︠30948442-739d-44a9-b0d1-eb2f6001b63es︠ load("tensegrity.sage") nodes = [1,2,3,4,5,6] h = 3 # Equation (1) in the paper: node_coordinates = {1:(1,0,0), 2:(-1/2, sqrt(3)/2, 0), 3:(-1/2, -sqrt(3)/2, 0), 4:(-sqrt(3)/2, -1/2, h), 5:(sqrt(3)/2, -1/2, h), 6:(0,1,h)} edges = [(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)] dim = 3 truss = Truss(nodes,edges,node_coordinates,dim) show(truss.before, frame=False) ︡be47dc0a-6336-4351-afc6-1dad4e02a3ad︡{"file":{"filename":"f3cecdae-496f-4e8d-8176-05db55797996.sage3d","uuid":"f3cecdae-496f-4e8d-8176-05db55797996"}}︡{"done":true} ︠75e00900-e338-4eb9-82b3-e978188a30ccs︠ # Equation (2) in the paper BCs = truss.bar_constraints g = [] for e in truss.edges: g.append(BCs[e]) for gij in g: show(gij) ︡4d1ffa64-50f1-48e7-9de5-46e30c221899︡{"html":"
$\\displaystyle {\\left(x_{11} - x_{21}\\right)}^{2} + {\\left(x_{12} - x_{22}\\right)}^{2} + {\\left(x_{13} - x_{23}\\right)}^{2} - 3$
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$\\displaystyle {\\left(x_{11} - x_{31}\\right)}^{2} + {\\left(x_{12} - x_{32}\\right)}^{2} + {\\left(x_{13} - x_{33}\\right)}^{2} - 3$
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$\\displaystyle {\\left(x_{11} - x_{41}\\right)}^{2} + {\\left(x_{12} - x_{42}\\right)}^{2} + {\\left(x_{13} - x_{43}\\right)}^{2} - \\frac{1}{4} \\, {\\left(\\sqrt{3} + 2\\right)}^{2} - \\frac{37}{4}$
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$\\displaystyle {\\left(x_{11} - x_{51}\\right)}^{2} + {\\left(x_{12} - x_{52}\\right)}^{2} + {\\left(x_{13} - x_{53}\\right)}^{2} - \\frac{1}{4} \\, {\\left(\\sqrt{3} - 2\\right)}^{2} - \\frac{37}{4}$
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$\\displaystyle {\\left(x_{21} - x_{31}\\right)}^{2} + {\\left(x_{22} - x_{32}\\right)}^{2} + {\\left(x_{23} - x_{33}\\right)}^{2} - 3$
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$\\displaystyle {\\left(x_{21} - x_{51}\\right)}^{2} + {\\left(x_{22} - x_{52}\\right)}^{2} + {\\left(x_{23} - x_{53}\\right)}^{2} - \\frac{1}{2} \\, {\\left(\\sqrt{3} + 1\\right)}^{2} - 9$
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$\\displaystyle {\\left(x_{21} - x_{61}\\right)}^{2} + {\\left(x_{22} - x_{62}\\right)}^{2} + {\\left(x_{23} - x_{63}\\right)}^{2} - \\frac{1}{4} \\, {\\left(\\sqrt{3} - 2\\right)}^{2} - \\frac{37}{4}$
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$\\displaystyle {\\left(x_{31} - x_{41}\\right)}^{2} + {\\left(x_{32} - x_{42}\\right)}^{2} + {\\left(x_{33} - x_{43}\\right)}^{2} - \\frac{1}{2} \\, {\\left(\\sqrt{3} - 1\\right)}^{2} - 9$
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$\\displaystyle {\\left(x_{31} - x_{61}\\right)}^{2} + {\\left(x_{32} - x_{62}\\right)}^{2} + {\\left(x_{33} - x_{63}\\right)}^{2} - \\frac{1}{4} \\, {\\left(\\sqrt{3} + 2\\right)}^{2} - \\frac{37}{4}$
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$\\displaystyle {\\left(x_{41} - x_{51}\\right)}^{2} + {\\left(x_{42} - x_{52}\\right)}^{2} + {\\left(x_{43} - x_{53}\\right)}^{2} - 3$
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$\\displaystyle {\\left(x_{41} - x_{61}\\right)}^{2} + {\\left(x_{42} - x_{62}\\right)}^{2} + {\\left(x_{43} - x_{63}\\right)}^{2} - 3$
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$\\displaystyle {\\left(x_{51} - x_{61}\\right)}^{2} + {\\left(x_{52} - x_{62}\\right)}^{2} + {\\left(x_{53} - x_{63}\\right)}^{2} - 3$
"}︡{"done":true} ︠bd2ae675-48b1-4ac3-bbcb-77b713aa6d6fs︠ # Equation (3) in the paper dg = jacobian(g,truss.X.list())/2 show(dg); print; # show(truss.rigidity_matrix) show(truss.rigidity_matrix_at_p) ︡cb2a5153-a520-4949-a37b-23fc6650e8fe︡{"html":"
$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrr}\nx_{11} - x_{21} & x_{12} - x_{22} & x_{13} - x_{23} & -x_{11} + x_{21} & -x_{12} + x_{22} & -x_{13} + x_{23} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\nx_{11} - x_{31} & x_{12} - x_{32} & x_{13} - x_{33} & 0 & 0 & 0 & -x_{11} + x_{31} & -x_{12} + x_{32} & -x_{13} + x_{33} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\nx_{11} - x_{41} & x_{12} - x_{42} & x_{13} - x_{43} & 0 & 0 & 0 & 0 & 0 & 0 & -x_{11} + x_{41} & -x_{12} + x_{42} & -x_{13} + x_{43} & 0 & 0 & 0 & 0 & 0 & 0 \\\\\nx_{11} - x_{51} & x_{12} - x_{52} & x_{13} - x_{53} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -x_{11} + x_{51} & -x_{12} + x_{52} & -x_{13} + x_{53} & 0 & 0 & 0 \\\\\n0 & 0 & 0 & x_{21} - x_{31} & x_{22} - x_{32} & x_{23} - x_{33} & -x_{21} + x_{31} & -x_{22} + x_{32} & -x_{23} + x_{33} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & x_{21} - x_{51} & x_{22} - x_{52} & x_{23} - x_{53} & 0 & 0 & 0 & 0 & 0 & 0 & -x_{21} + x_{51} & -x_{22} + x_{52} & -x_{23} + x_{53} & 0 & 0 & 0 \\\\\n0 & 0 & 0 & x_{21} - x_{61} & x_{22} - x_{62} & x_{23} - x_{63} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -x_{21} + x_{61} & -x_{22} + x_{62} & -x_{23} + x_{63} \\\\\n0 & 0 & 0 & 0 & 0 & 0 & x_{31} - x_{41} & x_{32} - x_{42} & x_{33} - x_{43} & -x_{31} + x_{41} & -x_{32} + x_{42} & -x_{33} + x_{43} & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & x_{31} - x_{61} & x_{32} - x_{62} & x_{33} - x_{63} & 0 & 0 & 0 & 0 & 0 & 0 & -x_{31} + x_{61} & -x_{32} + x_{62} & -x_{33} + x_{63} \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{41} - x_{51} & x_{42} - x_{52} & x_{43} - x_{53} & -x_{41} + x_{51} & -x_{42} + x_{52} & -x_{43} + x_{53} & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{41} - x_{61} & x_{42} - x_{62} & x_{43} - x_{63} & 0 & 0 & 0 & -x_{41} + x_{61} & -x_{42} + x_{62} & -x_{43} + x_{63} \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & x_{51} - x_{61} & x_{52} - x_{62} & x_{53} - x_{63} & -x_{51} + x_{61} & -x_{52} + x_{62} & -x_{53} + x_{63}\n\\end{array}\\right)$
