Moving frame calculations

General idea

Fundamental thing to start with $$ f(z) = \bar{f}(\bar{z}) $$

Then you need a general group transformation. Which you should simplify as far as possible. First: translations can be removed. This is always true, since you have choice of cross-section. It is an extremely good idea since it means that you will now evaluate everything at (x,y) = (0,0). Second: Remove any other group parameters you can.

Now prolong the group action. Just write it out, don't think. The code below should help.

Turn the prolonged action into a matrix up to the appropriate number of derivatives. Remember that you are solving for the entries of the matrix, not for the vectors.

Now comes the art. You need to find the rest of the cross-section. Choose values for sets of the barred derivatives in order to get all the parameters. What is left over is an invariant.

Mobius

Fundamental thing to start with $$ f(z) = \bar{f}(\bar{z}) $$ A general Mobius transformation is $$ \bar{z} = \frac{\alpha z + \beta}{\gamma z + \delta} $$

Assuming $\delta \neq 0$, we can normalise it: $\delta = 1$. For our cross-section, we'll choose $\bar{z} = 0$. From any point $z$, this determines $\beta$, so wlog assume we start at $z = 0$, i.e. that $\beta = 0$. So $z = 0$ from now on!!!.

$$ \bar{x} + i\bar{y} = \frac{(a + ib)(x + iy)}{1 + (c + id)(x + iy)} $$

After the zeroth-order frame translates the general point $\bar{x}$ to $0$. So all derivative calculations will be evaluated at $x = y = 0$.



In [1]:

    
from sympy import Function, Symbol, symbols, init_printing, expand, I, re, im
from IPython.display import Math, display

init_printing()
from transvectants import *

def disp(expr):
    display(Math(my_latex(expr)))



In [62]:









    



---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-62-4c851033d21e> in <module>
      1 from importlib import reload
----> 2 reload(transvectants)

NameError: name 'transvectants' is not defined



In [2]:

    
# p and q are \bar{x} \bar{y}
x, y = symbols('x y')
p, q = symbols('p q')
a, b, c, d = symbols('a b c d')
p = ((a*x - b*y)*(1 + c*x - d*y) + (b*x + a*y)*(d*x + c*y))/((1 + c*x - d*y)**2 + (d*x + c*y)**2)
q = ((b*x + a*y)*(1 + c*x - d*y) - (a*x - b*y)*(d*x + c*y))/((1 + c*x - d*y)**2 + (d*x + c*y)**2)

# Can we tidy this later - but this does work
g = Function('g')(p, q)
g









    Out[2]:





$$g{\left (\frac{\left(a x - b y\right) \left(c x - d y + 1\right) + \left(a y + b x\right) \left(c y + d x\right)}{\left(c y + d x\right)^{2} + \left(c x - d y + 1\right)^{2}},\frac{- \left(a x - b y\right) \left(c y + d x\right) + \left(a y + b x\right) \left(c x - d y + 1\right)}{\left(c y + d x\right)^{2} + \left(c x - d y + 1\right)^{2}} \right )}$$



In [3]:

    
# In the below, interpret fb_blah as the the f derivative
foo = diff(g, x).subs([(x, 0), (y, 0)])



In [4]:

    
foo









    Out[4]:





$$a \left. \frac{d}{d \xi_{1}} g{\left (\xi_{1},0 \right )} \right|_{\substack{ \xi_{1}=0 }} + b \left. \frac{d}{d \xi_{2}} g{\left (0,\xi_{2} \right )} \right|_{\substack{ \xi_{2}=0 }}$$



In [5]:

    
disp(diff(g, x).subs([(x, 0), (y, 0)]))
disp(diff(g, x).subs([(x, 0), (y, 0)]))
disp(diff(g, x, x).subs([(x, 0), (y, 0)]))
disp(diff(g, x, y).subs([(x, 0), (y, 0)]))
disp(diff(g, y, y).subs([(x, 0), (y, 0)]))
disp(diff(g, x, x, x).subs([(x, 0), (y, 0)]))
disp(diff(g, x, x, y).subs([(x, 0), (y, 0)]))
disp(diff(g, x, y, y).subs([(x, 0), (y, 0)]))
print('boo')
disp(diff(g, y, y, y).subs([(x, 0), (y, 0)]))









    





$\displaystyle a g_{\xi_{1}} + b g_{\xi_{2}}$






    





$\displaystyle a g_{\xi_{1}} + b g_{\xi_{2}}$






    





$\displaystyle a \left(a g_{\xi_{1}\xi_{1}} + b g_{\xi_{1}\xi_{2}}\right) + b \left(a g_{\xi_{1}\xi_{2}} + b g_{\xi_{2}\xi_{2}}\right) - 2 \left(a c - b d\right) g_{\xi_{1}} + 2 \left(- a d - b c\right) g_{\xi_{2}}$






    





$\displaystyle a \left(a g_{\xi_{1}\xi_{2}} - b g_{\xi_{1}\xi_{1}}\right) + b \left(a g_{\xi_{2}\xi_{2}} - b g_{\xi_{1}\xi_{2}}\right) - 2 \left(a c - b d\right) g_{\xi_{2}} - 2 \left(- a d - b c\right) g_{\xi_{1}}$






    





$\displaystyle a \left(a g_{\xi_{2}\xi_{2}} - b g_{\xi_{1}\xi_{2}}\right) - b \left(a g_{\xi_{1}\xi_{2}} - b g_{\xi_{1}\xi_{1}}\right) - 2 \left(- a c + b d\right) g_{\xi_{1}} - 2 \left(- a d - b c\right) g_{\xi_{2}}$






    





