((III)I) = (sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 - R2^2 = 0
(((II)I)I) = (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 - R3^2 = 0
((II)II) = (sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 - R2^2 = 0
((II)(II)) = (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2^2 = 0
how does one go about making them polynomials? Or, is there an easier way using the torus quartic? Maybe, actually.
ICN5D wrote:((III)I) = (sqrt(x^2 + y^2 + z^2) - R1)^2 + w^2 - R2^2 = 0
(((II)I)I) = (sqrt((sqrt(x^2 + y^2) - R1)^2 + z^2) - R2)^2 + w^2 - R3^2 = 0
((II)II) = (sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 - R2^2 = 0
((II)(II)) = (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2^2 = 0
PWrong wrote:Actually this will be fairly important, because the definitions with square roots aren't really accurate. Without a +/- sign, they only give half of the shape, or worse.
ICN5D wrote:I think using just the cartesian implicit function overcomes the square root problem. If converting into a polynomial, then solving for exact roots, the +/- sign has to be used. It allows two shapes to exist, as translated copies equidistant from a cut open higher diameter.
That's why I used
(x^2+y^2+x+y-(R1+R2))*(x^2+y^2+x+y-(R1-R2)) = z
for the circle intercepts for a quartic torus polynomial. The circles have a radius R1 set to exact values from the sliced open R2, as +/- from the square root. Though, if graphing as an explicit function, four simultaneous equations will have to be used, as only half the circles will appear. That's the square root problem with graphing. Luckily, using just the implicit function in Cartesian coordinates in CalcPlot3D overcomes this obstacle, and makes exactly what it should be. I tried to graph the torisphere implicit in Mathematica 10, but there's errors in the calculation.
NoRoots[f_] :=
Flatten[CoefficientList[Expand[f],
Cases[Expand[f], Power[_, 1/2], 2]]][[1]]
NewEq[f_] := (NoRoots[f])^2 - Expand[(Expand[f] - NoRoots[f])^2]
ExpandToratope[f_] :=
If[Count[Expand[f], Power[_, 1/2], 2] == 0, f,
ExpandToratope[NewEq[f]]]
In[82]:= tiger = (-a + Sqrt[x^2 + y^2])^2 + (-b + Sqrt[w^2 + z^2])^2;
ExpandToratope[tiger]
Out[83]= -64 a^2 b^2 w^2 x^2 - 64 a^2 b^2 w^2 y^2 -
64 a^2 b^2 x^2 z^2 -
64 a^2 b^2 y^2 z^2 + (a^4 + 2 a^2 b^2 + b^4 + 2 a^2 w^2 - 2 b^2 w^2 +
w^4 - 2 a^2 x^2 + 2 b^2 x^2 + 2 w^2 x^2 + x^4 - 2 a^2 y^2 +
2 b^2 y^2 + 2 w^2 y^2 + 2 x^2 y^2 + y^4 + 2 a^2 z^2 - 2 b^2 z^2 +
2 w^2 z^2 + 2 x^2 z^2 + 2 y^2 z^2 + z^4)^2
In[87]:= FullSimplify[
ExpandToratope[
Sqrt[(-a + Sqrt[x^2 + y^2])^2 + (-b + Sqrt[w^2 + z^2])^2] - R]]
Out[87]= -64 a^2 b^2 w^2 x^2 - 64 a^2 b^2 w^2 y^2 - 64 a^2 b^2 x^2 z^2 - 64 a^2 b^2 y^2 z^2
+ (a^4 + b^4 + 2 a^2 (b^2 - R^2 + w^2 - x^2 - y^2 + z^2) -
2 b^2 (R^2 + w^2 - x^2 - y^2 + z^2) + (-R^2 + w^2 + x^2 + y^2 + z^2)^2)^2
ICN5D wrote:There it is. The degree 8 polynomial for a tiger. Very cool! It takes on an interesting form, as polynomials within polynomials. Thats probably the combinatorics playing its part. I also imagine that it has a few levels of symmetric subgroups, too. Nice work.
PWrong wrote:We can simplify it some more:
-64 a^2 b^2 ( x^2 + y^2) ( z^2 + w^2) + ((a^2 + b^2)^2 - 2 (a^2 + b^2) R^2 + 2 (b^2 - a^2) (x^2 + y^2 - z^2 - w^2) + (-R^2 + x^2 + y^2 + z^2 + w^2)^2)^2
Some substitutions (e.g. p^2 = x^2 + y^2, q^2 = z^2 + w^2) might make the patterns clearer, if there are any.
ICN5D wrote:I might be using the wrong word for it. My idea is to use an algebraic approach to deriving intercepts, in the fully expanded polynomial form. Keeping it in implicit form is very easy, no doubt! I was curious in exploring the polynomials a little deeper, and deriving the number, type, and arrangement of intercept toratopes there would be. Trivial of course, since we can easily do it with the implicit. I like new challenges, if it actually is one!
PWrong wrote:This isn't really related, but I was curious about the "complex" aspect of this and I decided to look at what these equations look like if all the variables are complex. To keep things simple I just looked at a circle, and it's actually really strange and interesting.
So the equation is just p^2 + q^2 = 1, where x and y are complex numbers.
(x + y i)^2 + (z + w i)^2 = 1
Expanding this out, we get two equations:
x^2 + z^2 - y^2 - w^2 = 1
x y + z w = 1
So this "complex circle" is actually a weird intersection of two four dimensional hyperbolic surfaces that I'm not very familiar with.
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