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$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrr}\n\\frac{3}{2} & -\\frac{1}{2} \\, \\sqrt{3} & 0 & -\\frac{3}{2} & \\frac{1}{2} \\, \\sqrt{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\frac{3}{2} & \\frac{1}{2} \\, \\sqrt{3} & 0 & 0 & 0 & 0 & -\\frac{3}{2} & -\\frac{1}{2} \\, \\sqrt{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\frac{1}{2} \\, \\sqrt{3} + 1 & \\frac{1}{2} & -3 & 0 & 0 & 0 & 0 & 0 & 0 & -\\frac{1}{2} \\, \\sqrt{3} - 1 & -\\frac{1}{2} & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n-\\frac{1}{2} \\, \\sqrt{3} + 1 & \\frac{1}{2} & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac{1}{2} \\, \\sqrt{3} - 1 & -\\frac{1}{2} & 3 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & \\sqrt{3} & 0 & 0 & -\\sqrt{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -\\frac{1}{2} \\, \\sqrt{3} - \\frac{1}{2} & \\frac{1}{2} \\, \\sqrt{3} + \\frac{1}{2} & -3 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac{1}{2} \\, \\sqrt{3} + \\frac{1}{2} & -\\frac{1}{2} \\, \\sqrt{3} - \\frac{1}{2} & 3 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & -\\frac{1}{2} & \\frac{1}{2} \\, \\sqrt{3} - 1 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac{1}{2} & -\\frac{1}{2} \\, \\sqrt{3} + 1 & 3 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & \\frac{1}{2} \\, \\sqrt{3} - \\frac{1}{2} & -\\frac{1}{2} \\, \\sqrt{3} + \\frac{1}{2} & -3 & -\\frac{1}{2} \\, \\sqrt{3} + \\frac{1}{2} & \\frac{1}{2} \\, \\sqrt{3} - \\frac{1}{2} & 3 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & -\\frac{1}{2} & -\\frac{1}{2} \\, \\sqrt{3} - 1 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac{1}{2} & \\frac{1}{2} \\, \\sqrt{3} + 1 & 3 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\\sqrt{3} & 0 & 0 & \\sqrt{3} & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\\frac{1}{2} \\, \\sqrt{3} & -\\frac{3}{2} & 0 & 0 & 0 & 0 & \\frac{1}{2} \\, \\sqrt{3} & \\frac{3}{2} & 0 \\\\\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac{1}{2} \\, \\sqrt{3} & -\\frac{3}{2} & 0 & -\\frac{1}{2} \\, \\sqrt{3} & \\frac{3}{2} & 0\n\\end{array}\\right)$
"}︡{"done":true} ︠8036cdc4-7a54-4481-a966-46ca5b6d2437s︠ # Figure 5 in the paper load("tensegrity.sage") nodes = [1,2,3,4,5,6] h = 1 node_coordinates = {1:(QQ.random_element(50,50), QQ.random_element(50,50), QQ.random_element(50,50)), 2:(QQ.random_element(50,50), QQ.random_element(50,50), QQ.random_element(50,50)), 3:(QQ.random_element(50,50), QQ.random_element(50,50), QQ.random_element(50,50)), 4:(QQ.random_element(50,50), QQ.random_element(50,50), QQ.random_element(50,50)), 5:(QQ.random_element(50,50), QQ.random_element(50,50), QQ.random_element(50,50)), 6:(QQ.random_element(50,50), QQ.random_element(50,50), QQ.random_element(50,50))} edges = [(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)] dim = 3 truss = Truss(nodes,edges,node_coordinates,dim) truss.plot_flexes() show(truss.flexes_plot, frame=False) print; rank(truss.A); rank(truss.rigidity_matrix_at_p) ︡132200f4-68ef-4e54-bf22-d2d75d87e223︡{"file":{"filename":"93f11fe0-44de-442d-a18c-a3a23f2dd27c.sage3d","uuid":"93f11fe0-44de-442d-a18c-a3a23f2dd27c"}}︡{"stdout":"\n"}︡{"stdout":"12\n12\n"}︡{"done":true} ︠376bc2ea-f8db-4d6f-bc4a-1032d662b8fcs︠ # Figure 6 electrical network load("tensegrity.sage") nodes = [1,2,3,4] node_coordinates = {1:(1/2,1), 2:(5/2, 1), 3:(0, 0), 4:(3,0)} edges = [(1,2),(1,3),(2,3),(2,4),(3,4)] dim = 2 truss = Truss(nodes,edges,node_coordinates,dim) show(truss.before, frame=False) -truss.get_incidence_matrix() # notice the minus sign convention is different. But A^T C A remains the same. ︡e5aabf5d-a6eb-4fbd-aade-48bc182306b1︡{"file":{"filename":"/home/user/.sage/temp/project-e3d8858f-633e-4199-98c4-2459b1781fb3/85594/tmp_TlTbQG.svg","show":true,"text":null,"uuid":"c1a776a7-7a24-485d-815e-08ca936d4e19"},"once":false}︡{"stdout":"[-1 1 0 0]\n[-1 0 1 0]\n[ 0 -1 1 0]\n[ 0 -1 0 1]\n[ 0 0 -1 1]\n"}︡{"done":true} ︠eeb4c2f0-8415-40ba-abe3-183af8b345bbs︠ # Figure 8 shows the second row corresponding to edge (1,3). show(truss.rigidity_matrix) ︡106e9efe-ca64-4587-8eed-01b771fea720︡{"html":"
$\\displaystyle \\left(\\begin{array}{rrrrrrrr}\nx_{11} - x_{21} & x_{12} - x_{22} & -x_{11} + x_{21} & -x_{12} + x_{22} & 0 & 0 & 0 & 0 \\\\\nx_{11} - x_{31} & x_{12} - x_{32} & 0 & 0 & -x_{11} + x_{31} & -x_{12} + x_{32} & 0 & 0 \\\\\n0 & 0 & x_{21} - x_{31} & x_{22} - x_{32} & -x_{21} + x_{31} & -x_{22} + x_{32} & 0 & 0 \\\\\n0 & 0 & x_{21} - x_{41} & x_{22} - x_{42} & 0 & 0 & -x_{21} + x_{41} & -x_{22} + x_{42} \\\\\n0 & 0 & 0 & 0 & x_{31} - x_{41} & x_{32} - x_{42} & -x_{31} + x_{41} & -x_{32} + x_{42}\n\\end{array}\\right)$
"}︡{"done":true} ︠143473cc-bf1b-44cd-b166-e82f4593adc5s︠ load("tensegrity.sage") nodes = range(1,12+1) # 12 vertices of icosahedron u = 1 v = 1/2*(sqrt(5)+1) - 1 # since we use EXACT computation, this takes a bit of time... node_coordinates = {1:(v,0,u), 2:(-v,0,u), 3:(0,u,v), 4:(u,v,0), 5:(u,-v,0), 6:(0,-u,v), 7:(-u,v,0), 8:(0,u,-v), 9:(v,0,-u), 10:(0,-u,-v), 11:(-u,-v,0), 12:(-v,0,-u)} edges = [(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,6),(2,7),(2,11),(3,4), (3,7),(3,8),(4,5),(4,8),(4,9),(5,6),(5,9),(5,10),(6,10),(6,11), (7,8),(7,11),(7,12),(8,9),(8,12),(9,12),(10,11),(10,12),(11,12)] # remove (9,10) dim = 3 truss = Truss(nodes,edges,node_coordinates,dim,orthonormal=True) truss.plot_arrow_scale = 0.07 truss.plot_flexes() show(truss.flexes_plot, frame=False) ︡c9bd40c0-fbab-4e34-b011-124afd46b5ce︡{"file":{"filename":"94e15c86-06d5-4fb9-9eee-ce16d01daeb0.sage3d","uuid":"94e15c86-06d5-4fb9-9eee-ce16d01daeb0"}}︡{"done":true} ︠5fe706b3-9d01-4da2-8cc1-2f1cf2a58c0b︠ load("tensegrity.sage") nodes = range(1,6+1) # 6 nodes in the plane node_coordinates = {1:(0,0),2:(4,0),3:(0,3),4:(4,3),5:(4/3,3/2), 6:(8/3,3/2)} edges = [(1,2),(1,3),(1,5),(2,6),(2,4),(3,4),(3,5),(4,6),(5,6)] dim = 2 truss = Truss(nodes,edges,node_coordinates,dim,orthonormal=True) truss.plot_arrow_scale = 2.0 truss.plot_mechanisms() show(truss.mechanism_plot, frame=False) ︡e6f07a8a-4ef1-4ca8-a9ec-1a6ed8e2c6bb︡{"file":{"filename":"/home/user/.sage/temp/project-e3d8858f-633e-4199-98c4-2459b1781fb3/85594/tmp_nj2RJR.svg","show":true,"text":null,"uuid":"1bb32c1c-652b-42a3-b139-a25eecc9c70f"},"once":false}︡{"done":true} ︠315e6e74-cd7c-4d67-b036-ed2ee51fec32︠ load("tensegrity.sage") nodes = [1,2,3,4,5,6] node_coordinates = {1:(0,0), 2:(0,2), 3:(1,2), 4:(0,1), 5:(1,1), 6:(1,0)} # The nodes are labelled as in Problem 1, Section 2.7. edges = [(2,3),(2,4),(3,5),(4,5),(4,6),(1,5)] # indices of the correct node in "nodes" dim = 2 truss = Truss(nodes,edges,node_coordinates,dim,orthonormal=True) #T.fix_nodes([1,6]) truss.plot_arrow_scale = 1.0 truss.plot_mechanisms() show(truss.mechanism_plot, frame=False) ︡8700d887-bab0-48a6-968a-8b39bbb271e9︡{"file":{"filename":"/home/user/.sage/temp/project-e3d8858f-633e-4199-98c4-2459b1781fb3/85594/tmp_XUNGvV.svg","show":true,"text":null,"uuid":"acb66e17-20ff-4b67-be55-2139c61172a2"},"once":false}︡{"done":true} ︠15c8ad84-fb8f-47d0-b1af-652cf1b6d0bes︠ # Figure 12 x,y = var('x y') f = y - x^2 r = 3.5 plt = implicit_plot(f, (x,-r,r), (y,-1,3*r), aspect_ratio=1, linestyle='--', color='black') df = jacobian(f,[x,y]) df; print; dfatp = df({x:2,y:4}) print dfatp; print; null = dfatp.right_kernel().basis_matrix() v = vector(null) + vector([2,4]) v = arrow(vector([2,4]),v, color='blue') plt += plot(v) show(plt) ︡d8e923ec-14dd-4cf6-90bc-fe6c8a16f3b2︡{"stdout":"[-2*x 1]\n\n"}︡{"stdout":"[-4 1]\n\n"}︡{"file":{"filename":"/home/user/.sage/temp/project-e3d8858f-633e-4199-98c4-2459b1781fb3/85594/tmp_0EIQxN.svg","show":true,"text":null,"uuid":"de775fef-c0d0-44db-8e4d-ca12efc2e9d2"},"once":false}︡{"done":true} ︠70f3babc-4ff3-474e-a6cd-9e058d9d80c1s︠ # Figure 13 load("tensegrity.sage") nodes = [1,2,3,4,5,6] h = 3 node_coordinates = {1:(1,0,0), 2:(-1/2, sqrt(3)/2, 0), 3:(-1/2, -sqrt(3)/2, 0), 4:(-sqrt(3)/2, -1/2, h), 5:(sqrt(3)/2, -1/2, h), 6:(0,1,h)} edges = [(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)] dim = 3 truss = Truss(nodes,edges,node_coordinates,dim) truss.plot_flexes() show(truss.flexes_plot, frame=False) ︡3bd9a26b-4950-423c-ab6a-5acaaec9457a︡{"file":{"filename":"3ac7e878-dc91-46c1-8ae6-76aaca5d9b32.sage3d","uuid":"3ac7e878-dc91-46c1-8ae6-76aaca5d9b32"}}︡{"done":true} ︠79598473-339e-4d2b-b0bd-c38b6d33a581s︠ # Figure 14 load("tensegrity.sage") nodes = [1,2,3,4,5,6] h = 3 node_coordinates = {1:(1,0,0), 2:(-1/2, sqrt(3)/2, 0), 3:(-1/2, -sqrt(3)/2, 0), 4:(-sqrt(3)/2, -1/2, h), 5:(sqrt(3)/2, -1/2, h), 6:(0,1,h)} edges = [(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)] dim = 3 truss = Truss(nodes,edges,node_coordinates,dim) truss.plot_rigid_motions() show(truss.rigid_motions_plot, frame=False) ︡e4484e0a-ddd3-49fe-85dd-629d2c748317︡{"file":{"filename":"749e75ba-c084-4c2d-81be-98e5f2f05d20.sage3d","uuid":"749e75ba-c084-4c2d-81be-98e5f2f05d20"}}︡{"done":true} ︠27e4cf6b-de1f-4db8-bbab-4ed4de1244a7︠ # Figure 15 and 16 R. = PolynomialRing(QQ,3,order='degrevlex') eqns = [x*y-x^3, x*z-x^4, x^2*y*z-x^4*z-x^5*y + x^7] I = R.ideal( eqns ) AP = I.associated_primes() for p in AP: print p.gens() ︡619a6d98-8a19-482a-ba11-e3534418c522︡{"stdout":"[x]\n[y^2 - x*z, x*y - z, x^2 - y]\n"}︡{"done":true} ︠b79dbef9-3e06-40ee-b173-f3fd5cc98cdds︠ df = jacobian(eqns,[x,y,z]) show(df) print df((0,0,0)); print; print df((0,5,3)); print; print df((1,1,1)); print; ︡32f162bb-e618-4019-8a9a-b8b64b4bdaf0︡{"html":"
$\\displaystyle \\left(\\begin{array}{rrr}\n-3 x^{2} + y & x & 0 \\\\\n-4 x^{3} + z & 0 & x \\\\\n7 x^{6} - 5 x^{4} y - 4 x^{3} z + 2 x y z & -x^{5} + x^{2} z & -x^{4} + x^{2} y\n\\end{array}\\right)$
"}︡{"stdout":"[0 0 0]\n[0 0 0]\n[0 0 0]\n\n"}︡{"stdout":"[5 0 0]\n[3 0 0]\n[0 0 0]\n\n"}︡{"stdout":"[-2 1 0]\n[-3 0 1]\n[ 0 0 0]\n\n"}︡{"done":true} ︠2f282a63-1ac2-42bf-b134-3ffd86a286b1s︠ # Figure 17. load("tensegrity.sage") nodes = range(1,3+1) # 6 nodes in the plane node_coordinates = {1:(0,0),2:(2,0),3:(4,0)} edges = [(1,2),(2,3)] dim = 2 truss = Truss(nodes,edges,node_coordinates,dim,orthonormal=True) show(truss.before, axes=False) A = truss.get_incidence_matrix() K = A.T * A; show(K); print; K.eigenspaces_right() # Equation (6) ︡d2681331-0650-4a8b-ba2d-786f00014b08︡{"file":{"filename":"/home/user/.sage/temp/project-e3d8858f-633e-4199-98c4-2459b1781fb3/85594/tmp_kb48no.svg","show":true,"text":null,"uuid":"bc8f0b85-0b88-40f2-b911-e294a3e0a4a1"},"once":false}︡{"html":"
$\\displaystyle \\left(\\begin{array}{rrr}\n1 & -1 & 0 \\\\\n-1 & 2 & -1 \\\\\n0 & -1 & 1\n\\end{array}\\right)$
"}︡{"stdout":"\n"}︡{"stdout":"[\n(3, Vector space of degree 3 and dimension 1 over Rational Field\nUser basis matrix:\n[ 1 -2 1]),\n(1, Vector space of degree 3 and dimension 1 over Rational Field\nUser basis matrix:\n[ 1 0 -1]),\n(0, Vector space of degree 3 and dimension 1 over Rational Field\nUser basis matrix:\n[1 1 1])\n]\n"}︡{"done":true} ︠bd07dafa-29d1-42f3-898d-b0e309537001s︠ # for fun G = SymmetricGroup(3) show([g.matrix() for g in G]) Q = matrix(AA,3,3,[1,1,1, 1,0,-1, -1,2,-1]).T; show(Q) L = matrix.diagonal([1/sqrt(3), 1/sqrt(2), 1/sqrt(6)]); show(L) C = Q*L; show(C) newG = [C.T*g.matrix()*C for g in G] show(newG) ︡29269f54-e3ce-4ca5-be23-53c308bf494c︡{"html":"
