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scott <sco### [at] scott com> wrote:
> His diagram lines are OK, but some of the angles are wrong. Actually so
> long as you know the angles of a hexagon (120 degrees), just use
> Pythagoras on the lower left triangle (the one with hypotenuse L/2).
Yes, apologies for that. It was a quick sketch I was drawing out at the very
end of the day. It should have clicked when I was aware of the difference in
length between the horizontal and vertical catheti - that the angles would be
what I originally hastily labeled them as, but that just didn't happen.
I'm learning more and more from my own experience that it's important to just
put it down for a while and come back to it with fresh eyes and a fresh mind.
It can then take mere seconds to see what I've been looking straight past for
hours.
Good catches, Scott. I had only just begun to familiarize myself with the
geometry of the algorithm, but you just homed right in on it all. That also
shows me where the sqrt(7) term comes from, which would have bothered me until I
figured it out. :)
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ADB wrote:
> the rise is sqrt(3)/2 or sin(60), the run is 0.5 (half the side length) +
> sin(30) ==> 1 so atan(sqrt(3)/2)) ~= 40.8934 degrees which makes the upper
> (downward) angle ~= 49.1066 degrees.
and
scott <sco### [at] scottcom> wrote:
> (L/2)^2 = (S/2 + S*sin(30))^2 + (Scos(30))^2
> (L/2)^2 = (S/2 + S/2 )^2 + 3/4*S^2
> (L/2)^2 = 7/4 * S^2
> L^2 = 7 * S^2
> L = sqrt(7) * S
You two seem to have a peculiar (to me) way of going about your trig.
Either they do it a different and, admittedly, interesting way on "The Other
Side of The Pond", than we do here in 'Murica, or -
"We" do it weird, or -
I just never really learned it very well.
I recall going through something like this puzzling out Friederich Lohmueller's
pulley/belt calculations with Jerome.
You pick what, to me, is the least obvious approach. :D
I'd like to hear and see more about all of this side/2+side*sin(30) business.
I did it what seems to me a more direct way, but got the same answer. {whew!}
:)
See my updated diagram. Hopefully it's correct - I haven't yet finished coffee
#1.
I'm going to ponder how this gets converted into code, since there's an ever
increasing number of line segments to be drawn at every next level.
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>> (L/2)^2 = (S/2 + S*sin(30))^2 + (Scos(30))^2
>> (L/2)^2 = (S/2 + S/2 )^2 + 3/4*S^2
>> (L/2)^2 = 7/4 * S^2
>> L^2 = 7 * S^2
>> L = sqrt(7) * S
>
>
> You two seem to have a peculiar (to me) way of going about your trig.
> Either they do it a different and, admittedly, interesting way on "The Other
> Side of The Pond", than we do here in 'Murica, or -
> "We" do it weird, or -
> I just never really learned it very well.
>
> I recall going through something like this puzzling out Friederich Lohmueller's
> pulley/belt calculations with Jerome.
> You pick what, to me, is the least obvious approach. :D
>
> I'd like to hear and see more about all of this side/2+side*sin(30) business.
> I did it what seems to me a more direct way, but got the same answer. {whew!}
> :)
The problem is, assuming the horizontal distance to be S requires you to
either know that (I didn't) or figure it out from S/2+Ssin(30) or some
other geometric method.
I think I did it because I have a habit to start from the minimum
possible number of assumptions, in this case a hexagon has side length S
and angle 120 degrees. That's enough information to solve it. I've found
in past situations that faulty assumptions are often what cause errors,
so I try not to make them if they are not needed.
> See my updated diagram. Hopefully it's correct - I haven't yet finished coffee
> #1.
> I'm going to ponder how this gets converted into code, since there's an ever
> increasing number of line segments to be drawn at every next level.
