Batman Curve Notes
The full Batman Curve writeup can be found here.
8/5/21
Need to debug:
https://math.stackexchange.com/questions/54506/is-this-batman-equation-for-real
Batman:
Looks good (outer wings):
(((x/7)^2)*sqrt((abs(abs(x)-3))/(abs(x)-3))+((y/3)^2)*sqrt((abs(y+3*sqrt(33)/7))/(y+3*sqrt(33)/7))-1) = 0
Looks good (bottom curves):
(abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(1-(abs(abs(x)-2)-1)^2)-y) = 0
Fixed version: (abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(abs(1-(abs(abs(x)-2)-1)^2))-y)^2 + min(1, ceil(abs(min(0, 4-x))))+min(1, ceil(abs(max(0, -4-x)))) =0
Looks right (outer ears):
(9*sqrt(abs((abs(x)-1)*(abs(x)-0.75))/((1-abs(x))*(abs(x)-0.75)))-8*abs(x)-y) = 0
Looks right (inner ears):
(3*abs(x)+0.75*sqrt(abs((abs(x)-0.75)*(abs(x)-0.5))/((0.75-abs(x))*(abs(x)-0.5)))-y) = 0
Looks right (top of head):
(2.25*sqrt(abs((x-0.5)*(x+0.5))/((0.5-x)*(0.5+x)))-y) = 0
Fixed version: (2.25-y)^2+min(1, ceil(abs(min(0, 0.5-x))))+min(1, ceil(abs(max(0, -0.5-x)))) = 0
Looks right (shoulders)
(6*sqrt(10)/7+(1.5-0.5*abs(x))*sqrt(abs(abs(x)-1)/(abs(x)-1))-6*sqrt(10)/14*sqrt(4-(abs(x)-1)^2)-y) = 0
But when I multiply them all together, I get no graph result.
(Quick proof to myself that I can generally compose shapes using multiplication like this:
(2.25-y) * (x^2+y^2-1) * (x^2+y^2-4) = 0
This works!)
Desmos yields at least some graph for:
(2.25*sqrt(abs((x-0.5)*(x+0.5))/((0.5-x)*(0.5+x)))-y) * (abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(1-(abs(abs(x)-2)-1)^2)-y)=0
Ah good FormulaPlot's result matches Desmos.
This is the combination of the top of head and bottom curves. The top of head formula restricts x to a small range. Then this also restricts the bottom curves to that same valid range of x.
This term: sqrt(abs((x-0.5)*(x+0.5))/((0.5-x)*(0.5+x)))
Makes the whole formula invalid for any x values for which this is negative:
abs((x-0.5)*(x+0.5))/((0.5-x)*(0.5+x)) -- seemingly anything outside of x=[-0.5, 0.5]
Instead of restricting values by creating complex numbers (thus breaking the actual graph), I might be able to create a gadget using max and/or min (similar to the graph below)
Take this segment for example:
(2.25*sqrt(abs((x-0.5)*(x+0.5))/((0.5-x)*(0.5+x)))-y) = 0
Can I restrict x to be between [-0.5,0.5] another way?
y = 2.25
x = [-0.5,0.5]
2.25 - y + min(0.5, (floor(abs(x)+0.5))) = 0
Close but not quite right
I think I want to find a term that evaluates to 0 iff x is in the desired range
First attempt:
x is less than 0.5:
min(0, 0.5-x)
min(1, ceil(abs(min(0, 0.5-x))))
x is greater than -0.5:
max(0, -0.5-x)
min(1, ceil(abs(max(0, -0.5-x))))
x is between -0.5 and 0.5:
(min(0, 0.5-x)+max(0, -0.5-x))
(min(1, ceil(abs(max(0, -0.5-x))))+min(1, ceil(abs(min(0, 0.5-x)))))
(2.25-y)*(min(0, 0.5-x)+max(0, -0.5-x)) = 0
Not quite right: we need x between these AND y needs to equal 2.25
(2.25-y)+(min(0, 0.5-x)+max(0, -0.5-x)) = 0
We're still finding valid x values outside the desired range
This seems to work:
(2.25-y)^2+min(1, ceil(abs(min(0, 0.5-x))))+min(1, ceil(abs(max(0, -0.5-x))))=0
Squaring the y term prevents it from having a negative value that would cancel out the 1 we get when x is out of range. Using abs instead of ^2 also would work.
