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Description
Once again, lack of a good title.
Four isosurfaces, in pairs that are 180 degrees out of phase. Nothing in this scene modeled by hand, it's all equations that I spent forever working with.
For the curious and mathematically oriented:
f(x,y,z) = sqrt((ay - b sin(cx + d))^2 + (az - b cos (cx + d))^2) . . . . a and b are...uhh...something with radius, c is frequency, d is phase. It's pretty straightforward: the parametric equation for a 3D spiral is (u, sin(u), cos(u)); add in some functional transforms and apply the distance formula. The intention was that I would get a sphere sweep as a result of using the distance formula, but that didn't work out.
Four isosurfaces, in pairs that are 180 degrees out of phase. Nothing in this scene modeled by hand, it's all equations that I spent forever working with.
For the curious and mathematically oriented:
f(x,y,z) = sqrt((ay - b sin(cx + d))^2 + (az - b cos (cx + d))^2) . . . . a and b are...uhh...something with radius, c is frequency, d is phase. It's pretty straightforward: the parametric equation for a 3D spiral is (u, sin(u), cos(u)); add in some functional transforms and apply the distance formula. The intention was that I would get a sphere sweep as a result of using the distance formula, but that didn't work out.
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Comments4
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Yeah, it's POV-Ray 3.6.1, on my flashy new computer that can render a LOT faster than a $15 dumpster Celeron. But, alas, that $15 computer died in the middle of the night and I smashed it very violently.
The equation part may not make sense until you read about how isosurfaces work, in POV-Ray or otherwise... you have a function of 3 variables, and it will have a value for any point in the 3D space... if the function evaluated at a point is equal to a certain threshold, the point is on the surface; if less than, it's outside; if greater than, it's inside.
The parametric equation (u, sin(u), cos(u)) gives a simple helix along the X axis - but it's just a 2D line wrapped around and around, basically. To get anywhere, you need to get some kind of value for any point in 3D space. So... take my oddball approach... and use the distance formula to find the distance between the given point and whatever part of the spiral happens to have the same X coordinate. Why this works, I'm not sure.
Not explained well enough? Probably not, I don't know what I'm talking about beyond this.
The equation part may not make sense until you read about how isosurfaces work, in POV-Ray or otherwise... you have a function of 3 variables, and it will have a value for any point in the 3D space... if the function evaluated at a point is equal to a certain threshold, the point is on the surface; if less than, it's outside; if greater than, it's inside.
The parametric equation (u, sin(u), cos(u)) gives a simple helix along the X axis - but it's just a 2D line wrapped around and around, basically. To get anywhere, you need to get some kind of value for any point in 3D space. So... take my oddball approach... and use the distance formula to find the distance between the given point and whatever part of the spiral happens to have the same X coordinate. Why this works, I'm not sure.
Not explained well enough? Probably not, I don't know what I'm talking about beyond this.