Also, it is computationally quite effective and can eliminate these lengthy pre-computation steps that are typically present in previous techniques.

同时，从计算方面上来讲这也相当有效率，能够省去漫长的计算前的步骤，这是存在于先前技术中的典型问题。

It's all been about creating a language for expressing ourselves computationally, a language for computational thinking.

这一切都是为了创建一种用于计算表达我们自己的语言，一种用于计算思维的语言。

In the ruliad as a whole, in a sense, everything computationally possible can happen.

在整个 ruliad 中，从某种意义上说，一切计算上可能发生的事情都可能发生。

It will follow that preset pattern and computationally find anything that's changed or appeared.

它将遵循该预设模式并通过计算找到任何已更改或出现的内容。

So the answer that we found earlier computationally, using that special notational trick, must correspond geometrically to this vector.

所以我们之前计算得到的答案 用那个特殊的符号技巧 一定在几何上对应于这个向量。

What we ideally want is a telescope that can systematically and computationally do what Oscar did with his mind.

我们理想中想要的是一台能够系统地和计算地完成奥斯卡用他的思想所做的事情的望远镜。

Overall Path Tracing's computationally impossible problems are solved by using bounding volume  hierarchies along with improvements in GPU hardware.

总体而言，路径追踪的计算上不可能的问题是通过使用边界体积层次结构以及 GPU 硬件的改进来解决的。

And computational models that resolve all of these scales, from the microscale all the way up to the system level, are computationally intractable.

解决所有这些尺度（从微观尺度一直到系统级别）的计算模型在计算上非常困难。

However, when a scene has millions of triangles, finding which one triangle  a single ray hits first becomes incredibly repetitive and computationally problematic.

然而， 当场景有数百万个三角形时，找到单条射线首先击中哪个三角形会变得非常重复且计算困难。

First, we're computationally bounded, our minds are limited, and second, we believe we're persistent in time, even though we're made of different atoms of space at every moment.

首先，我们的计算能力是有限的，我们的思维是有限的，其次，我们相信我们在时间上是持久的，尽管我们每时每刻都是由不同的空间原子组成的。

And computing this projection transformation for arbitrary vectors in space, which requires multiplying that matrix by those vectors, is computationally identical to taking a dot product with you hat.

计算空间中任意向量的投影变换 需要用这些向量乘以这个矩阵 在计算上和做点积是一样的。

In this way, a matrix represents a specific linear transformation, and multiplying a matrix by a vector is what it means computationally to apply that transformation to that vector.

这样 一个矩阵表示一个特定的线性变换 用一个矩阵乘以一个向量在计算上就是把这个变换应用到那个向量上。

And multiplying this matrix by some vector V is computationally identical to taking the dot product between V and the vector you get by turning that matrix on its side.

这个矩阵乘以某个向量V在计算上是一样的V和向量的点积你把这个矩阵翻到一边。

Furthermore,   animations typically have 24 frames a second, so a 20-minute animation requires over a  quadrillion rays, and that's why path tracing was considered computationally impossible for  TV shows and movies for decades.

此外， 动画通常每秒有 24 帧，因此 20 分钟的动画需要超过一千万亿条射线，这就是为什么几十年来电视节目和电影中的路径追踪被认为在计算上是不可能的。