$\displaystyle - a \left(a \left(- a g_{\xi_{1}\xi_{1}\xi_{1}} - b g_{\xi_{1}\xi_{1}\xi_{2}}\right) - b^{2} g_{\xi_{1}\xi_{2}\xi_{2}} + 2 \left(a c - b d\right) g_{\xi_{1}\xi_{1}} + 2 \left(a d + b c\right) g_{\xi_{1}\xi_{2}}\right) + b \left(a^{2} g_{\xi_{1}\xi_{1}\xi_{2}} + b \left(a g_{\xi_{1}\xi_{2}\xi_{2}} + b g_{\xi_{2}\xi_{2}\xi_{2}}\right) - 2 \left(a c - b d\right) g_{\xi_{1}\xi_{2}} + 2 \left(- a d - b c\right) g_{\xi_{2}\xi_{2}}\right) - 4 \left(a c - b d\right) \left(a g_{\xi_{1}\xi_{1}} + b g_{\xi_{1}\xi_{2}}\right) + 4 \left(- a d - b c\right) \left(a g_{\xi_{1}\xi_{2}} + b g_{\xi_{2}\xi_{2}}\right) + 6 \left(4 a c^{2} - a \left(c^{2} + d^{2}\right) - 2 c \left(a c + b d\right)\right) g_{\xi_{1}} - 6 \left(- 4 b c^{2} + b \left(c^{2} + d^{2}\right) - 2 c \left(a d - b c\right)\right) g_{\xi_{2}}$






    





$\displaystyle - a \left(- a b g_{\xi_{1}\xi_{2}\xi_{2}} + a \left(- a g_{\xi_{1}\xi_{1}\xi_{2}} + b g_{\xi_{1}\xi_{1}\xi_{1}}\right) + 2 \left(a c - b d\right) g_{\xi_{1}\xi_{2}} + 2 \left(- a d - b c\right) g_{\xi_{1}\xi_{1}}\right) + b \left(- a b g_{\xi_{1}\xi_{1}\xi_{2}} + b \left(a g_{\xi_{2}\xi_{2}\xi_{2}} - b g_{\xi_{1}\xi_{2}\xi_{2}}\right) - 2 \left(a c - b d\right) g_{\xi_{2}\xi_{2}} - 2 \left(- a d - b c\right) g_{\xi_{1}\xi_{2}}\right) - 2 \left(a c - b d\right) \left(a g_{\xi_{1}\xi_{2}} + b g_{\xi_{2}\xi_{2}}\right) - 2 \left(a c - b d\right) \left(a g_{\xi_{1}\xi_{2}} - b g_{\xi_{1}\xi_{1}}\right) + 2 \left(- a d - b c\right) \left(a g_{\xi_{2}\xi_{2}} - b g_{\xi_{1}\xi_{2}}\right) - 2 \left(- a d - b c\right) \left(a g_{\xi_{1}\xi_{1}} + b g_{\xi_{1}\xi_{2}}\right) - 2 \left(- 4 a c^{2} + a \left(c^{2} + d^{2}\right) + 8 b c d + 2 d \left(a d - b c\right)\right) g_{\xi_{2}} + 2 \left(- 8 a c d - 4 b c^{2} + b \left(c^{2} + d^{2}\right) + 2 d \left(a c + b d\right)\right) g_{\xi_{1}}$






    





$\displaystyle - a \left(- a^{2} g_{\xi_{1}\xi_{2}\xi_{2}} - b \left(- a g_{\xi_{1}\xi_{1}\xi_{2}} + b g_{\xi_{1}\xi_{1}\xi_{1}}\right) + 2 \left(- a c + b d\right) g_{\xi_{1}\xi_{1}} + 2 \left(- a d - b c\right) g_{\xi_{1}\xi_{2}}\right) + b \left(a \left(a g_{\xi_{2}\xi_{2}\xi_{2}} - b g_{\xi_{1}\xi_{2}\xi_{2}}\right) + b^{2} g_{\xi_{1}\xi_{1}\xi_{2}} - 2 \left(- a c + b d\right) g_{\xi_{1}\xi_{2}} - 2 \left(- a d - b c\right) g_{\xi_{2}\xi_{2}}\right) - 4 \left(a c - b d\right) \left(a g_{\xi_{2}\xi_{2}} - b g_{\xi_{1}\xi_{2}}\right) - 4 \left(- a d - b c\right) \left(a g_{\xi_{1}\xi_{2}} - b g_{\xi_{1}\xi_{1}}\right) - 2 \left(- 4 a d^{2} + a \left(c^{2} + d^{2}\right) - 8 b c d + 2 c \left(a c + b d\right)\right) g_{\xi_{1}} + 2 \left(- 8 a c d + 4 b d^{2} - b \left(c^{2} + d^{2}\right) + 2 c \left(a d - b c\right)\right) g_{\xi_{2}}$






    



boo






    





$\displaystyle a \left(a \left(a g_{\xi_{2}\xi_{2}\xi_{2}} - b g_{\xi_{1}\xi_{2}\xi_{2}}\right) + b^{2} g_{\xi_{1}\xi_{1}\xi_{2}} - 2 \left(- a c + b d\right) g_{\xi_{1}\xi_{2}} - 2 \left(- a d - b c\right) g_{\xi_{2}\xi_{2}}\right) + b \left(- a^{2} g_{\xi_{1}\xi_{2}\xi_{2}} - b \left(- a g_{\xi_{1}\xi_{1}\xi_{2}} + b g_{\xi_{1}\xi_{1}\xi_{1}}\right) + 2 \left(- a c + b d\right) g_{\xi_{1}\xi_{1}} + 2 \left(- a d - b c\right) g_{\xi_{1}\xi_{2}}\right) - 4 \left(- a c + b d\right) \left(a g_{\xi_{1}\xi_{2}} - b g_{\xi_{1}\xi_{1}}\right) - 4 \left(- a d - b c\right) \left(a g_{\xi_{2}\xi_{2}} - b g_{\xi_{1}\xi_{2}}\right) - 6 \left(- 4 a d^{2} + a \left(c^{2} + d^{2}\right) + 2 d \left(a d - b c\right)\right) g_{\xi_{2}} + 6 \left(- 4 b d^{2} + b \left(c^{2} + d^{2}\right) + 2 d \left(a c + b d\right)\right) g_{\xi_{1}}$

