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Miloslav Ciz 2022-01-14 23:55:13 -06:00
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exercises.md
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@ -5,6 +5,7 @@ Here let be listed exercises for the readers of this wiki. You can allow yoursel
1. What's the difference between [free software](free_software.md) and [open source](open_source.md)?
2. Write a program in [C](c.md) that computes the value of [pi](pi.md) without using float/double and any libraries except for `stdio.h` and `stdint.h` -- you can only use built-in integer types and those from `stdint.h`. The program must compute pi as accurately as possible (at least 2 decimals) and write the value out as base 10 decimal.
3. Say we have an algorithm that finds all pairs of equal numbers in an array of numbers of length *N* and adds all of these (unordered) pairs to a set *S*. The algorithm is: `for i := 0 to N: for j := 0 to N: if numbers[i] == numbers[j]: add(S,set(i,j))`. How can we optimize the algorithm in terms of its execution speed (i.e. make it perform fewer operations)? How did the asymptotic time complexity ("big O") class change?
4. In computer graphics, what is the difference between ray casting, ray tracing and path tracing?
## Solutions
@ -77,4 +78,8 @@ for i := 0 to N:
add(S,set(i,j))
```
While the first algorithm performs N^2 comparisons, the new one only needs N - 1 + N - 2 + N - 3 + ... ~= N * N / 2 = N^2 / 2 comparisons. Even though the new version is always twice as fast, its time complexity class remains the same, that is O(N^2).
While the first algorithm performs N^2 comparisons, the new one only needs N - 1 + N - 2 + N - 3 + ... ~= N * N / 2 = N^2 / 2 comparisons. Even though the new version is always twice as fast, its time complexity class remains the same, that is O(N^2).
**solution 4**:
They are all image-order methods of 3D [rendering](rendering.md). [Ray casting](ray_casting.md) casts a single ray for each screen pixel and determines the pixel color from a single hit of the ray. [Ray tracing](ray_tracing.md) is a [recursive](recursion.md) form of ray casting -- it recursively spawns secondary rays from the first hit to more accurately determine the pixel color, allowing for effects such as shadows, reflections or refractions. Path tracing is a method also based on casting rays, but except for the primary rays the rays are cast at random (i.e. it is a [Monte Carlo](monte_carlo.md) method) to approximately solve the rendering equation, progressively computing a more accurate version of the image (i.e. the image contains significant noise at the beginning which lowers with more iterations performed) -- this allows computing [global illumination](global_illumination.md), i.e. a very realistic lighting that the two previous methods can't achieve.