[$\\displaystyle \\left(\\begin{array}{rrr}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1\n\\end{array}\\right)$, $\\displaystyle \\left(\\begin{array}{rrr}\n0 & 0 & 1 \\\\\n1 & 0 & 0 \\\\\n0 & 1 & 0\n\\end{array}\\right)$, $\\displaystyle \\left(\\begin{array}{rrr}\n0 & 1 & 0 \\\\\n0 & 0 & 1 \\\\\n1 & 0 & 0\n\\end{array}\\right)$, $\\displaystyle \\left(\\begin{array}{rrr}\n1 & 0 & 0 \\\\\n0 & 0 & 1 \\\\\n0 & 1 & 0\n\\end{array}\\right)$, $\\displaystyle \\left(\\begin{array}{rrr}\n0 & 0 & 1 \\\\\n0 & 1 & 0 \\\\\n1 & 0 & 0\n\\end{array}\\right)$, $\\displaystyle \\left(\\begin{array}{rrr}\n0 & 1 & 0 \\\\\n1 & 0 & 0 \\\\\n0 & 0 & 1\n\\end{array}\\right)$]
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$\\displaystyle \\left(\\begin{array}{rrr}\n1 & 1 & -1 \\\\\n1 & 0 & 2 \\\\\n1 & -1 & -1\n\\end{array}\\right)$
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$\\displaystyle \\left(\\begin{array}{rrr}\n\\frac{1}{3} \\, \\sqrt{3} & 0 & 0 \\\\\n0 & \\frac{1}{2} \\, \\sqrt{2} & 0 \\\\\n0 & 0 & \\frac{1}{6} \\, \\sqrt{6}\n\\end{array}\\right)$
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$\\displaystyle \\left(\\begin{array}{rrr}\n\\frac{1}{3} \\, \\sqrt{3} & \\frac{1}{2} \\, \\sqrt{2} & -\\frac{1}{6} \\, \\sqrt{6} \\\\\n\\frac{1}{3} \\, \\sqrt{3} & 0 & \\frac{1}{3} \\, \\sqrt{6} \\\\\n\\frac{1}{3} \\, \\sqrt{3} & -\\frac{1}{2} \\, \\sqrt{2} & -\\frac{1}{6} \\, \\sqrt{6}\n\\end{array}\\right)$
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[$\\displaystyle \\left(\\begin{array}{rrr}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1\n\\end{array}\\right)$, $\\displaystyle \\left(\\begin{array}{rrr}\n1 & 0 & 0 \\\\\n0 & -\\frac{1}{2} & -\\frac{1}{4} \\, \\sqrt{6} \\sqrt{2} \\\\\n0 & \\frac{1}{4} \\, \\sqrt{6} \\sqrt{2} & -\\frac{1}{2}\n\\end{array}\\right)$, $\\displaystyle \\left(\\begin{array}{rrr}\n1 & 0 & 0 \\\\\n0 & -\\frac{1}{2} & \\frac{1}{4} \\, \\sqrt{6} \\sqrt{2} \\\\\n0 & -\\frac{1}{4} \\, \\sqrt{6} \\sqrt{2} & -\\frac{1}{2}\n\\end{array}\\right)$, $\\displaystyle \\left(\\begin{array}{rrr}\n1 & 0 & 0 \\\\\n0 & \\frac{1}{2} & -\\frac{1}{4} \\, \\sqrt{6} \\sqrt{2} \\\\\n0 & -\\frac{1}{4} \\, \\sqrt{6} \\sqrt{2} & -\\frac{1}{2}\n\\end{array}\\right)$, $\\displaystyle \\left(\\begin{array}{rrr}\n1 & 0 & 0 \\\\\n0 & -1 & 0 \\\\\n0 & 0 & 1\n\\end{array}\\right)$, $\\displaystyle \\left(\\begin{array}{rrr}\n1 & 0 & 0 \\\\\n0 & \\frac{1}{2} & \\frac{1}{4} \\, \\sqrt{6} \\sqrt{2} \\\\\n0 & \\frac{1}{4} \\, \\sqrt{6} \\sqrt{2} & -\\frac{1}{2}\n\\end{array}\\right)$]
"}︡{"done":true} ︠fd1a2912-b024-4b43-855f-a73e709e0296s︠ # Equation (7) load("tensegrity.sage") nodes = [1,2,3,4,5,6] h = 3 node_coordinates = {1:(1,0,0), 2:(-1/2, sqrt(3)/2, 0), 3:(-1/2, -sqrt(3)/2, 0), 4:(-sqrt(3)/2, -1/2, h), 5:(sqrt(3)/2, -1/2, h), 6:(0,1,h)} edges = [(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)] dim = 3 truss = Truss(nodes,edges,node_coordinates,dim) truss.flexes ︡e69027fa-1afe-41e1-a436-96d4aff1337f︡{"stdout":"[(0, 1.577350269189626?, 0.2628917115316043?, -1.366025403784439?, -0.7886751345948129?, 0.2628917115316043?, 1.366025403784439?, -0.7886751345948129?, 0.2628917115316043?, -0.7886751345948129?, 1.366025403784439?, -0.2628917115316043?, -0.7886751345948129?, -1.366025403784439?, -0.2628917115316043?, 1.577350269189626?, 0, -0.2628917115316043?)]\n"}︡{"done":true} ︠be5916c2-0093-4dbd-b1dd-a7794dc0e4aas︠ # Equation (8) Q1 = sum([truss.bar_constraints[e] for e in truss.edges]) 1/2 * jacobian(jacobian(Q1,truss.X.list()), truss.X.list()) ︡c2ba8c76-02ab-41e2-9043-af71d1284e0f︡{"stdout":"[ 4 0 0 -1 0 0 -1 0 0 -1 0 0 -1 0 0 0 0 0]\n[ 0 4 0 0 -1 0 0 -1 0 0 -1 0 0 -1 0 0 0 0]\n[ 0 0 4 0 0 -1 0 0 -1 0 0 -1 0 0 -1 0 0 0]\n[-1 0 0 4 0 0 -1 0 0 0 0 0 -1 0 0 -1 0 0]\n[ 0 -1 0 0 4 0 0 -1 0 0 0 0 0 -1 0 0 -1 0]\n[ 0 0 -1 0 0 4 0 0 -1 0 0 0 0 0 -1 0 0 -1]\n[-1 0 0 -1 0 0 4 0 0 -1 0 0 0 0 0 -1 0 0]\n[ 0 -1 0 0 -1 0 0 4 0 0 -1 0 0 0 0 0 -1 0]\n[ 0 0 -1 0 0 -1 0 0 4 0 0 -1 0 0 0 0 0 -1]\n[-1 0 0 0 0 0 -1 0 0 4 0 0 -1 0 0 -1 0 0]\n[ 0 -1 0 0 0 0 0 -1 0 0 4 0 0 -1 0 0 -1 0]\n[ 0 0 -1 0 0 0 0 0 -1 0 0 4 0 0 -1 0 0 -1]\n[-1 0 0 -1 0 0 0 0 0 -1 0 0 4 0 0 -1 0 0]\n[ 0 -1 0 0 -1 0 0 0 0 0 -1 0 0 4 0 0 -1 0]\n[ 0 0 -1 0 0 -1 0 0 0 0 0 -1 0 0 4 0 0 -1]\n[ 0 0 0 -1 0 0 -1 0 0 -1 0 0 -1 0 0 4 0 0]\n[ 0 0 0 0 -1 0 0 -1 0 0 -1 0 0 -1 0 0 4 0]\n[ 0 0 0 0 0 -1 0 0 -1 0 0 -1 0 0 -1 0 0 4]\n"}︡{"done":true} ︠6d871280-581f-4b8e-ae30-0bac5c35f615s︠ # Equation (8) Qw = truss.potential; show(Qw) show(1/2 * jacobian(jacobian(Qw,truss.X.list()), truss.X.list())) ︡6db64672-0fc0-41a0-8085-cb7abb788869︡{"html":"
$\\displaystyle {\\left({\\left(x_{11} - x_{21}\\right)}^{2} + {\\left(x_{12} - x_{22}\\right)}^{2} + {\\left(x_{13} - x_{23}\\right)}^{2} - 3\\right)} w_{12} + {\\left({\\left(x_{11} - x_{31}\\right)}^{2} + {\\left(x_{12} - x_{32}\\right)}^{2} + {\\left(x_{13} - x_{33}\\right)}^{2} - 3\\right)} w_{13} + \\frac{1}{4} \\, {\\left(4 \\, {\\left(x_{11} - x_{41}\\right)}^{2} + 4 \\, {\\left(x_{12} - x_{42}\\right)}^{2} + 4 \\, {\\left(x_{13} - x_{43}\\right)}^{2} - {\\left(\\sqrt{3} + 2\\right)}^{2} - 37\\right)} w_{14} + \\frac{1}{4} \\, {\\left(4 \\, {\\left(x_{11} - x_{51}\\right)}^{2} + 4 \\, {\\left(x_{12} - x_{52}\\right)}^{2} + 4 \\, {\\left(x_{13} - x_{53}\\right)}^{2} - {\\left(\\sqrt{3} - 2\\right)}^{2} - 37\\right)} w_{15} + {\\left({\\left(x_{21} - x_{31}\\right)}^{2} + {\\left(x_{22} - x_{32}\\right)}^{2} + {\\left(x_{23} - x_{33}\\right)}^{2} - 3\\right)} w_{23} + \\frac{1}{2} \\, {\\left(2 \\, {\\left(x_{21} - x_{51}\\right)}^{2} + 2 \\, {\\left(x_{22} - x_{52}\\right)}^{2} + 2 \\, {\\left(x_{23} - x_{53}\\right)}^{2} - {\\left(\\sqrt{3} + 1\\right)}^{2} - 18\\right)} w_{25} + \\frac{1}{4} \\, {\\left(4 \\, {\\left(x_{21} - x_{61}\\right)}^{2} + 4 \\, {\\left(x_{22} - x_{62}\\right)}^{2} + 4 \\, {\\left(x_{23} - x_{63}\\right)}^{2} - {\\left(\\sqrt{3} - 2\\right)}^{2} - 37\\right)} w_{26} + \\frac{1}{2} \\, {\\left(2 \\, {\\left(x_{31} - x_{41}\\right)}^{2} + 2 \\, {\\left(x_{32} - x_{42}\\right)}^{2} + 2 \\, {\\left(x_{33} - x_{43}\\right)}^{2} - {\\left(\\sqrt{3} - 1\\right)}^{2} - 18\\right)} w_{34} + \\frac{1}{4} \\, {\\left(4 \\, {\\left(x_{31} - x_{61}\\right)}^{2} + 4 \\, {\\left(x_{32} - x_{62}\\right)}^{2} + 4 \\, {\\left(x_{33} - x_{63}\\right)}^{2} - {\\left(\\sqrt{3} + 2\\right)}^{2} - 37\\right)} w_{36} + {\\left({\\left(x_{41} - x_{51}\\right)}^{2} + {\\left(x_{42} - x_{52}\\right)}^{2} + {\\left(x_{43} - x_{53}\\right)}^{2} - 3\\right)} w_{45} + {\\left({\\left(x_{41} - x_{61}\\right)}^{2} + {\\left(x_{42} - x_{62}\\right)}^{2} + {\\left(x_{43} - x_{63}\\right)}^{2} - 3\\right)} w_{46} + {\\left({\\left(x_{51} - x_{61}\\right)}^{2} + {\\left(x_{52} - x_{62}\\right)}^{2} + {\\left(x_{53} - x_{63}\\right)}^{2} - 3\\right)} w_{56}$
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$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrr}\nw_{12} + w_{13} + w_{14} + w_{15} & 0 & 0 & -w_{12} & 0 & 0 & -w_{13} & 0 & 0 & -w_{14} & 0 & 0 & -w_{15} & 0 & 0 & 0 & 0 & 0 \\\\\n0 & w_{12} + w_{13} + w_{14} + w_{15} & 0 & 0 & -w_{12} & 0 & 0 & -w_{13} & 0 & 0 & -w_{14} & 0 & 0 & -w_{15} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & w_{12} + w_{13} + w_{14} + w_{15} & 0 & 0 & -w_{12} & 0 & 0 & -w_{13} & 0 & 0 & -w_{14} & 0 & 0 & -w_{15} & 0 & 0 & 0 \\\\\n-w_{12} & 0 & 0 & w_{12} + w_{23} + w_{25} + w_{26} & 0 & 0 & -w_{23} & 0 & 0 & 0 & 0 & 0 & -w_{25} & 0 & 0 & -w_{26} & 0 & 0 \\\\\n0 & -w_{12} & 0 & 0 & w_{12} + w_{23} + w_{25} + w_{26} & 0 & 0 & -w_{23} & 0 & 0 & 0 & 0 & 0 & -w_{25} & 0 & 0 & -w_{26} & 0 \\\\\n0 & 0 & -w_{12} & 0 & 0 & w_{12} + w_{23} + w_{25} + w_{26} & 0 & 0 & -w_{23} & 0 & 0 & 0 & 0 & 0 & -w_{25} & 0 & 0 & -w_{26} \\\\\n-w_{13} & 0 & 0 & -w_{23} & 0 & 0 & w_{13} + w_{23} + w_{34} + w_{36} & 0 & 0 & -w_{34} & 0 & 0 & 0 & 0 & 0 & -w_{36} & 0 & 0 \\\\\n0 & -w_{13} & 0 & 0 & -w_{23} & 0 & 0 & w_{13} + w_{23} + w_{34} + w_{36} & 0 & 0 & -w_{34} & 0 & 0 & 0 & 0 & 0 & -w_{36} & 0 \\\\\n0 & 0 & -w_{13} & 0 & 0 & -w_{23} & 0 & 0 & w_{13} + w_{23} + w_{34} + w_{36} & 0 & 0 & -w_{34} & 0 & 0 & 0 & 0 & 0 & -w_{36} \\\\\n-w_{14} & 0 & 0 & 0 & 0 & 0 & -w_{34} & 0 & 0 & w_{14} + w_{34} + w_{45} + w_{46} & 0 & 0 & -w_{45} & 0 & 0 & -w_{46} & 0 & 0 \\\\\n0 & -w_{14} & 0 & 0 & 0 & 0 & 0 & -w_{34} & 0 & 0 & w_{14} + w_{34} + w_{45} + w_{46} & 0 & 0 & -w_{45} & 0 & 0 & -w_{46} & 0 \\\\\n0 & 0 & -w_{14} & 0 & 0 & 0 & 0 & 0 & -w_{34} & 0 & 0 & w_{14} + w_{34} + w_{45} + w_{46} & 0 & 0 & -w_{45} & 0 & 0 & -w_{46} \\\\\n-w_{15} & 0 & 0 & -w_{25} & 0 & 0 & 0 & 0 & 0 & -w_{45} & 0 & 0 & w_{15} + w_{25} + w_{45} + w_{56} & 0 & 0 & -w_{56} & 0 & 0 \\\\\n0 & -w_{15} & 0 & 0 & -w_{25} & 0 & 0 & 0 & 0 & 0 & -w_{45} & 0 & 0 & w_{15} + w_{25} + w_{45} + w_{56} & 0 & 0 & -w_{56} & 0 \\\\\n0 & 0 & -w_{15} & 0 & 0 & -w_{25} & 0 & 0 & 0 & 0 & 0 & -w_{45} & 0 & 0 & w_{15} + w_{25} + w_{45} + w_{56} & 0 & 0 & -w_{56} \\\\\n0 & 0 & 0 & -w_{26} & 0 & 0 & -w_{36} & 0 & 0 & -w_{46} & 0 & 0 & -w_{56} & 0 & 0 & w_{26} + w_{36} + w_{46} + w_{56} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & -w_{26} & 0 & 0 & -w_{36} & 0 & 0 & -w_{46} & 0 & 0 & -w_{56} & 0 & 0 & w_{26} + w_{36} + w_{46} + w_{56} & 0 \\\\\n0 & 0 & 0 & 0 & 0 & -w_{26} & 0 & 0 & -w_{36} & 0 & 0 & -w_{46} & 0 & 0 & -w_{56} & 0 & 0 & w_{26} + w_{36} + w_{46} + w_{56}\n\\end{array}\\right)$
$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrr}\nw_{12} + w_{13} + w_{14} + w_{15} & 0 & 0 & -w_{12} & 0 & 0 & -w_{13} & 0 & 0 & -w_{14} & 0 & 0 & -w_{15} & 0 & 0 & 0 & 0 & 0 \\\\\n0 & w_{12} + w_{13} + w_{14} + w_{15} & 0 & 0 & -w_{12} & 0 & 0 & -w_{13} & 0 & 0 & -w_{14} & 0 & 0 & -w_{15} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & w_{12} + w_{13} + w_{14} + w_{15} & 0 & 0 & -w_{12} & 0 & 0 & -w_{13} & 0 & 0 & -w_{14} & 0 & 0 & -w_{15} & 0 & 0 & 0 \\\\\n-w_{12} & 0 & 0 & w_{12} + w_{23} + w_{25} + w_{26} & 0 & 0 & -w_{23} & 0 & 0 & 0 & 0 & 0 & -w_{25} & 0 & 0 & -w_{26} & 0 & 0 \\\\\n0 & -w_{12} & 0 & 0 & w_{12} + w_{23} + w_{25} + w_{26} & 0 & 0 & -w_{23} & 0 & 0 & 0 & 0 & 0 & -w_{25} & 0 & 0 & -w_{26} & 0 \\\\\n0 & 0 & -w_{12} & 0 & 0 & w_{12} + w_{23} + w_{25} + w_{26} & 0 & 0 & -w_{23} & 0 & 0 & 0 & 0 & 0 & -w_{25} & 0 & 0 & -w_{26} \\\\\n-w_{13} & 0 & 0 & -w_{23} & 0 & 0 & w_{13} + w_{23} + w_{34} + w_{36} & 0 & 0 & -w_{34} & 0 & 0 & 0 & 0 & 0 & -w_{36} & 0 & 0 \\\\\n0 & -w_{13} & 0 & 0 & -w_{23} & 0 & 0 & w_{13} + w_{23} + w_{34} + w_{36} & 0 & 0 & -w_{34} & 0 & 0 & 0 & 0 & 0 & -w_{36} & 0 \\\\\n0 & 0 & -w_{13} & 0 & 0 & -w_{23} & 0 & 0 & w_{13} + w_{23} + w_{34} + w_{36} & 0 & 0 & -w_{34} & 0 & 0 & 0 & 0 & 0 & -w_{36} \\\\\n-w_{14} & 0 & 0 & 0 & 0 & 0 & -w_{34} & 0 & 0 & w_{14} + w_{34} + w_{45} + w_{46} & 0 & 0 & -w_{45} & 0 & 0 & -w_{46} & 0 & 0 \\\\\n0 & -w_{14} & 0 & 0 & 0 & 0 & 0 & -w_{34} & 0 & 0 & w_{14} + w_{34} + w_{45} + w_{46} & 0 & 0 & -w_{45} & 0 & 0 & -w_{46} & 0 \\\\\n0 & 0 & -w_{14} & 0 & 0 & 0 & 0 & 0 & -w_{34} & 0 & 0 & w_{14} + w_{34} + w_{45} + w_{46} & 0 & 0 & -w_{45} & 0 & 0 & -w_{46} \\\\\n-w_{15} & 0 & 0 & -w_{25} & 0 & 0 & 0 & 0 & 0 & -w_{45} & 0 & 0 & w_{15} + w_{25} + w_{45} + w_{56} & 0 & 0 & -w_{56} & 0 & 0 \\\\\n0 & -w_{15} & 0 & 0 & -w_{25} & 0 & 0 & 0 & 0 & 0 & -w_{45} & 0 & 0 & w_{15} + w_{25} + w_{45} + w_{56} & 0 & 0 & -w_{56} & 0 \\\\\n0 & 0 & -w_{15} & 0 & 0 & -w_{25} & 0 & 0 & 0 & 0 & 0 & -w_{45} & 0 & 0 & w_{15} + w_{25} + w_{45} + w_{56} & 0 & 0 & -w_{56} \\\\\n0 & 0 & 0 & -w_{26} & 0 & 0 & -w_{36} & 0 & 0 & -w_{46} & 0 & 0 & -w_{56} & 0 & 0 & w_{26} + w_{36} + w_{46} + w_{56} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & -w_{26} & 0 & 0 & -w_{36} & 0 & 0 & -w_{46} & 0 & 0 & -w_{56} & 0 & 0 & w_{26} + w_{36} + w_{46} + w_{56} & 0 \\\\\n0 & 0 & 0 & 0 & 0 & -w_{26} & 0 & 0 & -w_{36} & 0 & 0 & -w_{46} & 0 & 0 & -w_{56} & 0 & 0 & w_{26} + w_{36} + w_{46} + w_{56}\n\\end{array}\\right)$