Just get the macro to call itself again (with some countdown to prevent
going forever):
#macro gosper_axiom(p0, p1,N)
#if(N>=0)
cylinder { p0, p1 0.0078125 }
#local V = p1 - p0;
#local vT = vnormalize(V);
#local len = vlength(V);
#local np1=p0+len/sqrt(7)*vec2rot( vT,asin(sqrt(3)/(2*sqrt(7))));
#local np2=p1-len/sqrt(7)*vec2rot( vT,asin(sqrt(3)/(2*sqrt(7))));
gosper_axiom(p0,np1,N-1)
gosper_axiom(np1,np2,N-1)
gosper_axiom(np2,p1,N-1)
#end
#end
Call with eg:
gosper_axiom(<-3,0>,<3,0>,5)
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scott <sco### [at] scottcom> wrote:
> >> (L/2)^2 = (S/2 + S*sin(30))^2 + (Scos(30))^2
> >> (L/2)^2 = (S/2 + S/2 )^2 + 3/4*S^2
> >> (L/2)^2 = 7/4 * S^2
> >> L^2 = 7 * S^2
> >> L = sqrt(7) * S
> >
> >
> > You two seem to have a peculiar (to me) way of going about your trig.
> > Either they do it a different and, admittedly, interesting way on "The Other
> > Side of The Pond", than we do here in 'Murica, or -
> > "We" do it weird, or -
> > I just never really learned it very well.
> >
> > I recall going through something like this puzzling out Friederich Lohmueller's
> > pulley/belt calculations with Jerome.
> > You pick what, to me, is the least obvious approach. :D
> >
> > I'd like to hear and see more about all of this side/2+side*sin(30) business.
> > I did it what seems to me a more direct way, but got the same answer. {whew!}
> > :)
>
> The problem is, assuming the horizontal distance to be S requires you to
> either know that (I didn't) or figure it out from S/2+Ssin(30) or some
> other geometric method.
>
> I think I did it because I have a habit to start from the minimum
> possible number of assumptions, in this case a hexagon has side length S
> and angle 120 degrees. That's enough information to solve it. I've found
> in past situations that faulty assumptions are often what cause errors,
> so I try not to make them if they are not needed.
>
> > See my updated diagram. Hopefully it's correct - I haven't yet finished coffee
> > #1.
> > I'm going to ponder how this gets converted into code, since there's an ever
> > increasing number of line segments to be drawn at every next level.
>
> Just get the macro to call itself again (with some countdown to prevent
> going forever):
>
> #macro gosper_axiom(p0, p1,N)
> #if(N>=0)
> cylinder { p0, p1 0.0078125 }
> #local V = p1 - p0;
> #local vT = vnormalize(V);
> #local len = vlength(V);
> #local np1=p0+len/sqrt(7)*vec2rot( vT,asin(sqrt(3)/(2*sqrt(7))));
> #local np2=p1-len/sqrt(7)*vec2rot( vT,asin(sqrt(3)/(2*sqrt(7))));
>
> gosper_axiom(p0,np1,N-1)
> gosper_axiom(np1,np2,N-1)
> gosper_axiom(np2,p1,N-1)
>
> #end
> #end
>
> Call with eg:
>
> gosper_axiom(<-3,0>,<3,0>,5)
I've tried that. I get "too many nested symbol tables"
More when I'm not posting from my iPod
AB
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I _ALMOST_ got it to work. It starts to spazz when I tell it to do 3 Reps.
1 Rep = 1 island
2 Reps = 6 islands around that - OK
3 Reps = 12 islands around those 7. But every other one is off.
It makes me sad.
In macro FullIsland put "yes" as the last argument to show all levels of
subdivision.