Write this out as a matrix

Now pick the cross-section

Conformal

OK, let's try again. This time we are gonna be awesome and do conformal. The Taylor expansion of a general conformal map up to third order is $$ \bar{z} = c_0 + c_1 z + c_2 z^2 + c_3 z^3 $$ Or in components, $$ \begin{align*} \bar{x} &= a_0 + a_1 x + a_2 (x^2 - y^2) + a_3 (x^3 - 3 xy^2) - b_1 y - 2b_2xy - 3b_3x^2y + b_3y^3 \\ 3 & = 4 \end{align*} $$



In [54]:

    
x, y = symbols('x y', real=True)
a0, a1, a2, a3, b0, b1, b2, b3 = symbols('a_0 a_1 a_2 a_3 b_0 b_1 b_2 b_3', real=True)
z = x + I*y
# We have removed the a_0 + I*b_0 term to take out the translation
w = (a1 + I*b1)*z + (a2 + I*b2)*z**2 + (a3 + I*b3)*z**3
p = re(w)
q = im(w)
p









    Out[54]:





$$a_{1} x + a_{2} \left(x^{2} - y^{2}\right) + a_{3} \left(x^{3} - 3 x y^{2}\right) - b_{1} y - 2 b_{2} x y - b_{3} \left(3 x^{2} y - y^{3}\right)$$



In [59]:

    
fb = Function('g')(p, q)
disp(diff(fb, x).subs([(x, 0), (y, 0)]))
disp(diff(fb, y).subs([(x, 0), (y, 0)]))
disp(diff(fb, x, x).subs([(x, 0), (y, 0)]))
disp(diff(fb, x, y).subs([(x, 0), (y, 0)]))
disp(diff(fb, y, y).subs([(x, 0), (y, 0)]))
disp(diff(fb, x, x, x).subs([(x, 0), (y, 0)]))
disp(diff(fb, x, x, y).subs([(x, 0), (y, 0)]))
disp(diff(fb, x, y, y).subs([(x, 0), (y, 0)]))
print('boo')
disp(diff(fb, y, y, y).subs([(x, 0), (y, 0)]))









    





$\displaystyle a_{1} \left. g_{\xi_{1}} \right|_{\substack{ \xi_{1}=0 }} + b_{1} \left. g_{\xi_{2}} \right|_{\substack{ \xi_{2}=0 }}$






    





$\displaystyle a_{1} \left. g_{\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. g_{\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}$






    





$\displaystyle a_{1} \left(a_{1} \left. g_{\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }} + b_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) + 2 a_{2} \left. g_{\xi_{1}} \right|_{\substack{ \xi_{1}=0 }} + b_{1} \left(a_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + b_{1} \left. g_{\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }}\right) + 2 b_{2} \left. g_{\xi_{2}} \right|_{\substack{ \xi_{2}=0 }}$






    





$\displaystyle a_{1} \left(a_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. g_{\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}\right) + 2 a_{2} \left. g_{\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} + b_{1} \left(a_{1} \left. g_{\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) - 2 b_{2} \left. g_{\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}$






    





$\displaystyle a_{1} \left(a_{1} \left. g_{\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) - 2 a_{2} \left. g_{\xi_{1}} \right|_{\substack{ \xi_{1}=0 }} - b_{1} \left(a_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. g_{\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}\right) - 2 b_{2} \left. g_{\xi_{2}} \right|_{\substack{ \xi_{2}=0 }}$






    





$\displaystyle a_{1} \left(a_{1} \left(a_{1} \left. g_{\xi_{1}\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }} + b_{1} \left. \left. g_{\xi_{1}\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) + 2 a_{2} \left. g_{\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }} + b_{1}^{2} \left. \left. g_{\xi_{1}\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} \right|_{\substack{ \xi_{1}=0 }} + 2 b_{2} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) + 4 a_{2} \left(a_{1} \left. g_{\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }} + b_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) + 6 a_{3} \left. g_{\xi_{1}} \right|_{\substack{ \xi_{1}=0 }} + b_{1} \left(a_{1}^{2} \left. \left. g_{\xi_{1}\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + 2 a_{2} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + b_{1} \left(a_{1} \left. \left. g_{\xi_{1}\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} \right|_{\substack{ \xi_{1}=0 }} + b_{1} \left. g_{\xi_{2}\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }}\right) + 2 b_{2} \left. g_{\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }}\right) + 4 b_{2} \left(a_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + b_{1} \left. g_{\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }}\right) + 6 b_{3} \left. g_{\xi_{2}} \right|_{\substack{ \xi_{2}=0 }}$






    





$\displaystyle - a_{1} \left(- a_{1} b_{1} \left. \left. g_{\xi_{1}\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} \right|_{\substack{ \xi_{1}=0 }} + a_{1} \left(- a_{1} \left. \left. g_{\xi_{1}\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + b_{1} \left. g_{\xi_{1}\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}\right) - 2 a_{2} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + 2 b_{2} \left. g_{\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}\right) + 2 a_{2} \left(a_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + b_{1} \left. g_{\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }}\right) + 2 a_{2} \left(a_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. g_{\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}\right) + 6 a_{3} \left. g_{\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} + b_{1} \left(- a_{1} b_{1} \left. \left. g_{\xi_{1}\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + 2 a_{2} \left. g_{\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} + b_{1} \left(a_{1} \left. g_{\xi_{2}\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. \left. g_{\xi_{1}\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} \right|_{\substack{ \xi_{1}=0 }}\right) - 2 b_{2} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) + 2 b_{2} \left(a_{1} \left. g_{\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) - 2 b_{2} \left(a_{1} \left. g_{\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }} + b_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) - 6 b_{3} \left. g_{\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}$