"}︡{"done":true} ︠0d18221b-38a3-4752-b452-613273a27d13s︠ # Equation (11) truss.leftnullspace ︡5abe69f7-1f2f-47b1-85ff-438be539726f︡{"stdout":"[(1, 1, -1.732050807568878?, 1.732050807568878?, 1, -1.732050807568878?, 1.732050807568878?, 1.732050807568878?, -1.732050807568878?, 1, 1, 1)]\n"}︡{"done":true} ︠56e34956-a75b-48d0-b608-19913d6bbe97s︠ v = truss.flexes[0] # pick the first and only flex vector dg v = 0 w = truss.leftnullspace[0] # pick the first and only self stress w^T dg = 0 show(v,w) omegaw = truss.stress_matrices[0] # pick the first and only stress matrix computed from "w" above. show(omegaw) print v*omegaw*v # a very small semidefinite program ︡2a227e87-456c-434b-8327-d98fa5241cb4︡{"html":"
$\\displaystyle \\left(0,\\,1.577350269189626?,\\,0.2628917115316043?,\\,-1.366025403784439?,\\,-0.7886751345948129?,\\,0.2628917115316043?,\\,1.366025403784439?,\\,-0.7886751345948129?,\\,0.2628917115316043?,\\,-0.7886751345948129?,\\,1.366025403784439?,\\,-0.2628917115316043?,\\,-0.7886751345948129?,\\,-1.366025403784439?,\\,-0.2628917115316043?,\\,1.577350269189626?,\\,0,\\,-0.2628917115316043?\\right)$ $\\displaystyle \\left(1,\\,1,\\,-1.732050807568878?,\\,1.732050807568878?,\\,1,\\,-1.732050807568878?,\\,1.732050807568878?,\\,1.732050807568878?,\\,-1.732050807568878?,\\,1,\\,1,\\,1\\right)$
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$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrr}\n2.000000000000000? & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 1.732050807568878? & 0 & 0 & -1.732050807568878? & 0 & 0 & 0 & 0 & 0 \\\\\n0 & 2 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 1.732050807568878? & 0 & 0 & -1.732050807568878? & 0 & 0 & 0 & 0 \\\\\n0 & 0 & 2 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 1.732050807568878? & 0 & 0 & -1.732050807568878? & 0 & 0 & 0 \\\\\n-1 & 0 & 0 & 2.000000000000000? & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1.732050807568878? & 0 & 0 & -1.732050807568878? & 0 & 0 \\\\\n0 & -1 & 0 & 0 & 2 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1.732050807568878? & 0 & 0 & -1.732050807568878? & 0 \\\\\n0 & 0 & -1 & 0 & 0 & 2 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 1.732050807568878? & 0 & 0 & -1.732050807568878? \\\\\n-1 & 0 & 0 & -1 & 0 & 0 & 2 & 0 & 0 & -1.732050807568878? & 0 & 0 & 0 & 0 & 0 & 1.732050807568878? & 0 & 0 \\\\\n0 & -1 & 0 & 0 & -1 & 0 & 0 & 2 & 0 & 0 & -1.732050807568878? & 0 & 0 & 0 & 0 & 0 & 1.732050807568878? & 0 \\\\\n0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 2 & 0 & 0 & -1.732050807568878? & 0 & 0 & 0 & 0 & 0 & 1.732050807568878? \\\\\n1.732050807568878? & 0 & 0 & 0 & 0 & 0 & -1.732050807568878? & 0 & 0 & 2.000000000000000? & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 \\\\\n0 & 1.732050807568878? & 0 & 0 & 0 & 0 & 0 & -1.732050807568878? & 0 & 0 & 2 & 0 & 0 & -1 & 0 & 0 & -1 & 0 \\\\\n0 & 0 & 1.732050807568878? & 0 & 0 & 0 & 0 & 0 & -1.732050807568878? & 0 & 0 & 2 & 0 & 0 & -1 & 0 & 0 & -1 \\\\\n-1.732050807568878? & 0 & 0 & 1.732050807568878? & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2.000000000000000? & 0 & 0 & -1 & 0 & 0 \\\\\n0 & -1.732050807568878? & 0 & 0 & 1.732050807568878? & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2 & 0 & 0 & -1 & 0 \\\\\n0 & 0 & -1.732050807568878? & 0 & 0 & 1.732050807568878? & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2 & 0 & 0 & -1 \\\\\n0 & 0 & 0 & -1.732050807568878? & 0 & 0 & 1.732050807568878? & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 2 & 0 & 0 \\\\\n0 & 0 & 0 & 0 & -1.732050807568878? & 0 & 0 & 1.732050807568878? & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 2 & 0 \\\\\n0 & 0 & 0 & 0 & 0 & -1.732050807568878? & 0 & 0 & 1.732050807568878? & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & 2\n\\end{array}\\right)$
"}︡{"stdout":"89.56921938165305?\n"}︡{"done":true} ︠806bfdb7-1377-4039-a21c-7cccc1cf8514s︠ wgl = truss.get_weighted_graph_laplacian(); show(wgl) ︡a577d962-5884-4918-a148-11307e45115f︡{"html":"
$\\displaystyle \\left(\\begin{array}{rrrrrr}\nw_{12} + w_{13} + w_{14} + w_{15} & -w_{12} & -w_{13} & -w_{14} & -w_{15} & 0 \\\\\n-w_{12} & w_{12} + w_{23} + w_{25} + w_{26} & -w_{23} & 0 & -w_{25} & -w_{26} \\\\\n-w_{13} & -w_{23} & w_{13} + w_{23} + w_{34} + w_{36} & -w_{34} & 0 & -w_{36} \\\\\n-w_{14} & 0 & -w_{34} & w_{14} + w_{34} + w_{45} + w_{46} & -w_{45} & -w_{46} \\\\\n-w_{15} & -w_{25} & 0 & -w_{45} & w_{15} + w_{25} + w_{45} + w_{56} & -w_{56} \\\\\n0 & -w_{26} & -w_{36} & -w_{46} & -w_{56} & w_{26} + w_{36} + w_{46} + w_{56}\n\\end{array}\\right)$
"}︡{"done":true} ︠1538fa2c-3969-4b8f-bf41-4a26b4c26c4fs︠ wgl = truss.get_weighted_graph_laplacian() show(wgl.tensor_product(matrix.identity(3))); print; show(wgl.tensor_product(matrix.identity(3), subdivide=False)) ︡ea79232a-75fb-4d00-ba04-8709881a1f7c︡{"html":"