There's probably a better, more elegant way, but this is what I've come up with
so far:
//####################################################################
#version 3.7;
global_settings { assumed_gamma 1.8 }
#include "colors.inc"
#include "math.inc"
#include "transforms.inc"
light_source { <0.0, 0.0, -10.0> rgb 1 }
#macro vec2rot(V, A)
#local _X_ = V.u*cos(A) - V.v*sin(A);
#local _Y_ = V.u*sin(A) + V.v*cos(A);
#local RESULT = <_X_,_Y_>;
RESULT
#end
/*
#macro gosper_axiom(p0, p1)
cylinder { p0, p1 0.02 }//cylinder { p0, p1 0.0078125 }
#local V = p1 - p0;
#local vT = vnormalize(V);
#local len = vlength(V);
#debug concat("len: "str(len,0,6) "\n")
#debug concat("vTlen: "str(vlength(vT),0,6) "\n")
#debug concat("newlen: " str(len/sqrt(7),0,6) "\n")
#local np1 = p0 + len/sqrt(7)*vec2rot( vT,asin(sqrt(3)/(2*sqrt(7))));
#local np2 = p1 - len/sqrt(7)*vec2rot( vT,asin(sqrt(3)/(2*sqrt(7))));
#debug concat("np1: " vstr(2, np1,", ",0,10) "\n")
#debug concat("np2: " vstr(2, np2,", ",0,10) "\n")
union {
cylinder { p0, np1 0.02 }
cylinder { np1, np2 0.02 }
cylinder { np2, p1 0.02 }
pigment { Red }
finish { ambient 0 diffuse 0.7 }
}
#end
*/
#macro gosper_axiom_r(p0, p1, N, DrawAllLevels)
#if (N >= 1)
#if (DrawAllLevels)
cylinder { p0, p1 0.002*Rep*N pigment {color srgbft CHSV2RGB(<(360/N), 1,
1>)} }
#elseif (N=1)
cylinder { p0, p1 0.002*Rep*N pigment {color srgbft CHSV2RGB(<(360/N), 1,
1>)} }
#end
#local V = p1 - p0;
#local vT = vnormalize(V);
#local len = vlength(V);
#local np1=p0+len/sqrt(7)*vec2rot( vT,asin(sqrt(3)/(2*sqrt(7))));
#local np2=p1-len/sqrt(7)*vec2rot( vT,asin(sqrt(3)/(2*sqrt(7))));
gosper_axiom_r(p0, np1, N-1, DrawAllLevels)
gosper_axiom_r(np1, np2, N-1, DrawAllLevels)
gosper_axiom_r(np2, p1, N-1, DrawAllLevels)
#declare N = N-1;
#end
#end
#declare TH = degrees( asin( sqrt(3)/(2*sqrt(7)) ) );
#declare L = 6.0/sqrt(7);
#debug concat("L: " str(L,0,6) "\n")
#debug concat("TH: " str(TH,0,6) "\n")
//####################################################################
#macro FullIsland (BeginSegment, EndSegment)
#for (i, 1, 6)
gosper_axiom_r(Endpoint1, Endpoint2, 4, no)
// Rotate endpoints
#declare RotatePoint = transform { rotate <0, 0, 60> }
#declare Endpoint1 = vtransform (Endpoint1, RotatePoint);
#declare Endpoint2 = vtransform (Endpoint2, RotatePoint);
//#debug concat("Endpoint1: " vstr(2, Endpoint1,", ", 2, 1) "\n")
//#debug concat("Endpoint2: " vstr(2, Endpoint2,", ", 2, 1) "\n")
#end
#end // end macro
//####################################################################
#declare Rep = 3;
#declare Center = <0, 0, 0>;
sphere {Center 0.1 pigment {Red}}
#declare Y1 = Center.y+sqrt(3)/2;
#declare Y2 = Center.y+sqrt(3)/2;
#declare X1 = Center.x-0.5;
#declare X2 = Center.x+0.5;
#declare Endpoint1 = <X1, Y1, 0>;
#declare Endpoint2 = <X2, Y2, 0>;
//#debug concat("Endpoint1: " vstr(2, Endpoint1,", ", 2, 1) "\n")
//#debug concat("Endpoint2: " vstr(2, Endpoint2,", ", 2, 1) "\n")
FullIsland (Endpoint1, Endpoint2)
//####################################################################
//Automatically move camera back far enough
