2108-Pl-Pl-Pl

︠fe990455-02e5-4b4c-984b-2f342c757b81i︠
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<h1>Pl&uuml;cker-Konoid</h1>

︡3a3a4a4f-9bc4-4ab9-ad60-fde4abda32bc︡{"html": "<h1>Pl&uuml;cker-Konoid</h1>"}︡
︠86a09ecc-ce3e-4848-bc3e-545c3f53f1b1︠
var('phi,v')
︡9b6428ec-3769-49a4-81f2-fe3ad28c67c7︡{"stdout": "(phi, v)"}︡
︠5bfd71a1-ed7b-4f46-a6a2-85adb87d8577i︠
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<p>Richtungs- und Momentenvektor:</p>

︡b1e38805-362b-4bf1-877c-abb1227a9f22︡{"html": "<p>Richtungs- und Momentenvektor:</p>"}︡
︠bfb6bec1-1fd4-4d96-948b-76a1caf682ba︠
r=vector([cos(phi),sin(phi),0])
r_=vector([-2*sin(phi)^2*cos(phi),2*sin(phi)*cos(phi)^2,0])
︡112d4bcc-433d-43b2-a238-24ceaac914bf︡︡
︠8e712dc0-15c8-4840-b3f3-c5d964078325i︠
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<p>Geradenkoordinaten:</p>

︡1b614f8e-c955-4e13-913d-a6ccddd5a25e︡{"html": "<p>Geradenkoordinaten:</p>"}︡
︠63666940-0e08-49e8-bf4b-a643237d82ae︠
R=vector([r[0],r[1],r[2],r_[0],r_[1],r_[2]])
show(R)
︡f67d8456-d010-437d-994f-b6040f1adab7︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\cos\\left(\\phi\\right),\\sin\\left(\\phi\\right),0,-2 \\, \\sin\\left(\\phi\\right)^{2} \\cos\\left(\\phi\\right),2 \\, \\sin\\left(\\phi\\right) \\cos\\left(\\phi\\right)^{2},0\\right)</div>"}︡
︠8663a439-92f6-47e1-9f2a-f9f4e7155503i︠
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<h2>Parameterdarstellung im R&sup3; (durch Leitkurve und Richtung):</h2>

︡cf5b71e9-b581-4a96-93ad-9625e7420883︡{"html": "<h2>Parameterdarstellung im R&sup3; (durch Leitkurve und Richtung):</h2>"}︡
︠2d20a8ae-fbd2-4fd1-ac71-31948be348da︠
show(r.cross_product(r_)+v*vector([cos(phi),sin(phi),0]))
S=vector([v*cos(phi),v*sin(phi),2*sin(phi)*cos(phi)])
show(S)
#Richtungsvektor:
rv=vector([cos(phi),sin(phi),0])
show(rv)
#Leitkurve:
l=vector([0,0,2*sin(phi)*cos(phi)])
show(l)
︡88af0b5e-0fc2-490e-ac02-50b7cf7bf6d8︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(v \\cos\\left(\\phi\\right),v \\sin\\left(\\phi\\right),2 \\, \\sin\\left(\\phi\\right)^{3} \\cos\\left(\\phi\\right) + 2 \\, \\sin\\left(\\phi\\right) \\cos\\left(\\phi\\right)^{3}\\right)</div>"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(v \\cos\\left(\\phi\\right),v \\sin\\left(\\phi\\right),2 \\, \\sin\\left(\\phi\\right) \\cos\\left(\\phi\\right)\\right)</div>"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\cos\\left(\\phi\\right),\\sin\\left(\\phi\\right),0\\right)</div>"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0,0,2 \\, \\sin\\left(\\phi\\right) \\cos\\left(\\phi\\right)\\right)</div>"}︡
︠2c50805a-e4c3-4fbd-b137-da8391f5c6f3i︠
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<h2>Singul&auml;re und torsale Erzeugende</h2>
<p>Erste Ableitung:</p>

