88 lines
4 KiB
Markdown
88 lines
4 KiB
Markdown
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geometry: margin=2cm
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output: pdf_document
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---
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# Raycaster
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Determining the viewing window for the raycaster for this assignment involved
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creating a "virtual" screen in world coordinates, mapping image pixels into that
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virtual screen, and then casting a ray through each pixel's world coordinate to
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see where it would intersect objects.
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### Creating a virtual screen
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The virtual screen is determined first using the eye's position and where it's
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looking. This gives us a single 3d vector, but it doesn't give us a 2d screen in
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the world. This is where the field of view (FOV) comes in; the FOV determines
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how many degrees the screen should take up.
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{width=180px}
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Changing the angle of the field of view would result in a wider or narrower
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screen, which when paired with the aspect ratio (width / height), would produce
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a bigger or smaller viewing screen, like the orange box in the above diagram
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shows. Simply put, FOV affects how _much_ of the frame you're able to see.
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Curiously, distance from the eye actually doesn't really affect the viewing
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screen very much. The reason is the screen is only used to determine how to
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project rays. As the two black rectangles in the diagram above demonstrates,
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changing the distance would still allow the viewer to see the same amount of the
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scene. (using the word _amount_ very loosely here to mean percentage of the
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landscape, rather than # of pixels, which is determined by the actual image
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dimensions)
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The up-direction vector controls the rotation of the scene. Without the
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up-direction, it would not be possible to tell which rotation the screen should
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be in:
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{width=240px}
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Together, all of these parameters can uniquely determine a virtual screen
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location, that we can use to cast rays through and fill pixels. We can change
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any of these to produce an image with a more exaggerated view of the scene for
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example; simply move the eye position to be incredibly close to the object that
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we are observing, and increase the field of view to cover the entire object.
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Because the rays are going in much different directions and travelling different
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distances, the corners of the image will seem more stretched than if we were
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observing the object from afar and all the rays are in approximately the same
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part of the virtual screen.
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One other point to make is that we're currently using a rectangle for our
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virtual screen, which automatically does a bit of the distortion. If instead we
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were to use a curved lens-like shape, then the rays pointing to any pixel of the
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screen would be travelling the same distance. Moving the eye position closer to
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the object would still generate distortion, but to a lesser extent.
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### Mapping image pixels
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After the rectangle has been determined, we can simply pick one corner to start
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as an anchor, and then find out what pixel values would correspond to it. For
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example, in the image below:
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{width=240px}
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I would pick a starting point like $A$, and then take the vector $B-A$ and
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subdivide it into 4 pieces, letting $\Delta x = \frac{B-A}{4}$. Then, same thing
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for the $y$ direction, I would set $\Delta y = \frac{D-A}{4}$. Taking $A +
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x_i \times \Delta x + y_i \times \Delta y$ yields the precise coordinate
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location for any pixel.
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(Technically really we would want the middle of the pixel, so just add
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$\frac{\Delta x + \Delta y}{2}$ to the point to get that)
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### Parallel Projection Notes
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Because of the way I implemented parallel projection, it's recommended to use
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`--distance` to force a much bigger distance from the eye for the raycaster.
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This is due to the size of the image.
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### Cylinder Intersection Notes
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First, we will transform the current point into the vector space of the
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cylinder, so that the cylinder location is $(0, 0, 0)$ and the direction vector
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is normalized into $(0, 0, 1)$.
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Then it's a matter of determining if the $x$ and $y$ coordinates fall into the
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space constrained by the equation $x^2 + y^2 = r^2$ and if $z \le L$.
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