camera {
perspective
location <0.0, 0.0, -3*Rep>
up y
right x*(image_width/image_height)
look_at <0.0, 0.0, 0.0>
}
#if (Rep > 1)
#for (j, 2, Rep)
#declare NewCenter = <0, sqrt(3)*(j-1), 0>;
#declare Y1 = NewCenter.y+sqrt(3)/2;
#declare Y2 = NewCenter.y+sqrt(3)/2;
#declare X1 = NewCenter.x-0.5;
#declare X2 = NewCenter.x+0.5;
#declare Endpoint1 = <X1, Y1, 0>;
#declare Endpoint2 = <X2, Y2, 0>;
#debug concat("j: " str(j, 0, 0) "\n")
#for (k, 1, j*6)
#debug concat("k: " str(k, 0, 0) "\n")
FullIsland (Endpoint1, Endpoint2)
#declare RotateCenter = transform { rotate <0, 0, 360/((j-1)*6)> }
#declare NewCenter = vtransform (NewCenter, RotateCenter);
#declare Y1 = NewCenter.y+sqrt(3)/2;
#declare Y2 = NewCenter.y+sqrt(3)/2;
#declare X1 = NewCenter.x-0.5;
#declare X2 = NewCenter.x+0.5;
#declare Endpoint1 = <X1, Y1, 0>;
#declare Endpoint2 = <X2, Y2, 0>;
//#debug concat("Endpoint1: " vstr(2, Endpoint1,", ", 2, 1) "\n")
//#debug concat("Endpoint2: " vstr(2, Endpoint2,", ", 2, 1) "\n")
#end // end k loop
#end // end J loop
#end // end if Reps
//gosper_axiom(<-3.0, 0.0>, <-0.7014146472, 0.7014146472>)
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"Bald Eagle" <cre### [at] netscapenet> wrote:
> I _ALMOST_ got it to work. It starts to spazz when I tell it to do 3 Reps.
> 1 Rep = 1 island
> 2 Reps = 6 islands around that - OK
> 3 Reps = 12 islands around those 7. But every other one is off.
> It makes me sad.
>
> In macro FullIsland put "yes" as the last argument to show all levels of
> subdivision.
>
> There's probably a better, more elegant way, but this is what I've come up with
> so far:
>
> //####################################################################
>
>
> #version 3.7;
>
> global_settings { assumed_gamma 1.8 }
>
> #include "colors.inc"
> #include "math.inc"
> #include "transforms.inc"
>
> light_source { <0.0, 0.0, -10.0> rgb 1 }
>
>
> #macro vec2rot(V, A)
> #local _X_ = V.u*cos(A) - V.v*sin(A);
> #local _Y_ = V.u*sin(A) + V.v*cos(A);
> #local RESULT = <_X_,_Y_>;
> RESULT
> #end
>
> /*
> #macro gosper_axiom(p0, p1)
> cylinder { p0, p1 0.02 }//cylinder { p0, p1 0.0078125 }
> #local V = p1 - p0;
> #local vT = vnormalize(V);
> #local len = vlength(V);
> #debug concat("len: "str(len,0,6) "\n")
> #debug concat("vTlen: "str(vlength(vT),0,6) "\n")
> #debug concat("newlen: " str(len/sqrt(7),0,6) "\n")
> #local np1 = p0 + len/sqrt(7)*vec2rot( vT,asin(sqrt(3)/(2*sqrt(7))));
> #local np2 = p1 - len/sqrt(7)*vec2rot( vT,asin(sqrt(3)/(2*sqrt(7))));
> #debug concat("np1: " vstr(2, np1,", ",0,10) "\n")
> #debug concat("np2: " vstr(2, np2,", ",0,10) "\n")
> union {
> cylinder { p0, np1 0.02 }
> cylinder { np1, np2 0.02 }
> cylinder { np2, p1 0.02 }
> pigment { Red }
> finish { ambient 0 diffuse 0.7 }
> }
> #end
> */
>
>
> #macro gosper_axiom_r(p0, p1, N, DrawAllLevels)
>
> #if (N >= 1)
> #if (DrawAllLevels)
> cylinder { p0, p1 0.002*Rep*N pigment {color srgbft CHSV2RGB(<(360/N), 1,
> 1>)} }
> #elseif (N=1)
> cylinder { p0, p1 0.002*Rep*N pigment {color srgbft CHSV2RGB(<(360/N), 1,
> 1>)} }
> #end
> #local V = p1 - p0;