    





$\displaystyle - a_{1} \left(a_{1} b_{1} \left. \left. g_{\xi_{1}\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + 2 a_{2} \left. g_{\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }} - b_{1} \left(- a_{1} \left. \left. g_{\xi_{1}\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + b_{1} \left. g_{\xi_{1}\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}\right) + 2 b_{2} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) + 4 a_{2} \left(a_{1} \left. g_{\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) - 6 a_{3} \left. g_{\xi_{1}} \right|_{\substack{ \xi_{1}=0 }} + b_{1} \left(a_{1} \left(a_{1} \left. g_{\xi_{2}\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. \left. g_{\xi_{1}\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} \right|_{\substack{ \xi_{1}=0 }}\right) - 2 a_{2} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + b_{1}^{2} \left. \left. g_{\xi_{1}\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} - 2 b_{2} \left. g_{\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }}\right) - 4 b_{2} \left(a_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. g_{\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}\right) - 6 b_{3} \left. g_{\xi_{2}} \right|_{\substack{ \xi_{2}=0 }}$






    



boo






    





$\displaystyle a_{1} \left(a_{1} \left(a_{1} \left. g_{\xi_{2}\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. \left. g_{\xi_{1}\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} \right|_{\substack{ \xi_{1}=0 }}\right) - 2 a_{2} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + b_{1}^{2} \left. \left. g_{\xi_{1}\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} - 2 b_{2} \left. g_{\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }}\right) - 4 a_{2} \left(a_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. g_{\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}\right) - 6 a_{3} \left. g_{\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} + b_{1} \left(a_{1} b_{1} \left. \left. g_{\xi_{1}\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + 2 a_{2} \left. g_{\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }} - b_{1} \left(- a_{1} \left. \left. g_{\xi_{1}\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }} + b_{1} \left. g_{\xi_{1}\xi_{1}\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}\right) + 2 b_{2} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) - 4 b_{2} \left(a_{1} \left. g_{\xi_{2}\xi_{2}} \right|_{\substack{ \xi_{2}=0 }} - b_{1} \left. \left. g_{\xi_{1}\xi_{2}} \right|_{\substack{ \xi_{1}=0 }} \right|_{\substack{ \xi_{2}=0 }}\right) + 6 b_{3} \left. g_{\xi_{1}} \right|_{\substack{ \xi_{1}=0 }}$

Write this out as a matrix

Now for the cross-section



In [19]:









    





$\displaystyle 2 f_{xxxx} f_{yyyy} - 8 f_{xyyy} f_{xxxy} + 6 \left(f_{xxyy}\right)^{2}$



In [14]:

    
disp(expand(partial_transvectDant((f, f, f), [[0, 1], [0, 1], [0, 2], [0, 2]])))









    





$\displaystyle \left(f_{xx}\right)^{2} f_{yyyy} + 2 f_{xx} f_{yy} f_{xxyy} - 4 f_{xx} f_{xy} f_{xyyy} + f_{xxxx} \left(f_{yy}\right)^{2} - 4 f_{yy} f_{xy} f_{xxxy} + 4 \left(f_{xy}\right)^{2} f_{xxyy}$



In [11]:

    
disp(expand(partial_transvectant((f, f, f, f, f), [[0, 1], [0, 1], [2, 3], [2, 3], [2, 4]]) ) -2*(expand(partial_transvectant((f, f, f, f, f), [[0, 1], [1, 2], [2, 3], [3, 0], [0, 4]]) )))









    





$\displaystyle 0$



In [16]:

    
disp(expand(partial_transvectant((f, f, f), [[0, 1], [0, 1], [0, 1], [0, 2]])))









    





$\displaystyle f_{x} f_{xxx} f_{yyyy} - f_{x} f_{yyy} f_{xxxy} + 3 f_{x} f_{xyy} f_{xxyy} - 3 f_{x} f_{xyyy} f_{xxy} - f_{xxx} f_{y} f_{xyyy} + f_{xxxx} f_{y} f_{yyy} - 3 f_{y} f_{xyy} f_{xxxy} + 3 f_{y} f_{xxy} f_{xxyy}$



In [18]:

    
disp(expand(partial_transvectant((f, f), [[0, 1], [0, 1], [0, 1]])))









    





$\displaystyle 0$



In [15]:

    
#C = transvectant(f, f, 2)
#D = -partial_transvectant((f, f, f), [[0, 1], [1, 2]])
# We are going to build these by weight, not degree. 
# Hence order does not match dispaper
# Weight 4 (2 of 'em)

I4_1 = partial_transvectant((f,f),[[0,1],[0,1]]) # = C
I4_2 = partial_transvectant((f, f, f), [[0, 1], [1, 2]]) # = -D

# Weight 6 (2 of 'em)
print('weight 3:')
I6_1 = partial_transvectant((f,f,f),[[0,1],[0,1],[0,2]]) # = transvectant(f, C, 1)
I6_2 = partial_transvectant((f,f,f,f),[[0,1],[0,2],[0,3]])

# Weight 8 (7 of 'em??)
print('weight 4:')
I8_1 = expand(partial_transvectant((f,f,f),[[0,1],[0,1],[1,2],[0,2]]))
I8_2 = expand(partial_transvectant((f,f,f,f),[[0,1],[0,1],[1,2],[2,3]]))
I8_3 = expand(partial_transvectant((f,f,f,f),[[0,1],[1,2],[2,3],[3,0]]))
I8_4 = expand(partial_transvectant((f,f,f,f),[[0,1],[1,2],[1,2],[2,3]]))