︡5f751053-31a5-4599-a4a9-84d0f6c087f9︡{"html": "<h2>Singul&auml;re und torsale Erzeugende</h2>\n<p>Erste Ableitung:</p>"}︡
︠ca2a131c-adbe-4045-8036-ed190ba76396︠
R_=diff(R,phi)
show(R); show(R_)
︡67362ab0-72d0-40d3-b2e9-50065e68a7eb︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\cos\\left(\\phi\\right),\\sin\\left(\\phi\\right),0,-2 \\, \\sin\\left(\\phi\\right)^{2} \\cos\\left(\\phi\\right),2 \\, \\sin\\left(\\phi\\right) \\cos\\left(\\phi\\right)^{2},0\\right)</div>"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(-\\sin\\left(\\phi\\right),\\cos\\left(\\phi\\right),0,2 \\, \\sin\\left(\\phi\\right)^{3} - 4 \\, \\sin\\left(\\phi\\right) \\cos\\left(\\phi\\right)^{2},-4 \\, \\sin\\left(\\phi\\right)^{2} \\cos\\left(\\phi\\right) + 2 \\, \\cos\\left(\\phi\\right)^{3},0\\right)</div>"}︡
︠8a9a465d-3eaa-40c5-ac2b-200babf62ab1i︠
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<p>Singul&auml;re Erzeugende: R,R_ l.a.</p>
<p>Es gibt keine singul&auml;ren Erzeugenden</p>
<p>Torsale Erzeugende: Omega(R_,R_)=0</p>

︡f5e4a504-dd81-47e6-9d3c-08d8099577f4︡{"html": "<p>Singul&auml;re Erzeugende: R,R_ l.a.</p>\n<p>Es gibt keine singul&auml;ren Erzeugenden</p>\n<p>Torsale Erzeugende: Omega(R_,R_)=0</p>"}︡
︠481223c2-06f4-4a10-91fe-2acc7a6873b3︠
solve(simplify(R_[0]*R_[3]+R_[1]*R_[4]+R_[2]*R_[5]),phi)
︡98ddd150-8201-4d33-9043-ed1647ace4d6︡{"stdout": "[sin(phi) == I*cos(phi), sin(phi) == -cos(phi), sin(phi) == -I*cos(phi), sin(phi) == cos(phi)]"}︡
︠3582fd4d-cf82-4fe3-a863-c45ea7ba04e3i︠
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<p>zu den torsalen Erzeugenden geh&ouml;rende reelle Parameterwerte:</p>
<p>pi/4; 3*pi/4; 5*pi/4; 7*pi/4</p>
<h2>Gleichung und Grad</h2>

︡41fc3455-9542-4cc4-92e7-2c56c588e123︡{"html": "<p>zu den torsalen Erzeugenden geh&ouml;rende reelle Parameterwerte:</p>\n<p>pi/4; 3*pi/4; 5*pi/4; 7*pi/4</p>\n<h2>Gleichung und Grad</h2>"}︡
︠1524c6ff-1175-4e8c-b605-a3703a3382c0︠
var('x,y,z')
︡6854c28f-10af-4107-be61-4580b17d2fc9︡{"stdout": "(x, y, z)"}︡
︠98ca05c1-9149-4ea9-acd2-524a5a66b19e︠
show(S)
︡a5820766-356f-4377-b2e0-f05c9aa53b66︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(v \\cos\\left(\\phi\\right),v \\sin\\left(\\phi\\right),2 \\, \\sin\\left(\\phi\\right) \\cos\\left(\\phi\\right)\\right)</div>"}︡
︠8b7da10b-f127-4d33-a5df-58e369be304ei︠
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<p><span style="font-family: terminal,monaco;">Es gilt:</span></p>
<p><span style="font-family: terminal,monaco;">x^2+y^2=v^2</span></p>
<p><span style="font-family: terminal,monaco;">2*x*y=z*v^2</span></p>
<p><span style="font-family: terminal,monaco;">&rarr; </span>2*x*y/(x^2+y^2)-z=0</p>
<p><span style="font-family: terminal,monaco;">&rarr; </span>2*x*y-z*(x^2+y^2)=0</p>

