On Sat, 2004-02-28 at 16:51, Hiram Berry wrote:
One thing has puzzled me thoughout, however, why is the slice generating formulae (ie. SliceJulibrot1..3) so complicated? AFAICT it uses direction cosines to take a 2D slice of an R4 space. Isn't the {Z0,C} Julibrot space really just a two dimensional phase space of the process Z->Z^2+C? Your formalae appear to map C2 to R4, then take an R2 slice of it. It seems to me that it would be simpler to work natively in the C2 space, take a C1 slice and parameterize the viewing area directly by the complex number contained in the "pixel" variable-- IOW use complex quantities directly, both in the p1..p5 and in the calculations. It looks to me like most of the geometric properties of the Euclidean plane translate to the C2 plane, including the transcendental functions, so that taking an arbitrary slice of the Julibrot is just the drawing of a C1 "line" curve with the parameter of variation equal to "pixel". Each of these curves has a unique point which either is the Julibrot origin or has the property that a vector drawn from that origin to the point will be orthogonal to a vector drawn from that point to any other point on the curve, just as in the Euclidean plane, so the slices can be uniquely specified and I think this approach allows all rather than most slices to be drawn (within the bounds of computational range and precision).
Hiram, Your approach to specifying a slice of a 4D object is very interesting. I'm not really able to say whether it works or not - my geometry's a bit hazy when it comes to this sort of thing. However it seems to me that you don't have enough parameters to specify any view in R4. I think you have p1, p2 and (implicitly) #pixel = 6 real parameters. However it takes 6 real parameters to uniquely specify a location + an orientation in 3D (or R3, as you put it) (x,y,z,yaw,pitch,roll). So I can't see how 6 would be sufficient in R4 - surely more information is required. Does your C2-based disallow some orientations which are possible in R4? FWIW I specify an image in R4 with 10 real parameters: (x,y,z,w) coords or the image center and angles around the xy, xz, xw, yz, yw and zw planes. The angles aren't all that intuitive so I'd be interested to find a more intuitively graspable way to specify a view. PS http://research.microsoft.com/~hollasch/thesis/ lists another method of specifying a view of an R4 object, using 4D "from" and "to" points and 2 4D "up" and "over" vectors. Regards, -- Edwin