$\\displaystyle \\left(\\begin{array}{rrr|rrr|rrr|rrr|rrr|rrr}\nw_{12} + w_{13} + w_{14} + w_{15} & 0 & 0 & -w_{12} & 0 & 0 & -w_{13} & 0 & 0 & -w_{14} & 0 & 0 & -w_{15} & 0 & 0 & 0 & 0 & 0 \\\\\n0 & w_{12} + w_{13} + w_{14} + w_{15} & 0 & 0 & -w_{12} & 0 & 0 & -w_{13} & 0 & 0 & -w_{14} & 0 & 0 & -w_{15} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & w_{12} + w_{13} + w_{14} + w_{15} & 0 & 0 & -w_{12} & 0 & 0 & -w_{13} & 0 & 0 & -w_{14} & 0 & 0 & -w_{15} & 0 & 0 & 0 \\\\\n\\hline\n -w_{12} & 0 & 0 & w_{12} + w_{23} + w_{25} + w_{26} & 0 & 0 & -w_{23} & 0 & 0 & 0 & 0 & 0 & -w_{25} & 0 & 0 & -w_{26} & 0 & 0 \\\\\n0 & -w_{12} & 0 & 0 & w_{12} + w_{23} + w_{25} + w_{26} & 0 & 0 & -w_{23} & 0 & 0 & 0 & 0 & 0 & -w_{25} & 0 & 0 & -w_{26} & 0 \\\\\n0 & 0 & -w_{12} & 0 & 0 & w_{12} + w_{23} + w_{25} + w_{26} & 0 & 0 & -w_{23} & 0 & 0 & 0 & 0 & 0 & -w_{25} & 0 & 0 & -w_{26} \\\\\n\\hline\n -w_{13} & 0 & 0 & -w_{23} & 0 & 0 & w_{13} + w_{23} + w_{34} + w_{36} & 0 & 0 & -w_{34} & 0 & 0 & 0 & 0 & 0 & -w_{36} & 0 & 0 \\\\\n0 & -w_{13} & 0 & 0 & -w_{23} & 0 & 0 & w_{13} + w_{23} + w_{34} + w_{36} & 0 & 0 & -w_{34} & 0 & 0 & 0 & 0 & 0 & -w_{36} & 0 \\\\\n0 & 0 & -w_{13} & 0 & 0 & -w_{23} & 0 & 0 & w_{13} + w_{23} + w_{34} + w_{36} & 0 & 0 & -w_{34} & 0 & 0 & 0 & 0 & 0 & -w_{36} \\\\\n\\hline\n -w_{14} & 0 & 0 & 0 & 0 & 0 & -w_{34} & 0 & 0 & w_{14} + w_{34} + w_{45} + w_{46} & 0 & 0 & -w_{45} & 0 & 0 & -w_{46} & 0 & 0 \\\\\n0 & -w_{14} & 0 & 0 & 0 & 0 & 0 & -w_{34} & 0 & 0 & w_{14} + w_{34} + w_{45} + w_{46} & 0 & 0 & -w_{45} & 0 & 0 & -w_{46} & 0 \\\\\n0 & 0 & -w_{14} & 0 & 0 & 0 & 0 & 0 & -w_{34} & 0 & 0 & w_{14} + w_{34} + w_{45} + w_{46} & 0 & 0 & -w_{45} & 0 & 0 & -w_{46} \\\\\n\\hline\n -w_{15} & 0 & 0 & -w_{25} & 0 & 0 & 0 & 0 & 0 & -w_{45} & 0 & 0 & w_{15} + w_{25} + w_{45} + w_{56} & 0 & 0 & -w_{56} & 0 & 0 \\\\\n0 & -w_{15} & 0 & 0 & -w_{25} & 0 & 0 & 0 & 0 & 0 & -w_{45} & 0 & 0 & w_{15} + w_{25} + w_{45} + w_{56} & 0 & 0 & -w_{56} & 0 \\\\\n0 & 0 & -w_{15} & 0 & 0 & -w_{25} & 0 & 0 & 0 & 0 & 0 & -w_{45} & 0 & 0 & w_{15} + w_{25} + w_{45} + w_{56} & 0 & 0 & -w_{56} \\\\\n\\hline\n 0 & 0 & 0 & -w_{26} & 0 & 0 & -w_{36} & 0 & 0 & -w_{46} & 0 & 0 & -w_{56} & 0 & 0 & w_{26} + w_{36} + w_{46} + w_{56} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & -w_{26} & 0 & 0 & -w_{36} & 0 & 0 & -w_{46} & 0 & 0 & -w_{56} & 0 & 0 & w_{26} + w_{36} + w_{46} + w_{56} & 0 \\\\\n0 & 0 & 0 & 0 & 0 & -w_{26} & 0 & 0 & -w_{36} & 0 & 0 & -w_{46} & 0 & 0 & -w_{56} & 0 & 0 & w_{26} + w_{36} + w_{46} + w_{56}\n\\end{array}\\right)$
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$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrrrrrrrr}\nw_{12} + w_{13} + w_{14} + w_{15} & 0 & 0 & -w_{12} & 0 & 0 & -w_{13} & 0 & 0 & -w_{14} & 0 & 0 & -w_{15} & 0 & 0 & 0 & 0 & 0 \\\\\n0 & w_{12} + w_{13} + w_{14} + w_{15} & 0 & 0 & -w_{12} & 0 & 0 & -w_{13} & 0 & 0 & -w_{14} & 0 & 0 & -w_{15} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & w_{12} + w_{13} + w_{14} + w_{15} & 0 & 0 & -w_{12} & 0 & 0 & -w_{13} & 0 & 0 & -w_{14} & 0 & 0 & -w_{15} & 0 & 0 & 0 \\\\\n-w_{12} & 0 & 0 & w_{12} + w_{23} + w_{25} + w_{26} & 0 & 0 & -w_{23} & 0 & 0 & 0 & 0 & 0 & -w_{25} & 0 & 0 & -w_{26} & 0 & 0 \\\\\n0 & -w_{12} & 0 & 0 & w_{12} + w_{23} + w_{25} + w_{26} & 0 & 0 & -w_{23} & 0 & 0 & 0 & 0 & 0 & -w_{25} & 0 & 0 & -w_{26} & 0 \\\\\n0 & 0 & -w_{12} & 0 & 0 & w_{12} + w_{23} + w_{25} + w_{26} & 0 & 0 & -w_{23} & 0 & 0 & 0 & 0 & 0 & -w_{25} & 0 & 0 & -w_{26} \\\\\n-w_{13} & 0 & 0 & -w_{23} & 0 & 0 & w_{13} + w_{23} + w_{34} + w_{36} & 0 & 0 & -w_{34} & 0 & 0 & 0 & 0 & 0 & -w_{36} & 0 & 0 \\\\\n0 & -w_{13} & 0 & 0 & -w_{23} & 0 & 0 & w_{13} + w_{23} + w_{34} + w_{36} & 0 & 0 & -w_{34} & 0 & 0 & 0 & 0 & 0 & -w_{36} & 0 \\\\\n0 & 0 & -w_{13} & 0 & 0 & -w_{23} & 0 & 0 & w_{13} + w_{23} + w_{34} + w_{36} & 0 & 0 & -w_{34} & 0 & 0 & 0 & 0 & 0 & -w_{36} \\\\\n-w_{14} & 0 & 0 & 0 & 0 & 0 & -w_{34} & 0 & 0 & w_{14} + w_{34} + w_{45} + w_{46} & 0 & 0 & -w_{45} & 0 & 0 & -w_{46} & 0 & 0 \\\\\n0 & -w_{14} & 0 & 0 & 0 & 0 & 0 & -w_{34} & 0 & 0 & w_{14} + w_{34} + w_{45} + w_{46} & 0 & 0 & -w_{45} & 0 & 0 & -w_{46} & 0 \\\\\n0 & 0 & -w_{14} & 0 & 0 & 0 & 0 & 0 & -w_{34} & 0 & 0 & w_{14} + w_{34} + w_{45} + w_{46} & 0 & 0 & -w_{45} & 0 & 0 & -w_{46} \\\\\n-w_{15} & 0 & 0 & -w_{25} & 0 & 0 & 0 & 0 & 0 & -w_{45} & 0 & 0 & w_{15} + w_{25} + w_{45} + w_{56} & 0 & 0 & -w_{56} & 0 & 0 \\\\\n0 & -w_{15} & 0 & 0 & -w_{25} & 0 & 0 & 0 & 0 & 0 & -w_{45} & 0 & 0 & w_{15} + w_{25} + w_{45} + w_{56} & 0 & 0 & -w_{56} & 0 \\\\\n0 & 0 & -w_{15} & 0 & 0 & -w_{25} & 0 & 0 & 0 & 0 & 0 & -w_{45} & 0 & 0 & w_{15} + w_{25} + w_{45} + w_{56} & 0 & 0 & -w_{56} \\\\\n0 & 0 & 0 & -w_{26} & 0 & 0 & -w_{36} & 0 & 0 & -w_{46} & 0 & 0 & -w_{56} & 0 & 0 & w_{26} + w_{36} + w_{46} + w_{56} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & -w_{26} & 0 & 0 & -w_{36} & 0 & 0 & -w_{46} & 0 & 0 & -w_{56} & 0 & 0 & w_{26} + w_{36} + w_{46} + w_{56} & 0 \\\\\n0 & 0 & 0 & 0 & 0 & -w_{26} & 0 & 0 & -w_{36} & 0 & 0 & -w_{46} & 0 & 0 & -w_{56} & 0 & 0 & w_{26} + w_{36} + w_{46} + w_{56}\n\\end{array}\\right)$
"}︡{"done":true} ︠95b5f31c-14d9-464b-9d31-f0bb5b27074e︠ xvarz = [var('x%s%s'%(i,j)) for i in range(1,2+1) for j in range(1,5+1)] adjacentminorz = [x11*x22 - x12*x21, x12*x23 - x13*x22, x13*x24 - x14*x23, x14*x25 - x15*x24] R = PolynomialRing(QQ,xvarz) I = R.ideal(adjacentminorz) for P in I.associated_primes(): print P; print; ︡33955931-2eee-4b6b-a6b3-a20921c03bca︡{"stdout":"Ideal (x23, x13, x15*x24 - x14*x25, x12*x21 - x11*x22) of Multivariate Polynomial Ring in x11, x12, x13, x14, x15, x21, x22, x23, x24, x25 over Rational Field"}︡{"stdout":"\n\nIdeal (x24, x14, x13*x22 - x12*x23, x13*x21 - x11*x23, x12*x21 - x11*x22) of Multivariate Polynomial Ring in x11, x12, x13, x14, x15, x21, x22, x23, x24, x25 over Rational Field\n\nIdeal (x24, x22, x14, x12) of Multivariate Polynomial Ring in x11, x12, x13, x14, x15, x21, x22, x23, x24, x25 over Rational Field\n\nIdeal (x22, x12, x15*x24 - x14*x25, x15*x23 - x13*x25, x14*x23 - x13*x24) of Multivariate Polynomial Ring in x11, x12, x13, x14, x15, x21, x22, x23, x24, x25 over Rational Field\n\nIdeal (x15*x24 - x14*x25, x15*x23 - x13*x25, x14*x23 - x13*x24, x15*x22 - x12*x25, x14*x22 - x12*x24, x13*x22 - x12*x23, x15*x21 - x11*x25, x14*x21 - x11*x24, x13*x21 - x11*x23, x12*x21 - x11*x22) of Multivariate Polynomial Ring in x11, x12, x13, x14, x15, x21, x22, x23, x24, x25 over Rational Field\n\n"}︡{"done":true} ︠9b557c37-d231-4332-88ac-01cd0801b9c1s︠ load("tensegrity.sage") nodes = range(1,5+1) # 5 nodes in the plane node_coordinates = {1:(0,0),2:(1,0),3:(2,-1),4:(2,1),5:(2,0)} edges = [(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)] dim = 2 truss = Truss(nodes,edges,node_coordinates,dim,orthonormal=True) truss.plot_arrow_scale = 2.0 truss.plot_mechanisms() show(truss.before, frame=False) ︡07902dcd-e887-4f50-83eb-e1c6b3fdd61f︡{"file":{"filename":"/home/user/.sage/temp/project-e3d8858f-633e-4199-98c4-2459b1781fb3/85594/tmp_05VsYw.svg","show":true,"text":null,"uuid":"00a7d709-3092-453b-9a38-0a8153d0727e"},"once":false}︡{"done":true} ︠e046fb4a-8e7b-491d-bd88-d109dce16423s︠ A = truss.rigidity_matrix; show(A) ︡e828772f-3983-4084-b383-af945321e47e︡{"html":"
