Counting the number of intersection points in  the diagram is a little bit trickier.

计算图中交数量要棘手一些。

Set the intersection points equal, the equations equal to each other and solve.

学生：使交相等，方程式相等并求解。

Then compare these two and find the intersection point, which is the determined opportunity point.

然后比较这两者，找出它们交集，这交集就是确定会。

No other choice of two intersection points is guaranteed to get all the robo-ants within five minutes.

没有其他选择两交可以保证在五分钟内得到所有蚂蚁。

Every intersection point corresponds  with some quadruplet of points.

交对应于某四组合。

The one that I landed on just because I feel like both of these teams are just at interesting intersection points.

我最终选择这，是因为我觉得这两支球队正处在有趣交汇上。

But the key is to treat this as a new graph  where those intersection points are them vertices.

但关键是要将这视为一新图，其中那些交就是顶。

So, in effect,   each intersection point adds two more edges.

因此，交实际上增加了两条边。

I claim that that line also passes through the intersection point between the external tangents of the circles that we care about.

我声称那条线也经过我们关心圆外切线交。

Now that we have a set of intersection points on separate planes, we find whether the point is inside each corresponding triangle.

现在我们有一组位于不同平面上交，我们需要判断这些是否在各自对应三角形内部。

One idea might be that it should be the number of pairs  of chords since every intersection point comes from two different chords.

一想法可能是它应该是弦对数，因为交都来自两条不同弦。

So the total number of vertices here  would not be N but N plus N choose four total intersection points.

所以这里顶总数不是N，而是N加上n选4总交数。

For example, look at this simple diagram where  we have three lines and two intersection points.

例如，看这简单示意图，我们有三条线和两交。

The claim, the theorem, the thing you need to prove is that those three intersection points of the external tangents always fall on the same line.

你需要证明命题是那些三条外切线交总是落在同一条直线上。

Maybe if you're a little bit lucky,   here's one thing you might notice; every intersection point is uniquely associated  with a quadruplet of points on the exterior.

也许如果你有幸运， 你会注意到一件事；交都唯一地与圆外四相关联。

For example, if you calculate 4 choose 4,   everything cancels and you just get one and indeed, there is a single intersection point  in this diagram.

例如，如果你计算4选4， 所有项都会约掉， 最后得到1，确实，在这图中只有一交。

You might mentally imagine turning them with a little knob to find the sweet spot angle where all those intersection points line up.

你可能会在心里想象用一小旋钮来调整它们，找到一所有交都对齐最佳角度。

If there are no objects between the intersection point and a light source, then that means that the point on the blue stripe is directly illuminated by that light source.

如果交和光源之间没有其他物体，那么这意味着蓝色条纹上那是直接被该光源照亮。

This light is called indirect illumination, and in order to calculate it, we take the intersection point from the primary ray and generate a secondary ray that bounces off it.

这种光线被称为间接照明，为了计算它， 我们从主光线交生成一二次光线， 该二次光线从该交弹出。

And then similarly, if you look at the lines passing through the tips of any other pair of cones, those lines will correspond to another one of those intersection points.

然后类似地，如果你看任意另一对圆锥顶所在直线，这些直线将对应另一交。