︡335534d7-03b4-4870-a1bc-6189f1170601︡{"html": "<p><span style=\"font-family: terminal,monaco;\">Es gilt:</span></p>\n<p><span style=\"font-family: terminal,monaco;\">x^2+y^2=v^2</span></p>\n<p><span style=\"font-family: terminal,monaco;\">2*x*y=z*v^2</span></p>\n<p><span style=\"font-family: terminal,monaco;\">&rarr; </span>2*x*y/(x^2+y^2)-z=0</p>\n<p><span style=\"font-family: terminal,monaco;\">&rarr; </span>2*x*y-z*(x^2+y^2)=0</p>"}︡
︠4d8733e7-6243-4be6-b698-5942cbcabd2c︠
var('x0,x1,x2,x3')
︡e1b49626-99b2-472d-8b05-75082d496ea0︡{"stdout": "(x0, x1, x2, x3)"}︡
︠e59a48a3-df7d-4481-afff-f85ce076f613︠
gl2=-(x1^2+x2^2)*x3+2*x1*x2*x0
show(gl2)
︡90649f32-22e4-4c81-9f45-11c85a499a49︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}2 \\, x_{0} x_{1} x_{2} - {\\left(x_{1}^{2} + x_{2}^{2}\\right)} x_{3}</div>"}︡
︠0f738ac9-abbd-41f0-a9f9-508487d29726i︠
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<p>&rarr; Die Fl&auml;che hat den algebraischen Grad 3</p>

<h2>Schnitt mit der Fernebene</h2>

︡8d75b138-2c39-42e1-b742-1ba517427cba︡{"html": "<p>&rarr; Die Fl&auml;che hat den algebraischen Grad 3</p>\n\n<h2>Schnitt mit der Fernebene</h2>"}︡
︠337f254f-2565-4c3a-b409-b402c7dad7ae︠
show(gl2.subs(x0=0))
︡30b013f6-3a56-4b83-8437-1e12c395f464︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}-{\\left(x_{1}^{2} + x_{2}^{2}\\right)} x_{3}</div>"}︡
︠a7f2d75f-b9ae-40de-b111-101320801fbei︠
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<h2>Drall</h2>

︡627a441e-b536-44a3-8c5d-89aa6626149b︡{"html": "<h2>Drall</h2>"}︡
︠2eaa6a56-dc28-4963-a89f-03b43884fdbd︠
transpose(matrix([diff(l,phi),rv,diff(rv,phi)]))
︡05fc086b-3a4d-47aa-9f50-7021c40232ff︡{"stdout": "[                           0                     cos(phi)                    -sin(phi)]\n[                           0                     sin(phi)                     cos(phi)]\n[-2*sin(phi)^2 + 2*cos(phi)^2                            0                            0]"}︡
︠6eb74a9c-54db-40f1-999e-f1290f0a802a︠
det(_)
︡6cb2df69-5790-403f-a951-3c8970977aca︡{"stdout": "-2*(sin(phi)^2 - cos(phi)^2)*(sin(phi)^2 + cos(phi)^2)"}︡
︠9222fdef-5fbd-47a3-938d-a7eec8884ba8︠
delta=-2*(sin(phi)^2-cos(phi)^2)
delta
︡e3a0bf58-c890-49c4-ab5f-b211c932c548︡{"stdout": "-2*sin(phi)^2 + 2*cos(phi)^2"}︡
︠b0cfb767-01ef-4e41-9eaf-bfd741575f26i︠
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<h2>Striktionskurve</h2>

︡f6d8ee8a-8273-41b1-a884-2d2f9726f0fb︡{"html": "<h2>Striktionskurve</h2>"}︡
︠2ab80f68-ccca-4b0a-8507-909fa6b99a7c︠
rv_=diff(rv,phi)
show(l)
show(rv_)
︡b94c553a-9921-4ed5-8634-b72a25ecc486︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0,0,2 \\, \\sin\\left(\\phi\\right) \\cos\\left(\\phi\\right)\\right)</div>"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(-\\sin\\left(\\phi\\right),\\cos\\left(\\phi\\right),0\\right)</div>"}︡
︠bbf1158a-0e2c-42c0-985f-05b83b8dc4c1︠
v_s=-l.dot_product(rv_)/rv_.dot_product(rv_)
v_s
︡ffdc0626-19dc-4f0b-9c03-bff8d638cecf︡{"stdout": "0"}︡
︠de102d67-b5c9-4bf9-8674-f0f01351db93︠
s=l
show(s)
︡41b47b5b-14c5-4c3d-9502-038e151f7d3e︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0,0,2 \\, \\sin\\left(\\phi\\right) \\cos\\left(\\phi\\right)\\right)</div>"}︡
︠b068af29-e3ec-42fd-a274-7339f678fd3di︠
%html
<h2>Ber&uuml;hrprojektivit&auml;t</h2>