I8_5 = expand(partial_transvectant((f,f,f,f,f),[[0,1],[1,2],[2,3],[3,4]]))
I8_6 = expand(partial_transvectant((f,f,f,f,f),[[0,1],[0,2],[0,3],[3,4]]))
I8_7 = expand(partial_transvectant((f,f,f,f,f),[[0,1],[0,2],[0,3],[0,4]]))



print('weight 2')
disp(I4_1)
disp(I4_2)

print('weight 3')
disp(I6_1)
disp(expand(I6_2))

print('weight 4')
disp(I8_1)
print('')
disp(I8_2)
print('')
disp(I8_3)
print('')
disp(I8_4)
print('')
disp(I8_5)
print('')
disp(I8_6)
print('')
disp(I8_7)









    



weight 3:
weight 4:
weight 2






    





$\displaystyle 2 f_{xx} f_{yy} - 2 \left(f_{xy}\right)^{2}$






    





$\displaystyle - \left(f_{x}\right)^{2} f_{yy} + 2 f_{x} f_{y} f_{xy} - f_{xx} \left(f_{y}\right)^{2}$






    



weight 3






    





$\displaystyle - \left(f_{xx} f_{yyy} + f_{yy} f_{xxy} - 2 f_{xy} f_{xyy}\right) f_{x} + \left(f_{xx} f_{xyy} + f_{xxx} f_{yy} - 2 f_{xy} f_{xxy}\right) f_{y}$






    





$\displaystyle - \left(f_{x}\right)^{3} f_{yyy} + 3 \left(f_{x}\right)^{2} f_{y} f_{xyy} - 3 f_{x} \left(f_{y}\right)^{2} f_{xxy} + f_{xxx} \left(f_{y}\right)^{3}$






    



weight 4






    





$\displaystyle 2 f_{xx} f_{yyy} f_{xxy} - 2 f_{xx} \left(f_{xyy}\right)^{2} + 2 f_{xxx} f_{yy} f_{xyy} - 2 f_{xxx} f_{yyy} f_{xy} - 2 f_{yy} \left(f_{xxy}\right)^{2} + 2 f_{xy} f_{xyy} f_{xxy}$






    









    





$\displaystyle - f_{x} f_{xx} f_{yy} f_{xyy} + f_{x} f_{xx} f_{yyy} f_{xy} - f_{x} f_{xxx} \left(f_{yy}\right)^{2} + 3 f_{x} f_{yy} f_{xy} f_{xxy} - 2 f_{x} \left(f_{xy}\right)^{2} f_{xyy} - \left(f_{xx}\right)^{2} f_{y} f_{yyy} - f_{xx} f_{y} f_{yy} f_{xxy} + 3 f_{xx} f_{y} f_{xy} f_{xyy} + f_{xxx} f_{y} f_{yy} f_{xy} - 2 f_{y} \left(f_{xy}\right)^{2} f_{xxy}$






    









    





$\displaystyle 2 \left(f_{xx}\right)^{2} \left(f_{yy}\right)^{2} - 4 f_{xx} f_{yy} \left(f_{xy}\right)^{2} + 2 \left(f_{xy}\right)^{4}$






    









    





$\displaystyle - 2 \left(f_{x}\right)^{2} f_{yyy} f_{xxy} + 2 \left(f_{x}\right)^{2} \left(f_{xyy}\right)^{2} + 2 f_{x} f_{xxx} f_{y} f_{yyy} - 2 f_{x} f_{y} f_{xyy} f_{xxy} - 2 f_{xxx} \left(f_{y}\right)^{2} f_{xyy} + 2 \left(f_{y}\right)^{2} \left(f_{xxy}\right)^{2}$






    









    





$\displaystyle \left(f_{x}\right)^{2} f_{xx} \left(f_{yy}\right)^{2} - \left(f_{x}\right)^{2} f_{yy} \left(f_{xy}\right)^{2} - 2 f_{x} f_{xx} f_{y} f_{yy} f_{xy} + 2 f_{x} f_{y} \left(f_{xy}\right)^{3} + \left(f_{xx}\right)^{2} \left(f_{y}\right)^{2} f_{yy} - f_{xx} \left(f_{y}\right)^{2} \left(f_{xy}\right)^{2}$






    









    





$\displaystyle - \left(f_{x}\right)^{3} f_{yy} f_{xyy} + \left(f_{x}\right)^{3} f_{yyy} f_{xy} - \left(f_{x}\right)^{2} f_{xx} f_{y} f_{yyy} + 2 \left(f_{x}\right)^{2} f_{y} f_{yy} f_{xxy} - \left(f_{x}\right)^{2} f_{y} f_{xy} f_{xyy} + 2 f_{x} f_{xx} \left(f_{y}\right)^{2} f_{xyy} - f_{x} f_{xxx} \left(f_{y}\right)^{2} f_{yy} - f_{x} \left(f_{y}\right)^{2} f_{xy} f_{xxy} - f_{xx} \left(f_{y}\right)^{3} f_{xxy} + f_{xxx} \left(f_{y}\right)^{3} f_{xy}$






    









    





$\displaystyle \left(f_{x}\right)^{4} f_{yyyy} - 4 \left(f_{x}\right)^{3} f_{y} f_{xyyy} + 6 \left(f_{x}\right)^{2} \left(f_{y}\right)^{2} f_{xxyy} - 4 f_{x} \left(f_{y}\right)^{3} f_{xxxy} + f_{xxxx} \left(f_{y}\right)^{4}$



In [162]:

    
# Only 'weight 4' affine invariant
disp(I4_2/I4_1)

# Only 'weight 6' affine invariant
disp(I6_2/I6_1)









    





$$\frac{- \left(f_{x}\right)^{2} f_{yy} + 2 f_{x} f_{y} f_{xy} - \left(f_{y}\right)^{2} f_{xx}}{2 f_{xx} f_{yy} - 2 \left(f_{xy}\right)^{2}}$$