Fixed gadgets:
x is less than 0.5:
min(1, ceil(abs(min(0, 0.5-x))))
x is greater than -0.5:
min(1, ceil(abs(max(0, -0.5-x))))
x is between -0.5 and 0.5:
(min(1, ceil(abs(max(0, -0.5-x))))+min(1, ceil(abs(min(0, 0.5-x)))))
Can I combine that with the bottom curves?
(abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(1-(abs(abs(x)-2)-1)^2)-y)=0
The desired x range for this segment is -4 to 4.
Examine whats in the sqrt:
(abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+ sqrt(1-(abs(abs(x)-2)-1)^2) -y)
sqrt(1-(abs(abs(x)-2)-1)^2)
I can adjust this to be valid for all x:
sqrt(abs(1-(abs(abs(x)-2)-1)^2))
Graphing this confirms it keeps the shape for x in [-4,4] while also being valid outside that range. Then I can constrain it with my x range gadgets separately.
Bringing this to the full formula for the bottom curves:
(abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(abs(1-(abs(abs(x)-2)-1)^2))-y)=0
Constraining x's values:
(abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(abs(1-(abs(abs(x)-2)-1)^2))-y)^2 + min(1, ceil(abs(min(0, 4-x))))+min(1, ceil(abs(max(0, -4-x)))) =0
Combine that with the top of head:
(2.25-y)^2+min(1, ceil(abs(min(0, 0.5-x))))+min(1, ceil(abs(max(0, -0.5-x))))=0
and
(abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(abs(1-(abs(abs(x)-2)-1)^2))-y)^2 + min(1, ceil(abs(min(0, 4-x))))+min(1, ceil(abs(max(0, -4-x)))) =0
((2.25-y)^2+min(1, ceil(abs(min(0, 0.5-x))))+min(1, ceil(abs(max(0, -0.5-x))))) * ((abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(abs(1-(abs(abs(x)-2)-1)^2))-y)^2 + min(1, ceil(abs(min(0, 4-x))))+min(1, ceil(abs(max(0, -4-x))))) = 0
That works!
Next steps: fix each term in the equation:
- Fix them so there are solutions for the full range of x
- Use my gadget to constrain formula F's x range to [L,U]:
(F^2 + min(1, ceil(abs(min(0, U-x))))+min(1, ceil(abs(max(0, L-x))))) = 0
- Combine segments by multiplying the sub-formulas together
Fixing outer wings:
(((x/7)^2)*sqrt((abs(abs(x)-3))/(abs(x)-3))+((y/3)^2)*sqrt((abs(y+3*sqrt(33)/7))/(y+3*sqrt(33)/7))-1)=0
The top has a different x range than the bottom and there are two sections. I might need to split this into several sub-sections... as many as 4.
I'll focus on the right side first and then might be able to just flip over the y-axis to get the left side.
x ranges from 3 to 7 for y>0
x ranges from 4 to 7 for y<0
Fix formula to be valid for full range of x:
(((x/7)^2)*sqrt( (abs(abs(x)-3))/(abs(x)-3) )+((y/3)^2)*sqrt((abs(y+3*sqrt(33)/7))/(y+3*sqrt(33)/7))-1)=0
The sqrt terms seem to only exist to limit the graph so I'll remove them. Ah, looks like the formula for a normal ellipse now.
(((x/7)^2)+((y/3)^2)-1)=0
Focus on the section where y>0:
(((x/7)^2)+((y/3)^2)-1)=0
min(1, ceil(abs(max(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)=0
Restrict x to [3,7]:
(min(1, ceil(abs(max(0, 3-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(max(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)=0
Now for y<0, restrict x to [4,7]:
(min(1, ceil(abs(max(0, 4-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(min(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)=0
Flip both of those over the y-axis by negating x terms:
(min(1, ceil(abs(max(0, 3-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(max(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)=0
(min(1, ceil(abs(max(0, 4-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(min(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)=0
Combine all four of those together:
(((min(1, ceil(abs(max(0, 3-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(max(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(min(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 3-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(max(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(min(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)))=0
Hmm this is giving me glitches close to (x,y)=(0,0)
I could add a term to ensure abs(x)>1?
(((min(1, ceil(abs(max(0, 3-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(max(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(min(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 3-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(max(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(min(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) + min(1, ceil(abs(max(0, 1-abs(x))))) )=0
That worked!