︡f3f264e0-15a8-4a42-b268-2956a537ccb8︡{"html": "<h2>Ber&uuml;hrprojektivit&auml;t</h2>"}︡
︠1b62eda0-05f9-4d9a-9799-fa86a122bfa9︠
S.subs(phi=0)
︡bbbe23cf-2b9e-4125-91e4-cdad5342c499︡{"stdout": "(v, 0, 0)"}︡
︠1d8f8a95-8872-46d2-b723-5396d6df68f7i︠
%html
<p>Koordinaten des Punktes: (1,v,0,0)</p>

︡89421e78-56ea-4826-ae44-7e83a7cb49b8︡{"html": "<p>Koordinaten des Punktes: (1,v,0,0)</p>"}︡
︠51ac1e14-a118-4764-a7dd-ef1ae5c915c6︠
S1=diff(S,phi)
S2=diff(S,v)
N=S1.cross_product(S2)
︡a75c620d-c40f-48c8-994d-b7829f3809af︡︡
︠d33e1f97-28ac-4151-a43e-b26431454897︠
N.subs(phi=0)
︡ee2170f5-d919-4d7d-ac0c-80f258b5b96d︡{"stdout": "(0, 2, -v)"}︡
︠e36c116f-36c8-423d-b799-f71e3d3c4db6i︠
%html
<p>Koordinaten der Tangentialebene: (0,0,2,-v)</p>

<p>Eine m&ouml;gliche Transformationsmatrix:</p>

︡745bc2e0-a533-47f6-81c6-f7f8ff546329︡{"html": "<p>Koordinaten der Tangentialebene: (0,0,2,-v)</p>\n\n<p>Eine m&ouml;gliche Transformationsmatrix:</p>"}︡
︠4dc0e982-b050-4794-8280-5ccaefd0402f︠
M=matrix([[0,0,2,0],[0,0,0,-1],[0,0,0,0],[0,0,0,0]])
︡a1c6c1eb-8022-4ef6-ac50-4d48ebdfbc5f︡︡
︠25328351-44b8-4570-85d8-f7c1401b66a0︠
v1=vector([1,v,0,0])
︡7f8fe6bf-7a48-4b27-8e69-2c4ba2b41c33︡︡
︠17dc3098-cc6a-4c6f-88f0-acd93b56fef9︠
v1*M
︡b78c0e64-c7f3-45c5-ac83-482713ad1d62︡{"stdout": "(0, 0, 2, -v)"}︡
︠e83edd5e-ca7c-4001-bff4-c061e10f74a3i︠
%html
<h2>Schmiegquadrik</h2>

