icebreaker wrote:I need to know what is an implicit equation of dicone in order to check out (by using CalcPlot) whether there is more cross-sections than I think.
ICN5D wrote:Yeah, man. I explored these shapes some time ago. For the dicone, you could add the slice where a large cone appears, and gradually shrinks to a point. If you're interested, I use a simple algorithm to derive these equations. They are the functions of STEMP notation (using I, >, O symbols).
generating tiger parametric equation, and converting to implicit polynomial
x = rcos(u)
y = rsin(u)
x = rcos(u)+R
y = rsin(u)
x = (rcos(u)+R)cos(v)
y = (rcos(u)+R)sin(v)
z = rsin(u)
x = (rcos(u)+R)cos(v)
y = (rcos(u)+R)sin(v)
z = rsin(u) + P
x = (rcos(u)+R)cos(v)
y = (rcos(u)+R)sin(v)
z = (rsin(u)+P)cos(t)
w = (rsin(u)+P)sin(t)
implicitize:
x = (rcos(u)+R)cos(v)
y = (rcos(u)+R)sin(v)
z = (rsin(u)+P)cos(t)
w = (rsin(u)+P)sin(t)
solve for variable t,
x = (rcos(u)+R)cos(v)
y = (rcos(u)+R)sin(v)
z/(rsin(u)+P) = cos(t)
arcsin(w/(rsin(u)+P)) = t
x = (rcos(u)+R)cos(v)
y = (rcos(u)+R)sin(v)
z/(rsin(u)+P) = cos(arcsin(w/(rsin(u)+P)))
solve for v,
x/(rcos(u)+R) = cos(v)
y/(rcos(u)+R) = sin(v)
z/(rsin(u)+P) = cos(arcsin(w/(rsin(u)+P)))
x/(rcos(u)+R) = cos(v)
arcsin(y/(rcos(u)+R)) = v
z/(rsin(u)+P) = cos(arcsin(w/(rsin(u)+P)))
x/(rcos(u)+R) = cos(arcsin(y/(rcos(u)+R)))
z/(rsin(u)+P) = cos(arcsin(w/(rsin(u)+P)))
x = (rcos(u)+R)cos(arcsin(y/(rcos(u)+R)))
z = (rsin(u)+P)cos(arcsin(w/(rsin(u)+P)))
solve for u, from x(u,y):
u = cos^(-1)((sqrt(r^2 x^2+r^2 y^2)-r R)/r^2)
substitute in z(u,w):
z = (rsin(cos^(-1)((sqrt(r^2 x^2+r^2 y^2)-r R)/r^2))+P)cos(arcsin(w/(rsin(cos^(-1)((sqrt(r^2 x^2+r^2 y^2)-r R)/r^2))+P)))
z = (P+r sqrt(1-(sqrt(r^2 x^2+r^2 y^2)-r R)^2/r^4)) sqrt(1-w^2/(P+r sqrt(1-(sqrt(r^2 x^2+r^2 y^2)-r R)^2/r^4))^2)
solving for r, 8 solutions:
r = -sqrt(-2 sqrt(-2 sqrt(P^2R^2w^2x^2 + P^2R^2w^2y^2 + P^2R^2x^2z^2 + P^2R^2y^2z^2) + P^2w^2+P^2z^2 + R^2x^2 + R^2y^2) + P^2 + R^2 + w^2 + x^2 + y^2 + z^2)
eliminating square roots,
r^2 = (-sqrt(-2 sqrt(-2 sqrt(P^2R^2((x^2 + y^2)(w^2 + z^2))) + P^2(w^2 + z^2) + R^2(x^2 + y^2)) + P^2 + R^2 + w^2 + x^2 + y^2 + z^2))^2
-r^2 = 2 sqrt(-2 sqrt(P^2R^2((x^2 + y^2)(w^2 + z^2))) + P^2(w^2 + z^2) + R^2(x^2 + y^2)) - (P^2 + R^2 + w^2 + x^2 + y^2 + z^2)
(x^2 +y^2 +z^2 +w^2 +P^2 +R^2 -r^2)^2 = (2 sqrt(-2 sqrt(P^2R^2((x^2 + y^2)(w^2 + z^2))) + P^2(w^2 + z^2) + R^2(x^2 + y^2)))^2
(x^2 +y^2 +z^2 +w^2 +P^2 +R^2 -r^2)^2 = 4(-2 sqrt(P^2R^2((x^2 + y^2)(w^2 + z^2))) + P^2(w^2 + z^2) + R^2(x^2 + y^2))
(x^2 +y^2 +z^2 +w^2 +P^2 +R^2 -r^2)^2 -4P^2(w^2 + z^2) -4R^2(x^2 + y^2) = -8sqrt(P^2R^2((x^2 + y^2)(w^2 + z^2)))
((x^2 +y^2 +z^2 +w^2 +P^2 +R^2 -r^2)^2 -4P^2(w^2 + z^2) -4R^2(x^2 + y^2))^2 = (-8sqrt(P^2R^2((x^2 + y^2)(w^2 + z^2))))^2
((x^2 +y^2 +z^2 +w^2 +P^2 +R^2 -r^2)^2 -4P^2(w^2 + z^2) -4R^2(x^2 + y^2))^2 = 64(P^2R^2((x^2 + y^2)(w^2 + z^2)))
((x^2 +y^2 +z^2 +w^2 +P^2 +R^2 -r^2)^2 -4P^2(w^2 + z^2) -4R^2(x^2 + y^2))^2 -64(P^2R^2((x^2 + y^2)(w^2 + z^2))) = 0
Degree-8 Polynomial with 4 variables, 3 coefficients
((x^2 +y^2 +z^2 +w^2 +a^2 +b^2 -c^2)^2 -4a^2(x^2 + y^2) -4b^2(z^2 + w^2))^2 -64a^2b^2(x^2 + y^2)(z^2 + w^2)
Rotate/Translate cross section:
x = (x*cos(t) + d*sin(t))
w = (x*sin(t) - d*cos(t))
(((x*cos(t) + d*sin(t))^2 +y^2 +z^2 +(x*sin(t) - d*cos(t))^2 +a^2 +b^2 -c^2)^2 -4a^2((x*sin(t) - d*cos(t))^2 + z^2) -4b^2((x*cos(t) + d*sin(t))^2 + y^2))^2 -64a^2b^2((x*cos(t) + d*sin(t))^2 + y^2)((x*sin(t) - d*cos(t))^2 + z^2) = 0
Graphs a tiger! Rotate/translate morphs, radius sizes all check out.