    





$$\frac{- \left(f_{x}\right)^{3} f_{yyy} + 3 \left(f_{x}\right)^{2} f_{y} f_{xyy} - 3 f_{x} \left(f_{y}\right)^{2} f_{xxy} + \left(f_{y}\right)^{3} f_{xxx}}{\left(f_{xx} f_{xyy} - 2 f_{xy} f_{xxy} + f_{yy} f_{xxx}\right) f_{y} - \left(f_{xx} f_{yyy} - 2 f_{xy} f_{xyy} + f_{yy} f_{xxy}\right) f_{x}}$$



In [24]:

    
disp(partial_transvectant((f,f,f,f,f),[[0,2],[1,2],[2,3],[3,4]]))









    





$$- \left(f_{x}\right)^{3} f_{xy} f_{yyy} + \left(f_{x}\right)^{3} f_{yy} f_{xyy} + \left(f_{x}\right)^{2} f_{y} f_{xx} f_{yyy} + \left(f_{x}\right)^{2} f_{y} f_{xy} f_{xyy} - 2 \left(f_{x}\right)^{2} f_{y} f_{yy} f_{xxy} - 2 f_{x} \left(f_{y}\right)^{2} f_{xx} f_{xyy} + f_{x} \left(f_{y}\right)^{2} f_{xy} f_{xxy} + f_{x} \left(f_{y}\right)^{2} f_{yy} f_{xxx} + \left(f_{y}\right)^{3} f_{xx} f_{xxy} - \left(f_{y}\right)^{3} f_{xy} f_{xxx}$$



In [35]:

    
disp(partial_transvectant((f,f,C),[[0,1],[1,2]]))









    





$$4 \left(\left(f_{x} f_{xy} - f_{y} f_{xx}\right) \left(f_{xx} f_{yyy} - 2 f_{xy} f_{xyy} + f_{yy} f_{xxy}\right) - \left(f_{x} f_{yy} - f_{y} f_{xy}\right) \left(f_{xx} f_{xyy} - 2 f_{xy} f_{xxy} + f_{yy} f_{xxx}\right)\right) f{\left (x,y \right )}$$



In [4]:

    
#disp(transvectant(C, C, 2))
funcs = (C, f**2)
pairs = [[0, 1]]
disp(partial_transvectant(funcs, pairs))









    





$$4 \left(\left(f_{xx} f_{xyy} - 2 f_{xy} f_{xxy} + f_{yy} f_{xxx}\right) f_{y} - \left(f_{xx} f_{yyy} - 2 f_{xy} f_{xyy} + f_{yy} f_{xxy}\right) f_{x}\right) f{\left (x,y \right )}$$



In [40]:

    
# Construct linear, quadratic, cubic forms
fx, fy, fxx, fxy, fyy, fxxx, fxxy, fxyy, fyyy = symbols('f_x, f_y, f_{xx}, f_{xy}, f_{yy}, f_{xxx}, f_{xxy}, f_{xyy}, f_{yyy}') 
l = fx*x + fy*y
q = fxx*x*x + 2*fxy*x*y + fyy*y*y
c = fxxx*x*x*x + 3*fxxy*x*x*y + 3*fxyy*x*y*y + fyyy*y*y*y



In [48]:

    
# I3 as a form (Robert's method to annoy us...)
disp(-expand(transvectant(q,transvectant(c,c,2),2)/288))
# I5
disp(expand(transvectant(transvectant(c,c,2),transvectant(c,c,2),2)/10368))
# I6
disp(transvectant(c,l**3,3)/36)









    





$$- f_{xxx} f_{xyy} f_{yy} + f_{xxx} f_{xy} f_{yyy} + f_{xxy}^{2} f_{yy} - f_{xxy} f_{xx} f_{yyy} - f_{xxy} f_{xyy} f_{xy} + f_{xx} f_{xyy}^{2}$$



In [ ]:



In [107]:









    





$$- 2 f_{xx} f_{xxy} f_{yyy} + 2 f_{xx} \left(f_{xyy}\right)^{2} + 2 f_{xy} f_{xxx} f_{yyy} - 2 f_{xy} f_{xxy} f_{xyy} - 2 f_{yy} f_{xxx} f_{xyy} + 2 f_{yy} \left(f_{xxy}\right)^{2}$$



In [102]:









    





$$2 f_{xx} f_{xxy} f_{yyy} - 2 f_{xx} \left(f_{xyy}\right)^{2} - 2 f_{xy} f_{xxx} f_{yyy} + 2 f_{xy} f_{xxy} f_{xyy} + 2 f_{yy} f_{xxx} f_{xyy} - 2 f_{yy} \left(f_{xxy}\right)^{2}$$



In [119]:









    





$$0$$



In [141]:

    
disp(simplify(partial_transvectant((f,f,f,f),[[0,1],[1,2],[2,3]])))
disp(simplify(partial_transvectant((f,f,f,f),[[0,1],[1,2],[2,3],[0,1]])))
disp(simplify(partial_transvectant((f,f,f,f),[[0,1],[1,2],[2,3],[0,2]])))
disp(simplify(partial_transvectant((f,f,f,f),[[0,1],[1,2],[2,3],[0,1],[0,2]])))

disp(simplify(partial_transvectant((f,f,f,f),[[0,1],[1,2],[2,3],[3,0]])))
disp(simplify(partial_transvectant((f,f,f,f),[[0,1],[1,2],[2,3],[3,0],[0,1]])))
disp(simplify(partial_transvectant((f,f,f,f),[[0,1],[1,2],[2,3],[3,0],[0,2]])))
disp(simplify(partial_transvectant((f,f,f,f),[[0,1],[0,1],[1,2],[1,2],[2,3]])))
disp(simplify(partial_transvectant((f,f,f,f),[[0,1],[0,1],[1,2],[1,2],[2,3],[2,3]])))
disp(simplify(partial_transvectant((f,f,f,f),[[0,1],[1,2],[2,3],[3,0],[0,1],[1,2]])))









