c******l
发帖数: 577 | 1 发信人: mrdi (不穿鞋的刀客终日乾乾，夕惕若，厉无咎。), 信区: Joke
标 题: 请给出点 （x,y,z) 到平面 ax + by + cz = d的距离公式
发信站: Unknown Space - 未名空间 (Tue Jul 20 12:49:22 2004) WWW-POST
10mins | c******l
发帖数: 577 | 2 哈哈哈哈，看来好多人不好好学习。：)
你是对的。用空间几何先求垂直线段表达式，然后求方程组解得交点，最后算距离，怎么
也是个大题，10分钟不一定做得出来。
要搁我现在，懒得算了，直接弄个算法让计算机狂试，求个最短距离就完了。哈哈。
【在 c******l 的大作中提到】
: 发信人: mrdi (不穿鞋的刀客终日乾乾，夕惕若，厉无咎。), 信区: Joke
: 标 题: 请给出点 （x,y,z) 到平面 ax + by + cz = d的距离公式
: 发信站: Unknown Space - 未名空间 (Tue Jul 20 12:49:22 2004) WWW-POST
: 10mins
| c******l
发帖数: 577 | 3
么
BTW, your equation of plane is wrong, if you really mean a "plane", that would
be a tough question. But now, you just give out a line equation, which is much
easier. haha.
【在 c******l 的大作中提到】
: 哈哈哈哈，看来好多人不好好学习。：)
: 你是对的。用空间几何先求垂直线段表达式，然后求方程组解得交点，最后算距离，怎么
: 也是个大题，10分钟不一定做得出来。
: 要搁我现在，懒得算了，直接弄个算法让计算机狂试，求个最短距离就完了。哈哈。
| m**i
发帖数: 724 | 4 其实也不太难。joke已经有人查到答案了。
我马后炮一下。
Let D = sqrt(a^2 + b^2 + c^2), then the plane is
(a, b, c)/D (x, y, z)' = d/D
That is the dot product of a generic point on the plane with the vector (a, b,
c)/D is d/D. Note that now (a, b, c)/D is a unit vector.
Now let (x, y, z) be any point, the the dot product of that (x, y, z) with (a,
b,c)/D is (ax + by + cz)/D.
In fact (a,b,c)/D is the vector that is perpendicular to the plane. So the
distance from (x, y, z) to the plane is
abs((ax + by + cz)/D - d/D).
么
【在 c******l 的大作中提到】
: 哈哈哈哈，看来好多人不好好学习。：)
: 你是对的。用空间几何先求垂直线段表达式，然后求方程组解得交点，最后算距离，怎么
: 也是个大题，10分钟不一定做得出来。
: 要搁我现在，懒得算了，直接弄个算法让计算机狂试，求个最短距离就完了。哈哈。
| m**i
发帖数: 724 | 5 No, it is not wrong.
ax + by + cz =d
means all the points that has dot product d with (a, b, c). In fact, that is
intuitively very clear. If you want a plane, you need a vector that is
perpendicular to it. In this case, it is (a, b, c). This will determine the
direction of the plane. And then you need do decide how far you want to put
this plane from the origin, which is determined by d. :)
Just think about that all the points satisfying this equation is a linear
space of 2 dimensions, which has
【在 c******l 的大作中提到】
:
: 么
: BTW, your equation of plane is wrong, if you really mean a "plane", that would
: be a tough question. But now, you just give out a line equation, which is much
: easier. haha.
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