Normal distance*)
]
];
策划的结果:
Plot3D[distance[{{0, 0}, {1, 0}}, {xp, yp}], {xp, -1, 2}, {yp, -1, 2}]
画出比截断距离更近的点:
等高线图:
这里是与c++答案相同的东西,但移植到pascal。点参数的顺序已经改变,以适应我的代码,但还是一样的东西。
function Dot(const p1, p2: PointF): double;
begin
Result := p1.x * p2.x + p1.y * p2.y;
end;
function SubPoint(const p1, p2: PointF): PointF;
begin
result.x := p1.x - p2.x;
result.y := p1.y - p2.y;
end;
function ShortestDistance2(const p,v,w : PointF) : double;
var
l2,t : double;
projection,tt: PointF;
begin
// Return minimum distance between line segment vw and point p
//l2 := length_squared(v, w); // i.e. |w-v|^2 - avoid a sqrt
l2 := Distance(v,w);
l2 := MPower(l2,2);
if (l2 = 0.0) then begin
result:= Distance(p, v); // v == w case
exit;
end;
// Consider the line extending the segment, parameterized as v + t (w - v).
// We find projection of point p onto the line.
// It falls where t = [(p-v) . (w-v)] / |w-v|^2
t := Dot(SubPoint(p,v),SubPoint(w,v)) / l2;
if (t < 0.0) then begin
result := Distance(p, v); // Beyond the 'v' end of the segment
exit;
end
else if (t > 1.0) then begin
result := Distance(p, w); // Beyond the 'w' end of the segment
exit;
end;
//projection := v + t * (w - v); // Projection falls on the segment
tt.x := v.x + t * (w.x - v.x);
tt.y := v.y + t * (w.y - v.y);
result := Distance(p, tt);
end;