    





$$0$$






    





$$f_{x} f_{xx} f_{xy} f_{yyy} - f_{x} f_{xx} f_{yy} f_{xyy} - 2 f_{x} \left(f_{xy}\right)^{2} f_{xyy} + 3 f_{x} f_{xy} f_{yy} f_{xxy} - f_{x} \left(f_{yy}\right)^{2} f_{xxx} - f_{y} \left(f_{xx}\right)^{2} f_{yyy} + 3 f_{y} f_{xx} f_{xy} f_{xyy} - f_{y} f_{xx} f_{yy} f_{xxy} - 2 f_{y} \left(f_{xy}\right)^{2} f_{xxy} + f_{y} f_{xy} f_{yy} f_{xxx}$$






    





$$0$$






    





$$0$$






    





$$2 \left(f_{xx}\right)^{2} \left(f_{yy}\right)^{2} - 4 f_{xx} \left(f_{xy}\right)^{2} f_{yy} + 2 \left(f_{xy}\right)^{4}$$






    





$$0$$






    





$$0$$






    





$$\left(\left(f_{xx} f_{xxyy} - 2 f_{xy} f_{xxxy} + f_{yy} f_{xxxx}\right) f_{xyy} - 2 \left(f_{xx} f_{xyyy} - 2 f_{xy} f_{xxyy} + f_{yy} f_{xxxy}\right) f_{xxy} + \left(f_{xx} f_{yyyy} - 2 f_{xy} f_{xyyy} + f_{yy} f_{xxyy}\right) f_{xxx}\right) f_{y} - \left(\left(f_{xx} f_{xxyy} - 2 f_{xy} f_{xxxy} + f_{yy} f_{xxxx}\right) f_{yyy} - 2 \left(f_{xx} f_{xyyy} - 2 f_{xy} f_{xxyy} + f_{yy} f_{xxxy}\right) f_{xyy} + \left(f_{xx} f_{yyyy} - 2 f_{xy} f_{xyyy} + f_{yy} f_{xxyy}\right) f_{xxy}\right) f_{x}$$






    





$$2 \left(f_{xx}\right)^{2} f_{xxyy} f_{yyyy} - 2 \left(f_{xx}\right)^{2} \left(f_{xyyy}\right)^{2} - 4 f_{xx} f_{xy} f_{xxxy} f_{yyyy} + 4 f_{xx} f_{xy} f_{xxyy} f_{xyyy} + 2 f_{xx} f_{yy} f_{xxxx} f_{yyyy} - 4 f_{xx} f_{yy} f_{xxxy} f_{xyyy} + 2 f_{xx} f_{yy} \left(f_{xxyy}\right)^{2} + 8 \left(f_{xy}\right)^{2} f_{xxxy} f_{xyyy} - 8 \left(f_{xy}\right)^{2} \left(f_{xxyy}\right)^{2} - 4 f_{xy} f_{yy} f_{xxxx} f_{xyyy} + 4 f_{xy} f_{yy} f_{xxxy} f_{xxyy} + 2 \left(f_{yy}\right)^{2} f_{xxxx} f_{xxyy} - 2 \left(f_{yy}\right)^{2} \left(f_{xxxy}\right)^{2}$$






    





$$- f_{xx} \left(f_{xxy}\right)^{2} f_{yyyy} + 4 f_{xx} f_{xxy} f_{xyy} f_{xyyy} - 2 f_{xx} f_{xxy} f_{yyy} f_{xxyy} - 4 f_{xx} \left(f_{xyy}\right)^{2} f_{xxyy} + 4 f_{xx} f_{xyy} f_{yyy} f_{xxxy} - f_{xx} \left(f_{yyy}\right)^{2} f_{xxxx} + 2 f_{xy} f_{xxx} f_{xxy} f_{yyyy} - 4 f_{xy} f_{xxx} f_{xyy} f_{xyyy} + 2 f_{xy} f_{xxx} f_{yyy} f_{xxyy} - 4 f_{xy} \left(f_{xxy}\right)^{2} f_{xyyy} + 10 f_{xy} f_{xxy} f_{xyy} f_{xxyy} - 4 f_{xy} f_{xxy} f_{yyy} f_{xxxy} - 4 f_{xy} \left(f_{xyy}\right)^{2} f_{xxxy} + 2 f_{xy} f_{xyy} f_{yyy} f_{xxxx} - f_{yy} \left(f_{xxx}\right)^{2} f_{yyyy} + 4 f_{yy} f_{xxx} f_{xxy} f_{xyyy} - 2 f_{yy} f_{xxx} f_{xyy} f_{xxyy} - 4 f_{yy} \left(f_{xxy}\right)^{2} f_{xxyy} + 4 f_{yy} f_{xxy} f_{xyy} f_{xxxy} - f_{yy} \left(f_{xyy}\right)^{2} f_{xxxx}$$



In [ ]:

    
disp(simplify(partial_transvectant((f,f,f),[[0,1],[1,2],[2,0]])))
disp(simplify(partial_transvectant((f,f,f),[[0,1],[1,2],[0,1]])))
disp(simplify(partial_transvectant((f,f,f),[[0,1],[1,2],[2,0],[0,1]])))



In [142]:

    
disp(simplify(partial_transvectant((f,f,f,f),[[0,1],[0,1],[1,2],[1,2],[2,3]])))
disp(simplify(partial_transvectant((f,f,f,f),[[0,1],[0,1],[1,2],[1,2],[2,3],[2,3]])))









