Into the Heart of the Mandelbulb

[kram1032]

when the logarithmic zoom was mentioned, something different came into my mind:

what about letting the camera follow 3D lines of magnetic flux, ie let it kinda be a particle with direction that is affected by magnetism...

Afaik, the simplest way to colour the Mandelbrot actually shows how the magnetic field around it would look like...

[David Makin]

when the logarithmic zoom was mentioned, something different came into my mind:

what about letting the camera follow 3D lines of magnetic flux, ie let it kinda be a particle with direction that is affected by magnetism...

Afaik, the simplest way to colour the Mandelbrot actually shows how the magnetic field around it would look like...

Good idea, but getting the field lines analytically is tricky - even on a standard complex 2D fractal

One way to do it would be have the camera simply follow the path of greatest iteration density.

[iq]

twinbee, you can probably use this code for spline camera interpolation: http://iquilezles.org/www/articles/minispline/minispline.htm  Well, for camera, light position, shading parameters, anything you want to animate over time. It's pretty easy tu use, much more than Bezier curves, and is probably enough to build simple animations. I copy the code here anyway, as it's very short:

Code:

static signed char coefs[16] = {
    -1, 2,-1, 0,
     3,-5, 0, 2,
    -3, 4, 1, 0,
     1,-1, 0, 0 };

void spline( const float *key, int num, int dim, float t, float *v )
{
    const int size = dim + 1;

    // find key
    int k = 0; while( key[k*size] < t ) k++;

    // interpolant
    const float h = (t-key[(k-1)*size])/(key[k*size]-key[(k-1)*size]);

    // init result
    for( int i=0; i < dim; i++ ) v[i] = 0.0f;

    // add basis functions
    for( int i=0; i<4; i++ )
    {
        int kn = k+i-2; if( kn<0 ) kn=0; else if( kn>(num-1) ) kn=num-1;

        const signed short *co = coefs + 4*i;

        const float b  = 0.5f*(((co[0]*h + co[1])*h + co[2])*h + co[3]);

        for( int j=0; j < dim; j++ ) v[j] += b * key[kn*size+j+1];
    }
}

"key" contains the time-data keyframes. Data can be 1D, 2D, 3D, 4D or any D you want. For cameras you probably want 6D, something like positionX, positionY, positionZ, lookAtPointX, lookAtPointY, lookAtPointZ, or a couple of 3D splines, dunno. Your global illumination algorithms seems pretty robust regarding different scale factors, but if you with to modify any parameter as you move deep into the fractal (like, more light bounces, or more contrast) you can bake that info in a spline too.

[twinbee]

Thanks for that code iq. It looks as though the Catmull-Rom spline has some useful properties including:

a: local control; only nearest part of curve (within 2 control points either side) is affected.
b: curves pass through points exactly.

Both are great for precise camera control. I suppose one thing it may be lacking is controlling the 'sharpness' of the curve (which I think NURBs allow), but one can always add more control points.

It always confuses me why traditional 2D paint programs such as Photoshop still use only Bezier curves, when there are probably better options available.

[Snakehand]

twinbee, you can probably use this code for spline camera interpolation.....
 For cameras you probably want 6D, something like positionX, positionY, positionZ, lookAtPointX, lookAtPointY, lookAtPointZ, or a couple of 3D splines, dunno...

For camera interpolation, I would suggest using quaternion, and SLERP interpolation. http://en.wikipedia.org/wiki/Slerp This should give you the necessary control of the camera, and steers clear of numerous common problems. At any rate this is what I will use when I get around to writing camera control. Those splines look neat though, I might try them out on a problem I have encountered at work  (thanks)

[kram1032]

afaik, there is some kind of extension of cardinal-splines where you can affect the roundness, bias and tension of the curves. It's kinda based on manipulating the gradient of the spline...

formulae aren't too complex but I don't remember the exact name so I can't find them