Add that to the portion of the graph I already had:
((2.25-y)^2+min(1, ceil(abs(min(0, 0.5-x))))+min(1, ceil(abs(max(0, -0.5-x))))) * ((abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(abs(1-(abs(abs(x)-2)-1)^2))-y)^2 + min(1, ceil(abs(min(0, 4-x))))+min(1, ceil(abs(max(0, -4-x))))) = 0
and
(((min(1, ceil(abs(max(0, 3-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(max(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(min(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 3-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(max(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(min(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) + min(1, ceil(abs(max(0, 1-abs(x))))) )=0
(((2.25-y)^2+min(1, ceil(abs(min(0, 0.5-x))))+min(1, ceil(abs(max(0, -0.5-x))))) * ((abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(abs(1-(abs(abs(x)-2)-1)^2))-y)^2 + min(1, ceil(abs(min(0, 4-x))))+min(1, ceil(abs(max(0, -4-x)))))) * (((min(1, ceil(abs(max(0, 3-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(max(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(min(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 3-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(max(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(min(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) + min(1, ceil(abs(max(0, 1-abs(x))))) )=0
Looking good!
Next segment:
Shoulders:
(6*sqrt(10)/7+(1.5-0.5*abs(x))*sqrt(abs(abs(x)-1)/(abs(x)-1))-6*sqrt(10)/14*sqrt(4-(abs(x)-1)^2)-y)
=0
Focus on the segment that runs from x=1 to x=3 and then I'll mirror it:
(6*sqrt(10)/7+(1.5-0.5*abs(x))* sqrt(abs(abs(x)-1)/(abs(x)-1)) -6*sqrt(10)/14*sqrt(4-(abs(x)-1)^2)-y)
=0
Remove term that just serves to restrict x:
(6*sqrt(10)/7+(1.5-0.5*abs(x)) -6*sqrt(10)/14*sqrt(4-(abs(x)-1)^2)-y)
=0
What about the other sqrt term?
(6*sqrt(10)/7+(1.5-0.5*abs(x)) -6*sqrt(10)/14* sqrt(abs(4-(abs(x)-1)^2)) -y)
=0
Looks kind of like a dress now
Now limit x's range to [1,3]:
(min(1, ceil(abs(min(0, 3-x))))+min(1, ceil(abs(max(0, 1-x))))) + ((6*sqrt(10)/7+(1.5-0.5*abs(x)) -6*sqrt(10)/14*sqrt(abs(4-(abs(x)-1)^2))-y))^2
=0
Reflect over the y axis too:
(min(1, ceil(abs(min(0, 3+x))))+min(1, ceil(abs(max(0, 1+x))))) + ((6*sqrt(10)/7+(1.5-0.5*abs(-x)) -6*sqrt(10)/14*sqrt(abs(4-(abs(-x)-1)^2))-y))^2
=0
Combine:
((min(1, ceil(abs(min(0, 3-x))))+min(1, ceil(abs(max(0, 1-x))))) + ((6*sqrt(10)/7+(1.5-0.5*abs(x)) -6*sqrt(10)/14*sqrt(abs(4-(abs(x)-1)^2))-y))^2) * ((min(1, ceil(abs(min(0, 3+x))))+min(1, ceil(abs(max(0, 1+x))))) + ((6*sqrt(10)/7+(1.5-0.5*abs(-x)) -6*sqrt(10)/14*sqrt(abs(4-(abs(-x)-1)^2))-y))^2)
=0
Combine with the rest of the batman:
(((2.25-y)^2+min(1, ceil(abs(min(0, 0.5-x))))+min(1, ceil(abs(max(0, -0.5-x))))) * ((abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(abs(1-(abs(abs(x)-2)-1)^2))-y)^2 + min(1, ceil(abs(min(0, 4-x))))+min(1, ceil(abs(max(0, -4-x)))))) * (((min(1, ceil(abs(max(0, 3-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(max(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(min(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 3-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(max(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(min(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) + min(1, ceil(abs(max(0, 1-abs(x))))) ) * ((min(1, ceil(abs(min(0, 3-x))))+min(1, ceil(abs(max(0, 1-x))))) + ((6*sqrt(10)/7+(1.5-0.5*abs(x)) -6*sqrt(10)/14*sqrt(abs(4-(abs(x)-1)^2))-y))^2) * ((min(1, ceil(abs(min(0, 3+x))))+min(1, ceil(abs(max(0, 1+x))))) + ((6*sqrt(10)/7+(1.5-0.5*abs(-x)) -6*sqrt(10)/14*sqrt(abs(4-(abs(-x)-1)^2))-y))^2)
=0
Almost done!!