> #local vT = vnormalize(V);
> #local len = vlength(V);
> #local np1=p0+len/sqrt(7)*vec2rot( vT,asin(sqrt(3)/(2*sqrt(7))));
> #local np2=p1-len/sqrt(7)*vec2rot( vT,asin(sqrt(3)/(2*sqrt(7))));
>
> gosper_axiom_r(p0, np1, N-1, DrawAllLevels)
> gosper_axiom_r(np1, np2, N-1, DrawAllLevels)
> gosper_axiom_r(np2, p1, N-1, DrawAllLevels)
> #declare N = N-1;
> #end
> #end
>
> #declare TH = degrees( asin( sqrt(3)/(2*sqrt(7)) ) );
> #declare L = 6.0/sqrt(7);
> #debug concat("L: " str(L,0,6) "\n")
> #debug concat("TH: " str(TH,0,6) "\n")
>
>
> //####################################################################
> #macro FullIsland (BeginSegment, EndSegment)
> #for (i, 1, 6)
> gosper_axiom_r(Endpoint1, Endpoint2, 4, no)
>
> // Rotate endpoints
> #declare RotatePoint = transform { rotate <0, 0, 60> }
> #declare Endpoint1 = vtransform (Endpoint1, RotatePoint);
> #declare Endpoint2 = vtransform (Endpoint2, RotatePoint);
> //#debug concat("Endpoint1: " vstr(2, Endpoint1,", ", 2, 1) "\n")
> //#debug concat("Endpoint2: " vstr(2, Endpoint2,", ", 2, 1) "\n")
> #end
> #end // end macro
> //####################################################################
>
>
> #declare Rep = 3;
> #declare Center = <0, 0, 0>;
> sphere {Center 0.1 pigment {Red}}
> #declare Y1 = Center.y+sqrt(3)/2;
> #declare Y2 = Center.y+sqrt(3)/2;
> #declare X1 = Center.x-0.5;
> #declare X2 = Center.x+0.5;
> #declare Endpoint1 = <X1, Y1, 0>;
> #declare Endpoint2 = <X2, Y2, 0>;
> //#debug concat("Endpoint1: " vstr(2, Endpoint1,", ", 2, 1) "\n")
> //#debug concat("Endpoint2: " vstr(2, Endpoint2,", ", 2, 1) "\n")
> FullIsland (Endpoint1, Endpoint2)
>
>
> //####################################################################
> //Automatically move camera back far enough
> camera {
> perspective
> location <0.0, 0.0, -3*Rep>
> up y
> right x*(image_width/image_height)
> look_at <0.0, 0.0, 0.0>
> }
>
>
> #if (Rep > 1)
> #for (j, 2, Rep)
> #declare NewCenter = <0, sqrt(3)*(j-1), 0>;
> #declare Y1 = NewCenter.y+sqrt(3)/2;
> #declare Y2 = NewCenter.y+sqrt(3)/2;
> #declare X1 = NewCenter.x-0.5;
> #declare X2 = NewCenter.x+0.5;
> #declare Endpoint1 = <X1, Y1, 0>;
> #declare Endpoint2 = <X2, Y2, 0>;
> #debug concat("j: " str(j, 0, 0) "\n")
>
> #for (k, 1, j*6)
> #debug concat("k: " str(k, 0, 0) "\n")
> FullIsland (Endpoint1, Endpoint2)
> #declare RotateCenter = transform { rotate <0, 0, 360/((j-1)*6)> }
> #declare NewCenter = vtransform (NewCenter, RotateCenter);
> #declare Y1 = NewCenter.y+sqrt(3)/2;
> #declare Y2 = NewCenter.y+sqrt(3)/2;
> #declare X1 = NewCenter.x-0.5;
> #declare X2 = NewCenter.x+0.5;
> #declare Endpoint1 = <X1, Y1, 0>;
> #declare Endpoint2 = <X2, Y2, 0>;
> //#debug concat("Endpoint1: " vstr(2, Endpoint1,", ", 2, 1) "\n")
> //#debug concat("Endpoint2: " vstr(2, Endpoint2,", ", 2, 1) "\n")
> #end // end k loop
>
> #end // end J loop
> #end // end if Reps
>
>
> //gosper_axiom(<-3.0, 0.0>, <-0.7014146472, 0.7014146472>)
wow... I didn't mean rewrite it. I need to use regular math because I'm refining
calculations for a c++ project I'm working on.
