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@ -1,15 +1,14 @@
---
slug: 2025/03/playing-with-fire-camera
title: "Playing with fire: The camera"
date: 2025-03-07 12:00:00
date: 2025-03-10 12:00:00
authors: [bspeice]
tags: []
---
Something that bugged me while writing the first three articles on fractal flames were the constraints on
output images. At the time, I had worked out how to render fractal flames by studying
the source code of [Apophysis](https://sourceforge.net/projects/apophysis/)
and [flam3](https://github.com/scottdraves/flam3). That was just enough to define a basic camera for displaying
[Apophysis](https://sourceforge.net/projects/apophysis/) and [flam3](https://github.com/scottdraves/flam3); just enough to display images
in a browser.
Having spent more time with fractal flames and computer graphics, it's time to implement
@ -26,33 +25,30 @@ To review, the restrictions we've had so far:
>
> -- [The fractal flame algorithm](/2024/11/playing-with-fire)
There are a couple problems here:
First, we've assumed that fractals get displayed in a square image. Ignoring aspect ratios simplifies
the render process, but we don't usually want square images. It's possible to render a large
square image and crop it to fit, but we'd rather render directly to the desired size.
First, the assumption that fractals get displayed in a square image. Ignoring aspect ratios simplifies
the render process, but we usually don't want square images. As a workaround, you could render
a large square image and crop it to fit an aspect ratio, but it's better to render the desired
image size to start with.
Second, the assumption that fractals use the range $[0, 1]$. My statement above is an over-simplification;
for Sierpinski's Gasket, the solution set is indeed defined on $[0, 1]$, but all other images in the series
use a display range of $[-2, 2]$.
Second, we've assumed that fractals have a pre-determined display range. For Sierpinski's Gasket,
that was $[0, 1]$ (the reference parameters used a range of $[-2, 2]$). However, if we could
control the display range, it would let us zoom in and out of the image.
## Parameters
For comparison, here are the camera controls available in Apophysis and [`flam3`](https://github.com/scottdraves/flam3/wiki/XML-File-Format):
For comparison, the camera controls available in Apophysis offer a lot of flexibility:
<center>![Screenshot of Apophysis camera controls](./camera-controls.png)</center>
There are four parameters yet to implement: position, rotation, zoom, and scale.
The remaining parameters to implement are: position (X and Y), rotation, zoom, and scale.
### Position
Fractal flames normally use the origin as the center of an image. The position parameters (X and Y) move
the center point, which effectively pans the image. A positive X position shifts the image left,
and a negative X position shifts the image right. Similarly, a positive Y position shifts the image up,
and a negative Y position shifts the image down.
Fractal flames normally use $(0, 0)$ as the image center. The position parameters (X and Y) move
the center point, which effectively pans the image. A positive X position shifts left, and a
negative X position shifts right. Similarly, a positive Y position shifts up, and a negative
Y position shifts the image down.
To apply the position, simply subtract the X and Y position from each point in the chaos game prior to plotting it:
To apply the position parameters, simply subtract them from each point in the chaos game prior to plotting it:
```typescript
[x, y] = [
@ -63,9 +59,9 @@ To apply the position, simply subtract the X and Y position from each point in t
### Rotation
After the position parameters are applied, we can rotate the image around the (potentially shifted) center point.
To do so, we'll go back to the [affine transformations](https://en.wikipedia.org/wiki/Affine_transformation)
we've been using. Specifically, the rotation angle $\theta$ gives us a transform matrix we can apply to our point:
After the position parameters, we can rotate the image around the (new) center point. To do so, we'll go back to the
[affine transformations](https://en.wikipedia.org/wiki/Affine_transformation) we've been using so far.
Specifically, the rotation angle $\theta$ gives us a transform matrix we can apply prior to plotting:
$$
\begin{bmatrix}
@ -79,7 +75,6 @@ y
\end{bmatrix}
$$
As a minor tweak, we also negate the rotation angle to match the behavior of Apophysis/`flam3`.
```typescript
[x, y] = [
@ -90,11 +85,17 @@ As a minor tweak, we also negate the rotation angle to match the behavior of Apo
];
```
:::note
To match the behavior of Apophysis/`flam3`, we need to negate the rotation angle.
:::
### Zoom
This parameter does what the name implies; zoom in and out of the image. Specifically, for a zoom parameter $z$,
every point in the chaos game is scaled by $\text{pow}(2, z)$ prior to plotting. For example, if the point is $(1, 1)$,
a zoom of 1 means we actually plot $(1, 1) \cdot \text{pow}(2, 1) = (2, 2)$.
This parameter does what the name implies; zoom in and out of the image. To do this, we multiply
the X and Y coordinates of each point by a zoom factor. For a zoom parameter $z$, the zoom factor
will be $\text{pow}(2, z)$.
For example, if the current point is $(1, 1)$, a zoom parameter of 1 means we actually plot $(1, 1) \cdot \text{pow}(2, 1) = (2, 2)$.
```
[x, y] = [
@ -105,25 +106,32 @@ a zoom of 1 means we actually plot $(1, 1) \cdot \text{pow}(2, 1) = (2, 2)$.