Next segment:
Outer ears:
(9* sqrt( abs((abs(x)-1)*(abs(x)-0.75)) / ((1-abs(x))*(abs(x)-0.75)) )-8*abs(x)-y) = 0
This runs from y=1 to y=3
Removing the sqrt term:
(9-8*abs(x)-y) = 0
Restricting y value to [1,3]:
((min(1, ceil(abs(max(0, 1-y))))+min(1, ceil(abs(min(0, 3-y))))) + (9-8*abs(x)-y)^2) = 0
Adding to the whole batman:
(((2.25-y)^2+min(1, ceil(abs(min(0, 0.5-x))))+min(1, ceil(abs(max(0, -0.5-x))))) * ((abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(abs(1-(abs(abs(x)-2)-1)^2))-y)^2 + min(1, ceil(abs(min(0, 4-x))))+min(1, ceil(abs(max(0, -4-x)))))) * (((min(1, ceil(abs(max(0, 3-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(max(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(min(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 3-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(max(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(min(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) + min(1, ceil(abs(max(0, 1-abs(x))))) ) * ((min(1, ceil(abs(min(0, 3-x))))+min(1, ceil(abs(max(0, 1-x))))) + ((6*sqrt(10)/7+(1.5-0.5*abs(x)) -6*sqrt(10)/14*sqrt(abs(4-(abs(x)-1)^2))-y))^2) * ((min(1, ceil(abs(min(0, 3+x))))+min(1, ceil(abs(max(0, 1+x))))) + ((6*sqrt(10)/7+(1.5-0.5*abs(-x)) -6*sqrt(10)/14*sqrt(abs(4-(abs(-x)-1)^2))-y))^2) * ((min(1, ceil(abs(max(0, 1-y))))+min(1, ceil(abs(min(0, 3-y))))) + (9-8*abs(x)-y)^2)
=0
Last segment:
Inner ears:
(3*abs(x)+0.75*sqrt(abs((abs(x)-0.75)*(abs(x)-0.5))/((0.75-abs(x))*(abs(x)-0.5)))-y) = 0
Removing the sqrt element:
(3*abs(x)+0.75-y) = 0
Restrict y to [2.25,3]:
((min(1, ceil(abs(max(0, 2.25-y))))+min(1, ceil(abs(min(0, 3-y))))) + (3*abs(x)+0.75-y)^2) = 0
Complete our batman equation:
(((2.25-y)^2+min(1, ceil(abs(min(0, 0.5-x))))+min(1, ceil(abs(max(0, -0.5-x))))) * ((abs(x/2)-((3*sqrt(33)-7)/112)*(x^2)-3+sqrt(abs(1-(abs(abs(x)-2)-1)^2))-y)^2 + min(1, ceil(abs(min(0, 4-x))))+min(1, ceil(abs(max(0, -4-x)))))) * (((min(1, ceil(abs(max(0, 3-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(max(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-x))))+min(1, ceil(abs(min(0, 7-x))))) + min(1, ceil(abs(min(0, 0-y))))+ (((x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 3-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(max(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) * ((min(1, ceil(abs(max(0, 4-(-x)))))+min(1, ceil(abs(min(0, 7-(-x)))))) + min(1, ceil(abs(min(0, 0-y))))+ (((-x/7)^2)+((y/3)^2)-1)) + min(1, ceil(abs(max(0, 1-abs(x))))) ) * ((min(1, ceil(abs(min(0, 3-x))))+min(1, ceil(abs(max(0, 1-x))))) + ((6*sqrt(10)/7+(1.5-0.5*abs(x)) -6*sqrt(10)/14*sqrt(abs(4-(abs(x)-1)^2))-y))^2) * ((min(1, ceil(abs(min(0, 3+x))))+min(1, ceil(abs(max(0, 1+x))))) + ((6*sqrt(10)/7+(1.5-0.5*abs(-x)) -6*sqrt(10)/14*sqrt(abs(4-(abs(-x)-1)^2))-y))^2) * ((min(1, ceil(abs(max(0, 1-y))))+min(1, ceil(abs(min(0, 3-y))))) + (9-8*abs(x)-y)^2) * ((min(1, ceil(abs(max(0, 2.25-y))))+min(1, ceil(abs(min(0, 3-y))))) + (3*abs(x)+0.75-y)^2) = 0
Looks perfect!
Bonus points... can I get the shape to fill in with an inequality? It looks like the wings and the body have different polarity so changing to "<=0" only fills in the wings. I wonder if I can make the formulas for the wings negative to get what I want.
I can't quite figure this out right now. It actually might not even be possible without other major rework because using ">=0" fills in far more than the body.