As for the math, the only assumption I used was that the outer radius of the
hexagons was 1 unit. Because a regular hexagon is composed of six equilateral
triangles, I could infer the other angles from bisection.
So most of it was geometry, really.
I'd say what I'm aiming for eventually, but then someone here would beat me to
it, and I really dislike that.
you're disturbingly close as it is.
Regards,
A.D.B.
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scott <sco### [at] scottcom> wrote:
> > So if anyone can tell me what I'm doing wrong below, I'll be grateful.
>
> Two mistakes...
>
> > #macro vec2rot(V, A)
> > #local _X_ = -V.u*cos(A)*sin(A) - V.v*cos(A)*sin(A);
> > #local _Y_ = V.u*sin(A)*cos(A) + V.v*sin(A)*cos(A);
> > #local RESULT = <_X_,_Y_>;
> > RESULT
> > #end
>
> This is wrong, should be:
>
> #macro vec2rot(V, A)
> #local _X_ = V.u*cos(A) - V.v*sin(A);
> #local _Y_ = V.u*sin(A) + V.v*cos(A);
> #local RESULT = <_X_,_Y_>;
> RESULT
> #end
>
> You seem to have made it too complicated, it's just a 2D rotation:
>
Ah... >.< I see it now. Stupid mistake.
which explains why the next part didn't work right.
Regards,
A.D.B.
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"Anthony D. Baye" <Sha### [at] spamnomorehotmailcom> wrote:
> wow... I didn't mean rewrite it.
Heh. Sorry. Sometimes I just can't help fleshing things out.
> I'd say what I'm aiming for eventually, but then someone here would beat me to
> it, and I really dislike that.
>
> you're disturbingly close as it is.
Yes, I had a feeling I might "steal your thunder" - and I hope I didn't overstep
here. That wasn't my intention.
I just got a little overzealous. Hope you work it out and make something
great.
Sorry again for ... yeah. :|
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"Bald Eagle" <cre### [at] netscapenet> wrote:
> "Anthony D. Baye" <Sha### [at] spamnomorehotmailcom> wrote:
>
> > wow... I didn't mean rewrite it.
>
> Heh. Sorry. Sometimes I just can't help fleshing things out.
>
>
> > I'd say what I'm aiming for eventually, but then someone here would beat me to
> > it, and I really dislike that.
> >
> > you're disturbingly close as it is.
>
> Yes, I had a feeling I might "steal your thunder" - and I hope I didn't overstep
> here. That wasn't my intention.
> I just got a little overzealous. Hope you work it out and make something
> great.
> Sorry again for ... yeah. :|
don't worry about it. I can be a little sensitive.
I did figure it out. Scott's comment pointed me in the right direction. Silly
me forgot how to do a dot product.
I'll post my results in the thread in P.B.I.
Regards,
A.D.B.
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"Anthony D. Baye" <Sha### [at] spamnomorehotmailcom> wrote:
> don't worry about it. I can be a little sensitive.
It's OK - I understand.
Didn't mean to step on any toes.
Glad it's working out for you though.
Believe it or not, someone was asking me about where to get lots of equilateral
triangles to make some tetra and octahedrons, and I found this looking for
"triangular tiles"
http://mathafou.free.fr/themes_en/fractzoo.html
Scroll down a little more than halfway to "FlowSnake"
More math. Less geometry :)
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