:::info
In addition to scaling the image, renderers also [scale the image quality](https://github.com/scottdraves/flam3/blob/f8b6c782012e4d922ef2cc2f0c2686b612c32504/rect.c#L796-L797)
to compensate for the zoom parameter.
to compensate for the reduced display range.
:::
### Scale
Finally, we need to convert from fractal flame coordinates to individual pixels. The scale parameter defines
how many pixels are in one unit of the fractal flame coordinate system. For example, if you open the
[reference parameters](../params.flame) in a text editor, you'll see the following:
how many pixels are in one unit of the fractal flame coordinate system, which gives us a mapping from one system
to the other.
If you open the [reference parameters](../params.flame) in a text editor, you'll see the following:
```xml
<flame name="final xform" size="600 600" center="0 0" scale="150">
```
This says that the final image should be 600 pixels wide and 600 pixels tall, centered at the point $(0, 0)$,
with 150 pixels per unit. Dividing 600 by 150 gives us an image that is 4 units wide and 4 units tall.
And because the center is at $(0, 0)$, the final image is effectively looking at the range $[-2, 2]$ in the
fractal coordinate system (as mentioned above).
Here's what each element means:
To go from the fractal coordinate system to a pixel coordinate system, we multiply by the scale,
- `size="600 600"`: The image should be 600 pixels wide and 600 pixels tall
- `center="0 0"`: The image is centered at the point $(0, 0)$
- `scale="150"`: The image has 150 pixels per unit
Let's break it down. Dividing the image width (600) by the image scale (150) gives us a value of 4.
This means the image should be 4 units wide (same for the height). Because the center is at $(0, 0)$,
the final image is effectively using the range $[-2, 2]$ in fractal coordinates.
Now, to go from fractal coordinates to pixel coordinates we multiply by the scale,
then subtract half the image width and height:
```typescript
@ -133,22 +141,23 @@ then subtract half the image width and height:
]
```
Scale can be used to implement a kind of "zoom" in images. If the reference parameters instead used `scale="300"`,
Scale and zoom have similar effects on images. If the reference parameters used `scale="300"`,
the same 600 pixels would instead be looking at the range $[-1, 1]$ in the fractal coordinate system.
Using `zoom="1"` would accomplish the same result.
However, this also demonstrates the biggest problem with using scale: it's a parameter that only controls the output image.
If the output image changed to `size="1200 1200"` and we kept `scale="150"`, the output image would
be looking at the range $[-4, 4]$ - nothing but white space. Because, using the zoom parameter
is the preferred way to zoom in and out of an image.
However, this also demonstrates the biggest problem with using scale: it only controls the output image.
For example, if the output image changed to `size="1200 1200"` and we kept `scale="150"`, it would
have a display range of $[-4, 4]$. There would be a lot of extra white space. Because the zoom parameter
has the same effect regardless of output image size, it is the preferred way to zoom in and out.
:::info
One final note about the camera controls: every step in this process (position, rotation, zoom, scale)
is an affine transformation. And because affine transformations can be chained together, it's possible to
express all of our camera controls as a single transformation matrix. This is important for software optimization;
rather than applying individual camera controls step-by-step, apply all of them at once.
express all the camera controls as a single transformation matrix. This is important for software optimization;
rather than applying parameters step-by-step, we can apply all of them at once.
Additionally, because the camera controls are an affine transformation, they could be implemented
as a transform after the final transform. In practice though, it's helpful to control them separately.
They could also be implemented as part of the final transform, but in practice, it's helpful
to control them separately.
:::
## Camera
@ -160,9 +169,9 @@ import cameraSource from "!!raw-loader!./camera"
<CodeBlock language="typescript">{cameraSource}</CodeBlock>
For demonstration, the output image has a 4:3 aspect ratio, removing the previous restriction of a square image.
In addition, the scale is automatically chosen to make sure the width of the image covers the range $[-2, 2]$.
As a result of the aspect ratio, the image height effectively covers the range $[-1.5, 1.5]$.
To demonstrate, this display has a 4:3 aspect ratio, removing the restriction of a square image.
In addition, the scale is automatically chosen so the image width covers the range $[-2, 2]$.
Because of the 4:3 aspect ratio, the image height now covers the range $[-1.5, 1.5]$.
import {SquareCanvas} from "../src/Canvas";
import FlameCamera from "./FlameCamera";
@ -171,9 +180,11 @@ import FlameCamera from "./FlameCamera";
## Summary
The fractal images so far relied on critical assumptions about the output format to make sure everything
looked correct. However, we can implement a 2D "camera" with a series of affine transformations - going from
the fractal flame coordinate system to pixel coordinates. Later implementations of fractal flame renderers like
[Fractorium](http://fractorium.com/) operate in 3D, and have to implement the camera slightly differently.
The previous fractal images relied on assumptions about the output format to make sure they looked correct.
Now, we have much more control. We can implement a 2D "camera" as a series of affine transformations, going from
fractal flame coordinates to pixel coordinates.
But for this blog series, it's nice to achieve feature parity with the existing code.
More recent fractal flame renderers like [Fractorium](http://fractorium.com/) can also operate in 3D,
and have to implement a more complex camera system to handle the extra dimension.
But for this blog series, it's nice to achieve feature parity with the reference implementation.