Ideas off the top of my head:

If you could find one point, say (3,4), you could then try small differences to find the next point, and continue that way following the curve in both directions.

To find the first point you could do a "hill climb" (move in the direction that reduces the error the most), or just a brute-force search dividing the space into (say) 1,000 x 1,000, and then narrowing down further as needed.

Continuous functions would be easiest.

ive figured out a good method that renders the objects perfectly, at arbitrary depth, and works for any implicit function

Basicly: i split the viewing area into equally sized squares.

for each square:

i evaluate the function for each corner, and also whether that point is inside (if the value is less than 0)
if all are inside, or all are outside, i finish here:

then i go through the following 6 cases:

top left is inside ONLY or outside ONLY
top right is inside ONLY or outside ONLY
bottom left is inside ONLY or outside ONLY
bottom right is inside ONLY or outside ONLY
(top is inside AND bottom is outside) OR (top is outside AND bottom is inside)
(left is inside AND right is outside) OR (left is outside AND right is inside)

then for each case, i use the function values to interpolate linearly for where it is assumed f(x,y) would be 0 from the two function values (not perfect, but a good approximation) then same for the other side for whatever case it is, and draw a line.

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the red graph is for the square size of 50pixels (pretty terrible)
the green graph is for the square size of 25pixels (not too bad)
the blue graph is for the square size of 10pixels (almost perfect)
the black graph is for teh square size of 5pixels (perfect for all itensive purposes)

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0.02x³ - 2x² + y² + 2xy - 100 = 0
http://denvish.net/ulf/090807/32088_laser.php
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20cos(0.08y) - 50sin(0.04x) - 0.004xy= 0
http://denvish.net/ulf/090807/32340_laser.php
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http://denvish.net/ulf/090807/33048_laser.php

i absolutely adore the second graph, and third graph is just immense

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thinking about it, this could be directly extended into 3D aswell, only you would have a few more cases to check for, but otherwise it is directly extendable

Last edited by luca-deltodesco (2007-08-09 18:12:48)

the narrowing in of boxes looks cool, but the only reason it 'narrows' in is because i am rendering the graphs at different accuracies, it draws a box whenever it detects that the graph passes through it. so having different accuracies, different sized boxes, they congrugate around where the graph parts are drawn

next i need to make it render the shapes as solids, shouldnt be too hard, only a bit of changing to do (have to seperate teh different cases a bit etc, but shouldnt be too hard)

my god, this is just plain cool:

http://denvish.net/ulf/090807/49336_laserrend.php

i had to make it render at a higher quality to pick up details:

You're a genuous!! Aside from the pun,  what if you set some equations equal to values close to zero, but not exactly, and compare the graphs you get to the zero set equations.
And if they look totally different, then set them even closer to zero until they look similar.