Easter eggs anybody?

[tenochtitlanuk]

With the egg-citing approach of Easter and its association with eggs- decorated ones ( pretty!)- and chocolate ones ( yes please!) you may have missed news that mathematicians have cracked ( !) the problem of describing all egg shapes with a single equation. Previously they could only do so for circles, ovals, or ovals with the long axis distorted to bias the fattest point towards one end. This unfortunately does not cover the more pear-shaped eggs.

We won't beat Anatoly's ( tsh) Faberge eggs, but still fun to have. Do a Google search and work from there.

Some ideas.. A single outline? Eggs in arrays or circles? Eggs in a basket? Varying sizes? Colour schemes- single colour or gradient fill? 3D eggs as rotatable 2d line projections?

[bplus]

Yes I see they have a formula now:

[bplus]

Just figured how to shade a little bit:

[bplus]

And a little background:

[bplus]

But can he draw more than one?


yes

[Carl Gundel]

Mar 26, 2022 14:33:58 GMT -5 bplus said:

But can he draw more than one?
View Attachment

yes

Now mix that with the ray tracing code that was recently posted.

[bplus]

hmm...I'll look into it.

[bplus]

Hint #1:

Don M. Jacobs, M.D., from Daly City, USA developped a nice egg shape by changing the circle equation x²+y²=1 a little: x² + [1.4^x*1.6y]² = 1. The egg equation is an exponential equation of the type t3.

I only needed the eq who knows what a type t3 exponential eq is?

Then I moved the 1 from right side of eq and I ran x and y loops and found when the plug in x, y was nearly 0 ie <= .01.

Then I scaled (x,y) and shifted the pixel sets by the screen center to graph results on JB/LB screen. That's when things shaped up.

Using a 500 X 500 screen setup, the code looks like this:


for x = -1 to 1 step .01 ' i reversed x, y order for coloring
for y = -1 to 1 step .01
   scan
   if ( x*x + (1.4 ^ x * 1.6 * y) ^ 2 - 1 ) <= .01 then
       #gr "color blue"
       #gr "set ";scale * x + 250;" ";scale * y + 250
   end if
   'if (x * x + y * y - 1) <= .01 then 'test a known figure a circle
       '#gr "color red"
       '#gr "set ";scale * x + 250;" ";scale * y + 250
   'end if
next
next

The 250's put x= 0 and y=0 right in the center of the graphics screen. Scale just expands or shrinks the plot.


Wouldn't it be nice if author of Ray Tracing could come and explain their code step by step?

[Brandon Parker]

A "T3" equation is most likely a "Third Taylor Polynomial" ...


{:0)

Brandon Parker

[Rod]

Might be difficult since the sphere to sphere collision is well understood. The egg is neither a plane nor a sphere and in 3d it would be hard to know what part of the egg you hit.

The best tutorial I have found is here raytracing.github.io/books/RayTracingInOneWeekend.html

[Carl Gundel]

Mar 27, 2022 12:59:44 GMT -5 Rod said:

Might be difficult since the sphere to sphere collision is well understood. The egg is neither a plane nor a sphere and in 3d it would be hard to know what part of the egg you hit.

The best tutorial I have found is here raytracing.github.io/books/RayTracingInOneWeekend.html

Just use brute force and make the eggs out of millions of microspheres.  ;P  It would take a loooooong time to do the processing. 

[tsh73]

Just use brute force and make the eggs out of millions of microspheres.

How about just 7?

Just put those DATA lines in


'nomainwin
open "egg" for graphics_nsb_nf as #gr
#gr "trapclose [quit]"
#gr "down; home; posxy cx cy"

pi=acs(-1)
pi2=2*pi
x0=70:xk=cx*2-100
print "data 7"
for i = 0 to 6
   x=x0+i/6*(xk-x0)
   #gr "place ";x;" ";cy
   r = sin((i+2)/pi2*1.5)*80
   #gr "circle ";r
   print "data ";x/80*.4-.5;",";-0.6;",";1.5;",";r/80*.4
next

#gr "flush"
wait

[quit]
close #gr
end


[Rod]

Wow. Falls over in disbelief!!

[Carl Gundel]

Mar 28, 2022 7:28:53 GMT -5 tsh73 said:

Just use brute force and make the eggs out of millions of microspheres.

How about just 7?

Just put those DATA lines in


'nomainwin
open "egg" for graphics_nsb_nf as #gr
#gr "trapclose [quit]"
#gr "down; home; posxy cx cy"

pi=acs(-1)
pi2=2*pi
x0=70:xk=cx*2-100
print "data 7"
for i = 0 to 6
   x=x0+i/6*(xk-x0)
   #gr "place ";x;" ";cy
   r = sin((i+2)/pi2*1.5)*80
   #gr "circle ";r
   print "data ";x/80*.4-.5;",";-0.6;",";1.5;",";r/80*.4
next

#gr "flush"
wait

[quit]
close #gr
end


Love it! Just need a higher sampling rate to make it more convincing, but wow, nice!

[bplus]

A little more brute force!

Modifying tsh73 egg drawing and data making:
'nomainwin
open "egg" for graphics_nsb_nf as #gr
#gr "trapclose [quit]"
#gr "down; home; posxy cx cy"

'#gr "size 2"
pi=acs(-1)
pi2=2*pi
x0=70:xk=cx*2-100
print "data 49"
#gr "color 255 160 200"
#gr "backcolor 255 160 200"
for i = 0 to 6 step .125
   x=x0+i/7*(xk-x0)
   #gr "place ";x;" ";cy
   r = sin((i+2)/pi2*1.5)*80
   #gr "circlefilled ";r
   print "data ";x/80*.4-.5;",";-0.6;",";1.5;",";r/80*.4
next
'#gr "color 0 0 200"
'#gr "backcolor 0 0 200"
'for i = 0 to 6
'    x=x0+i/6*(xk-x0)
'    #gr "place ";x;" ";cy
'    r = sin((i+2)/pi2*1.5)*80
'    #gr "circle ";r
'    print "data ";x/80*.4-.5;",";-0.6;",";1.5;",";r/80*.4
'next

#gr "flush"
wait

[quit]
close #gr
end