    





$$\left(\left(f_{xx} f_{xxyy} - 2 f_{xy} f_{xxxy} + f_{yy} f_{xxxx}\right) f_{xyy} - 2 \left(f_{xx} f_{xyyy} - 2 f_{xy} f_{xxyy} + f_{yy} f_{xxxy}\right) f_{xxy} + \left(f_{xx} f_{yyyy} - 2 f_{xy} f_{xyyy} + f_{yy} f_{xxyy}\right) f_{xxx}\right) f_{y} - \left(\left(f_{xx} f_{xxyy} - 2 f_{xy} f_{xxxy} + f_{yy} f_{xxxx}\right) f_{yyy} - 2 \left(f_{xx} f_{xyyy} - 2 f_{xy} f_{xxyy} + f_{yy} f_{xxxy}\right) f_{xyy} + \left(f_{xx} f_{yyyy} - 2 f_{xy} f_{xyyy} + f_{yy} f_{xxyy}\right) f_{xxy}\right) f_{x}$$






    





$$2 \left(f_{xx}\right)^{2} f_{xxyy} f_{yyyy} - 2 \left(f_{xx}\right)^{2} \left(f_{xyyy}\right)^{2} - 4 f_{xx} f_{xy} f_{xxxy} f_{yyyy} + 4 f_{xx} f_{xy} f_{xxyy} f_{xyyy} + 2 f_{xx} f_{yy} f_{xxxx} f_{yyyy} - 4 f_{xx} f_{yy} f_{xxxy} f_{xyyy} + 2 f_{xx} f_{yy} \left(f_{xxyy}\right)^{2} + 8 \left(f_{xy}\right)^{2} f_{xxxy} f_{xyyy} - 8 \left(f_{xy}\right)^{2} \left(f_{xxyy}\right)^{2} - 4 f_{xy} f_{yy} f_{xxxx} f_{xyyy} + 4 f_{xy} f_{yy} f_{xxxy} f_{xxyy} + 2 \left(f_{yy}\right)^{2} f_{xxxx} f_{xxyy} - 2 \left(f_{yy}\right)^{2} \left(f_{xxxy}\right)^{2}$$



In [147]:

    
disp(expand(partial_transvectant((f,f,f,f,f),[[0,1],[1,2],[2,3],[3,4]])))









    





$$\left(f_{x}\right)^{2} f_{xx} \left(f_{yy}\right)^{2} - \left(f_{x}\right)^{2} \left(f_{xy}\right)^{2} f_{yy} - 2 f_{x} f_{y} f_{xx} f_{xy} f_{yy} + 2 f_{x} f_{y} \left(f_{xy}\right)^{3} + \left(f_{y}\right)^{2} \left(f_{xx}\right)^{2} f_{yy} - \left(f_{y}\right)^{2} f_{xx} \left(f_{xy}\right)^{2}$$



In [153]:

    
disp(expand(partial_transvectant((f,f,f,f,f),[[0,1],[1,2],[2,3],[3,4]])))
disp(expand(partial_transvectant((f,f,f,f,f),[[0,1],[0,2],[0,3],[3,4]])))
disp(expand(partial_transvectant((f,f,f,f,f),[[0,1],[0,2],[0,3],[0,4]])))









    





$$\left(f_{x}\right)^{2} f_{xx} \left(f_{yy}\right)^{2} - \left(f_{x}\right)^{2} \left(f_{xy}\right)^{2} f_{yy} - 2 f_{x} f_{y} f_{xx} f_{xy} f_{yy} + 2 f_{x} f_{y} \left(f_{xy}\right)^{3} + \left(f_{y}\right)^{2} \left(f_{xx}\right)^{2} f_{yy} - \left(f_{y}\right)^{2} f_{xx} \left(f_{xy}\right)^{2}$$






    





$$- \left(f_{x}\right)^{2} f_{xx} \left(f_{yy}\right)^{2} + \left(f_{x}\right)^{2} \left(f_{xy}\right)^{2} f_{yy} + 2 f_{x} f_{y} f_{xx} f_{xy} f_{yy} - 2 f_{x} f_{y} \left(f_{xy}\right)^{3} - \left(f_{y}\right)^{2} \left(f_{xx}\right)^{2} f_{yy} + \left(f_{y}\right)^{2} f_{xx} \left(f_{xy}\right)^{2}$$






    





$$\left(f_{x}\right)^{3} f_{xy} f_{yyy} - \left(f_{x}\right)^{3} f_{yy} f_{xyy} - \left(f_{x}\right)^{2} f_{y} f_{xx} f_{yyy} - \left(f_{x}\right)^{2} f_{y} f_{xy} f_{xyy} + 2 \left(f_{x}\right)^{2} f_{y} f_{yy} f_{xxy} + 2 f_{x} \left(f_{y}\right)^{2} f_{xx} f_{xyy} - f_{x} \left(f_{y}\right)^{2} f_{xy} f_{xxy} - f_{x} \left(f_{y}\right)^{2} f_{yy} f_{xxx} - \left(f_{y}\right)^{3} f_{xx} f_{xxy} + \left(f_{y}\right)^{3} f_{xy} f_{xxx}$$






    





$$\left(f_{x}\right)^{4} f_{yyyy} - 4 \left(f_{x}\right)^{3} f_{y} f_{xyyy} + 6 \left(f_{x}\right)^{2} \left(f_{y}\right)^{2} f_{xxyy} - 4 f_{x} \left(f_{y}\right)^{3} f_{xxxy} + \left(f_{y}\right)^{4} f_{xxxx}$$



In [ ]:

Transvectants:

(1) We can use transvectants to create lots of invariants. More than Robert can. (2) We understand the role of weight and degree, and have a nice graph-based picture for all the invariants up to weight 8 (3) There is no issue with SA(2), but for A(2) the weight 4 invariants have non-removable singularities. There are three possible solutions: (i) claim you only do this on parts of images, (ii) look at the weight 8 invariants instead, particularly I8_1 and I8_3 which do not have f_x in at all, or I4^2/I8, (iii) consider projection as Robert did. Our preference is (ii) but we haven't done it yet.

Moving frame:

(1) We understand it, but it is still ugly (2) So we just write it as is

Hence:

We can build all the invariants we need now. We still have a projection issue. And hence a high weight invariant issue. And hence a serious noise problem.

But we probably now have enough for a paper. So we should write it.