<h2></h2>

︡f9a4e6a5-afa2-468c-a5ab-5bc72a234165︡{"html": "<h2>Schmiegquadrik</h2>\n\n<h2></h2>"}︡
︠52d42a5d-8b31-4ca6-a88e-5a489663c9d5︠
show(R.subs(phi=0))
show(R_.subs(phi=0))
R__=diff(R_,phi)
show(R__.subs(phi=0))
︡918b7836-52d5-4052-92b9-6c39aefb7ac7︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1,0,0,0,0,0\\right)</div>"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0,1,0,0,2,0\\right)</div>"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(-1,0,0,-4,0,0\\right)</div>"}︡
︠fc6a210b-c147-4498-b41c-e813e5728230︠
#Aufgespannte Ebene im R6:
var('l,m,n')
e=l*vector([1,0,0,0,0,0])+m*vector([0,1,0,0,2,0])+n*vector([-1,0,0,-4,0,0])
e
︡431b7fe9-f0a5-4f8f-a87f-f218730a9a26︡{"stdout": "(l - n, m, 0, -4*n, 2*m, 0)"}︡
︠70ec4963-9d7d-48f2-b856-10d7d899e7e8︠
#e geschnitten m42:
solve((l-n)*(-4*n)+m*2*m,l)
︡d20dcb7a-45aa-4482-873a-1e288715d5e2︡{"stdout": "[l == 1/2*(m^2 + 2*n^2)/n]"}︡
︠4faf3695-7f83-46ee-8720-9d45387fa930︠
e1=e.subs(l=(m^2+2*n^2)/2/n)
︡78ffdb60-65d0-408f-b88f-683ccde96ada︡︡
︠fe5e1bc4-afb4-4d31-a59a-82561a2214f5︠
e1*2*n
︡a3e5e53d-fcd4-4f2d-9450-687e7f148669︡{"stdout": "(-(2*n - (m^2 + 2*n^2)/n)*n, 2*m*n, 0, -8*n^2, 4*m*n, 0)"}︡
︠ec2e3f86-8922-42dd-81e8-3e947efc0ccd︠
Lambda=vector([-2*n^2-(m^2+2*n^2),2*m*n,0,-8*n^2,4*m*n,0])
show(Lambda/n^2)
#setze t=m/n:
var('t')
vector([-t^2+4,2*t,0,-8,4*t,0])
gamma= vector([-t^2+4,2*t,0])
gamma_=vector([-8,4*t,0])
var('u')
cc=gamma.cross_product(gamma_)/ (gamma.dot_product(gamma)) + u*gamma
show(cc)
︡a8a19ee5-1eb5-42c5-86be-ef901ca0d9c5︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\frac{-{\\left(m^{2} + 4 \\, n^{2}\\right)}}{n^{2}},\\frac{2 \\, m}{n},0,-8,\\frac{4 \\, m}{n},0\\right)</div>"}︡{"html": "<div class=\"math\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(-{\\left(t^{2} - 4\\right)} u,2 \\, t u,\\frac{-4 \\, {\\left({\\left(t^{2} - 4\\right)} t - 4 \\, t\\right)}}{{\\left({\\left(t^{2} - 4\\right)}^{2} + 4 \\, t^{2}\\right)}}\\right)</div>"}︡
︠3dfe084d-50f6-483f-b27d-c5ede1ec9a2a︠
c=parametric_plot3d(cc,(u,-0.5,0.5),(t,-0.5,0.5),viewer='tachyon',rgbcolor='white')
︡031c4291-3d2d-4b0a-a331-dadedf8c4b24︡︡
︠721c711f-b953-4b8f-86be-67f63abe9e84i︠
%html
<h2>Visualisierung</h2>

<h2></h2>

︡4229226e-6818-4e6c-99f0-35227ece8121︡{"html": "<h2>Visualisierung</h2>\n\n<h2></h2>"}︡
︠0970540d-7b49-4027-b06b-f4ae83d0e0a0︠
s_p=parametric_plot3d(s,(phi,-1,1),rgbcolor='red',thickness=3)
︡bdb902e3-b903-46a9-bf32-29a847e7fdd2︡︡
︠b86c93be-44d4-4a8f-ba12-1a62fc073688︠
a=plot3d(2*x*y/(x^2+y^2),(x,-1,1),(y,-1,1),rgbcolor='red')
b=parametric_plot3d(S,(phi,0,2*pi),(v,-1,1))
︡3944c46b-44fc-42f9-b1c8-8d6fb373eb99︡︡
︠97964665-c68c-449a-9c3b-4a731cb9b398︠
e1=parametric_plot3d(S.substitute(phi=0),(v,-1,1),thickness=3,rgbcolor='green')
e2=parametric_plot3d(S.substitute(phi=pi/8),(v,-1,1),thickness=3,rgbcolor='green')
e3=parametric_plot3d(S.substitute(phi=2*pi/8),(v,-1,1),thickness=3,rgbcolor='yellow')
e4=parametric_plot3d(S.substitute(phi=3*pi/8),(v,-1,1),thickness=3,rgbcolor='green')
e5=parametric_plot3d(S.substitute(phi=4*pi/8),(v,-1,1),thickness=3,rgbcolor='green')
e6=parametric_plot3d(S.substitute(phi=5*pi/8),(v,-1,1),thickness=3,rgbcolor='green')
e7=parametric_plot3d(S.substitute(phi=6*pi/8),(v,-1,1),thickness=3,rgbcolor='yellow')
e8=parametric_plot3d(S.substitute(phi=7*pi/8),(v,-1,1),thickness=3,rgbcolor='green')
︡96bc89d4-c0a1-49d5-889f-5fb04ac84f7c︡︡
︠314e059a-7adb-41a4-856c-b253fd2a494f︠
a=show(b+s_p+e1+e2+e3+e4+e5+e6+e7+e8,figsize=6,viewer='tachyon')
︡55ac6f09-b090-45db-81b2-483f4d5b5b1b︡︡