\begin{align*}
& \left(\left(x^2 +y^2 +z^2 +w^2 +a^2 +b^2 -c^2\right)^2 -4a^2\left(x^2 + y^2\right) -4b^2\left(z^2 + w^2\right)\right)^2 -64a^2b^2\left(x^2 + y^2\right)\left(z^2 + w^2\right) \\
\end{align*}
Generate 3-torus parametric equation, and convert to implicit
x = rcos(u)
y = rsin(u)
x = rcos(u)+R
y = rsin(u)
x = (rcos(u)+R)cos(v)
y = (rcos(u)+R)sin(v)
z = rsin(u)
x = (rcos(u)+R)cos(v)+P
y = (rcos(u)+R)sin(v)
z = rsin(u)
x = ((rcos(u)+R)cos(v)+P)cos(t)
w = ((rcos(u)+R)cos(v)+P)sin(t)
y = (rcos(u)+R)sin(v)
z = rsin(u)
R1 > R2 > r
x(t,u,v) = ((rcos(u)+R2)cos(v)+R1)cos(t)
y(t,u,v) = ((rcos(u)+R2)cos(v)+R1)sin(t)
z(u,v) = (rcos(u)+R2)sin(v)
w(u) = rsin(u)
Convert to Implicit Equation:
x = ((rcos(u)+R)cos(v)+P)cos(t)
y = ((rcos(u)+R)cos(v)+P)sin(t)
z = (rcos(u)+R)sin(v)
w = rsin(u)
Using the cos^2(t) + sin^2(t) = 1 identity
x = ((rcos(u)+R)cos(v)+P)cos(t)
y = ((rcos(u)+R)cos(v)+P)sin(t)
z = (rcos(u)+R)sin(v)
w = rsin(u)
x/((rcos(u)+R)cos(v)+P) = cos(t)
y/((rcos(u)+R)cos(v)+P) = sin(t)
z = (rcos(u)+R)sin(v)
w = rsin(u)
(x/((rcos(u)+R)cos(v)+P))^2 = cos^2(t)
(y/((rcos(u)+R)cos(v)+P))^2 = sin^2(t)
z = (rcos(u)+R)sin(v)
w = rsin(u)
(x/((rcos(u)+R)cos(v)+P))^2 + (y/((rcos(u)+R)cos(v)+P))^2 = 1
z = (rcos(u)+R)sin(v)
w = rsin(u)
x^2 + y^2 = ((rcos(u)+R)cos(v)+P)^2
z = (rcos(u)+R)sin(v)
w = rsin(u)
sqrt(x^2 + y^2) = (rcos(u)+R)cos(v)+P
z = (rcos(u)+R)sin(v)
w = rsin(u)
sqrt(x^2 + y^2)-P = (rcos(u)+R)cos(v)
z = (rcos(u)+R)sin(v)
w = rsin(u)
(sqrt(x^2 + y^2)-P)/(rcos(u)+R) = cos(v)
z/(rcos(u)+R) = sin(v)
w = rsin(u)
((sqrt(x^2 + y^2)-P)/(rcos(u)+R))^2 = cos^2(v)
(z/(rcos(u)+R))^2 = sin^2(v)
w = rsin(u)
((sqrt(x^2 + y^2)-P)/(rcos(u)+R))^2 + (z/(rcos(u)+R))^2 = 1
w = rsin(u)
(sqrt(x^2 + y^2)-P)^2 + z^2 = (rcos(u)+R)^2
w = rsin(u)
sqrt((sqrt(x^2 + y^2)-P)^2 + z^2) = rcos(u)+R
w = rsin(u)
sqrt((sqrt(x^2 + y^2)-P)^2 + z^2)-R = rcos(u)
w = rsin(u)
(sqrt((sqrt(x^2 + y^2)-P)^2 + z^2)-R)/r = cos(u)
w/r = sin(u)
((sqrt((sqrt(x^2 + y^2)-P)^2 + z^2)-R)/r)^2 = cos^2(u)
(w/r)^2 = sin^2(u)
((sqrt((sqrt(x^2 + y^2)-P)^2 + z^2)-R)/r)^2 + (w/r)^2 = 1
(sqrt((sqrt(x^2 + y^2)-P)^2 + z^2)-R)^2 + w^2 = r^2
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