$\\displaystyle \\left(\\begin{array}{rrrrrrrrrr}\nx_{11} - x_{21} & x_{12} - x_{22} & -x_{11} + x_{21} & -x_{12} + x_{22} & 0 & 0 & 0 & 0 & 0 & 0 \\\\\nx_{11} - x_{31} & x_{12} - x_{32} & 0 & 0 & -x_{11} + x_{31} & -x_{12} + x_{32} & 0 & 0 & 0 & 0 \\\\\nx_{11} - x_{41} & x_{12} - x_{42} & 0 & 0 & 0 & 0 & -x_{11} + x_{41} & -x_{12} + x_{42} & 0 & 0 \\\\\n0 & 0 & x_{21} - x_{31} & x_{22} - x_{32} & -x_{21} + x_{31} & -x_{22} + x_{32} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & x_{21} - x_{41} & x_{22} - x_{42} & 0 & 0 & -x_{21} + x_{41} & -x_{22} + x_{42} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & x_{31} - x_{51} & x_{32} - x_{52} & 0 & 0 & -x_{31} + x_{51} & -x_{32} + x_{52} \\\\\n0 & 0 & 0 & 0 & 0 & 0 & x_{41} - x_{51} & x_{42} - x_{52} & -x_{41} + x_{51} & -x_{42} + x_{52}\n\\end{array}\\right)$
"}︡{"done":true} ︠f6fc803f-63dd-4051-86cf-ac495ac46935s︠ subz = {var('x%s%s'%(i,k)):QQ.random_element(50,50) for i in range(1,6+1) for k in range(1,2+1)}; subz B = A.subs(subz); show(B) rank(B) # generic rank ︡32cd1e9f-45c8-4de6-929a-1d0674c0123d︡{"stdout":"{x12: -31/48, x61: -13/23, x22: 15/19, x11: -45/46, x32: -37/24, x21: 47/44, x42: 46/47, x31: 25/24, x52: 7/5, x41: -13/17, x62: -35/8, x51: 1/27}\n"}︡{"html":"
$\\displaystyle \\left(\\begin{array}{rrrrrrrrrr}\n-\\frac{2071}{1012} & -\\frac{1309}{912} & \\frac{2071}{1012} & \\frac{1309}{912} & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n-\\frac{1115}{552} & \\frac{43}{48} & 0 & 0 & \\frac{1115}{552} & -\\frac{43}{48} & 0 & 0 & 0 & 0 \\\\\n-\\frac{167}{782} & -\\frac{3665}{2256} & 0 & 0 & 0 & 0 & \\frac{167}{782} & \\frac{3665}{2256} & 0 & 0 \\\\\n0 & 0 & \\frac{7}{264} & \\frac{1063}{456} & -\\frac{7}{264} & -\\frac{1063}{456} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & \\frac{1371}{748} & -\\frac{169}{893} & 0 & 0 & -\\frac{1371}{748} & \\frac{169}{893} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & \\frac{217}{216} & -\\frac{353}{120} & 0 & 0 & -\\frac{217}{216} & \\frac{353}{120} \\\\\n0 & 0 & 0 & 0 & 0 & 0 & -\\frac{368}{459} & -\\frac{99}{235} & \\frac{368}{459} & \\frac{99}{235}\n\\end{array}\\right)$
"}︡{"stdout":"7\n"}︡{"done":true} ︠a0febff3-d74f-477b-b541-cb3071afd1e0s︠ subz = {var('x11'):0, var('x12'):0, var('x22'):0} # moving frame B = A.subs(subz) show(B) ︡4ce43634-4543-46e4-b237-058b460cb41a︡{"html":"
$\\displaystyle \\left(\\begin{array}{rrrrrrrrrr}\n-x_{21} & 0 & x_{21} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n-x_{31} & -x_{32} & 0 & 0 & x_{31} & x_{32} & 0 & 0 & 0 & 0 \\\\\n-x_{41} & -x_{42} & 0 & 0 & 0 & 0 & x_{41} & x_{42} & 0 & 0 \\\\\n0 & 0 & x_{21} - x_{31} & -x_{32} & -x_{21} + x_{31} & x_{32} & 0 & 0 & 0 & 0 \\\\\n0 & 0 & x_{21} - x_{41} & -x_{42} & 0 & 0 & -x_{21} + x_{41} & x_{42} & 0 & 0 \\\\\n0 & 0 & 0 & 0 & x_{31} - x_{51} & x_{32} - x_{52} & 0 & 0 & -x_{31} + x_{51} & -x_{32} + x_{52} \\\\\n0 & 0 & 0 & 0 & 0 & 0 & x_{41} - x_{51} & x_{42} - x_{52} & -x_{41} + x_{51} & -x_{42} + x_{52}\n\\end{array}\\right)$
"}︡{"done":true} ︠02dede80-c08d-4534-8b99-e1e6c58bd8b7s︠ minorz = B.minors(7) # generic rank 7 minorz = [minor for minor in minorz if minor!=0] print len(minorz), binomial(10,7) ︡0caa6976-3662-402b-a90c-2fe8c04c34da︡{"stdout":"95 120\n"}︡{"done":true} ︠333db155-2b9b-4a7d-8ad7-8b60cff1b4b2s︠ subz = {var('x11'):0, var('x12'):0, var('x22'):0} # moving frame eqns = [truss.bar_constraints[e].subs(subz) for e in truss.edges] for eqn in eqns: print eqn ︡2c26bcf0-6d0d-4b7f-bd36-90450d99508d︡{"stdout":"x21^2 - 1\nx31^2 + x32^2 - 5\nx41^2 + x42^2 - 5\n(x21 - x31)^2 + x32^2 - 2\n(x21 - x41)^2 + x42^2 - 2\n(x31 - x51)^2 + (x32 - x52)^2 - 1\n(x41 - x51)^2 + (x42 - x52)^2 - 1\n"}︡{"done":true} ︠d29c2650-926a-4321-bc31-de8dc9a40c23s︠ subz = {var('x11'):0, var('x12'):0, var('x22'):0} # moving frame eqns = [truss.bar_constraints[e].subs(subz) for e in truss.edges] eqnz = eqns + minorz xvarz = [var('x%s%s'%(i,k)) for i in range(1,6+1) for k in range(1,2+1) if i > k] R = PolynomialRing(QQ,xvarz) eqnz = [R(eqn) for eqn in eqnz] I = R.ideal(eqnz) AP = I.associated_primes() ︡abbb5b89-2fc1-436c-8821-eb99cf200ee5︡{"done":true} ︠c40b8391-1f9b-4d09-ade6-a9ee33c5c072s︠ for P in AP: print P.gens() ︡ad1d14db-c4dc-4747-a901-823d377e6481︡{"stdout":"[x52, x51 + 2, x42 - 1, x41 + 2, x32 + 1, x31 + 2, x21 + 1]\n[x52, x51 + 2, x42 + 1, x41 + 2, x32 - 1, x31 + 2, x21 + 1]\n[x42 + 1, x41 + 2, x32 + 1, x31 + 2, x21 + 1, x51^2 + x52^2 + 4*x51 + 2*x52 + 4]\n[x42 - 1, x41 + 2, x32 - 1, x31 + 2, x21 + 1, x51^2 + x52^2 + 4*x51 - 2*x52 + 4]\n[x52, x51 - 2, x42 - 1, x41 - 2, x32 + 1, x31 - 2, x21 - 1]\n[x52, x51 - 2, x42 + 1, x41 - 2, x32 - 1, x31 - 2, x21 - 1]\n[x42 + 1, x41 - 2, x32 + 1, x31 - 2, x21 - 1, x51^2 + x52^2 - 4*x51 + 2*x52 + 4]\n[x42 - 1, x41 - 2, x32 - 1, x31 - 2, x21 - 1, x51^2 + x52^2 - 4*x51 - 2*x52 + 4]\n"}︡{"done":true} ︠90f863f9-ca56-4ca3-a5d7-1c4c4c508181︠