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

学生：使点相等，方程式相等并求解。

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

因此，实际上，每个点又添加了两条边。

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

然后比较两者，找出它们的集点，个集点就是确定的机会点。

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

有其他选择的两个点可以保证在五分钟内得到所有的机器蚂蚁。

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

计算图中点的数量有点棘手。

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

但关键是将其视为一个新图，其中点是顶点。

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

例如，看看个简单的图表，其中有三条线和两个点。

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

因此，里的顶点总数不是 N，而是 N 加 N 选择四个点。

Every intersection point corresponds  with some quadruplet of points.

每个点都对应于一四组点。

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

您可能会在脑海中想象用一个小旋钮转动它们，找到所有叉点对齐的最佳点角度。

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

一种想法可能是它应该是和弦对的数量，因为每个点都来自两个不同的和弦。

But one way you can think about it is that every intersection point  takes what started as two separate lines and then turns it into four lines.

但你可以考虑的一种方式是，每个叉点都将最初的两条单独的线变成四条线。

But of course that can't happen, whether you're looking at it from one side, or looking at it from the other, those intersection points are all the same.

但当然不可能发生种情况，无论你从一侧看，还是从另一侧看，那点都是一样的。

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.

种光被称为间接照明，为了计算它， 我们取主射线的点并生成从该点反射的次级射线。

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

如果点和光源之间有物体，则意味着蓝色条纹上的该点被该光源直接照亮。

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，如果您计算 4 选择 4， 则所有内容都会取消， 您只得到 1，实际上，该图中只有一个点。

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.

也许如果您幸运一点，您可能会注意到以下一件事：每个点都与外部的四组点唯一关联。

If it was the case that no bending happened, then because the wavelength is smaller in there, when you look at all those intersection points, they would have to be closer together.

如果有发生弯曲，那么因为那里的波长较小，所以当你观察所有叉点时，它们必须靠得更近。

And we'll label that intersection point D.  And then next I will draw the angle bisector at A, which means by definition this little angle here  is the same as this little angle here.

我们将标记该点 D。 然后接下来我将在 A 处绘制角平分线，意味着根据定义， 里的个小角与里的个小角相同。