Plane Sections of Real and Complex ToriorWhy the Graph of is a Torus()221yxx=−Based on a presentation by David Sklar and Bruce Cohenat Asilomar in December 2004Sonoma State - February 2006Part I - Slicing a Real Circular Torus The Spiric Sections of PerseusOvals of Cassini and The Lemniscate of BernoulliEquations for the torus in R3Other SlicesThe Villarceau CirclesA Characterization of the torusThe Spiric Sections of Perseus:The sectioning planes are parallel to the axis of rotationMore Spiric SectionsEquations of a Circular Torusϕθabxyz(),,xyzParametric equations:( cos )cosxabθϕ=+(cos)sinyabθϕ=+sinzbθ=Cartesian equations:()()()( )()2 2222 22222 222 2222 0xyz abxyz abz ab++ − + ++ − − + − =Note: we can get a cartesian equation for a spiric section by setting y equal to a constant. In general the left hand side equation will be an irreducible fourth degree polynomial, but for y = 0, it factors.()()()()()2 222 2222 222 2222 0xz abxz abz ab+− + +− − +−=() ()2222 220xa zb xa zb⎡⎤⎡⎤++− − +− =⎣⎦⎣⎦()()()( )()2 2222 22222 222 2222 0 with 1.2, 0.4, and 1.6xyz abxyz abz abab y++ − + ++ − − + − === ≤xzyzxy=xy=xyzxy=xzyxy=xzyxy=xzyxy=xzySections with planes rotating about the x-axisθ21More sections with planes rotating about the x-axisVillarceau circlesA Characterization of the TorusA complete, sufficiently smooth surface with the property that through each point on the surface there exist exactly four distinct circles (that lie on the surface) is a circular torus.Villarceau Circles about 500 years before his 1848 paperPart II - Slicing a Complex Torus22(1)yxx c=−+Some graphs ofHints of toric sectionsTwo closures: Algebraic and GeometricGeometric closure, Projective spacesP1(R), P2(R), P1(C), and P2(C)22(1)yxx=−Algebraic closure, C2, R4, and the graph of The graphs of22(1),yxx=−222(),yxxn=−and 2222222(1)(2)( )yxx x xg=− − −"BibliographyGraphs of 22(1)yxx c=−+: Hints of Toric Sectionsxyxy22(1)yxx=−If we close up the geometry to include points at infinity and the algebra to include the complex numbers, we can argue that the graph of is a torus.22(1)yxx=−22(1)0.3yxx=−+22(1)1yxx=−+22(1)0.385yxx=−+xyxyGeometric Closure: an Introduction to Projective GeometryPart I – Real Projective GeometryOne-Dimension - the Real Projective Line P1(R)The real (affine) line R is theordinary real number lineThe real projective line P1(R) isthe set {} ∪∞R0It is topologically equivalent to the open interval (-1, 1) by the map(1 )xx x+601− 1and topologically equivalent to a punctured circle by stereographic projection0It is topologically equivalent to a closed interval with the endpoints identified0PP0and topologically equivalent to a circle by stereographic projection∞∞∞∞0P∞Geometric Closure: an Introduction to Projective GeometryPart I – Real Projective GeometryTwo-Dimensions - the Real Projective Plane P2(R)The real (affine) plane R2isthe ordinary x, y -planeIt is topologically equivalent to a closed disk with antipodal points on the boundary circle identified.xy22 22 (, ) ,11xyxyxyxy++ ++6It is topologically equivalent to the open unit disk by the map( )xyxyThe real projective plane P2(R) is the set . It is R2together with a “line at infinity”, . Every line in R2intersects , parallel lines meet at the same point on , and nonparallel lines intersect at distinct points.2 L∞∪RL∞L∞L∞L∞Two distinct lines intersect at one and only one point.Every line in P2(R) is a P1(R).A Projective View of the ConicsxyEllipsexyxyParabolaxyxyHyperbolaA Projective View of the ConicsParabolaEllipseHyperbolaGraphs of 22(1)yxx c=−+: Hints of Toric Sections22(1)yxx=−22(1)0.3yxx=−+22(1)1yxx=−+22(1)0.385yxx=−+including point topological view at infinityxyxyxyxyIf we close up the algebra, by extending to the , and the geometry, complex numbers22by including points at infinity we can argue that the graph of ( 1) is a torus.yxx=−Graph of with x and y complexAlgebraic closure22(1)yxx=−12 12Let and x x ix y y iy=+ =+22 2 212 1212Then ( 1) becomes ( ) ( )[( ) 1]y x x y iy x ix x ix=− +=+ +−22 3 2 3 212 12 1112 2212()(2)( 3)( 3)yyiyyxxxx ixx xx−+ =−− +−−+Expanding and collecting terms we have Equating real and imaginery parts we have 223 2 3 2121112 12 22123 and 2 3yyxxxxyyxx xx−=−− =−−+is a system of two equations in four real 22 3 212 11123yyxxxx−= −−unknowns whose graph is a 2-dimensional surface in real 4-dimensional space 3212 2 2 1 223yyxx xx=− − + It's not so easy to graph a 2-surface in 4-space, but we can look at intersections of the graph with some convenient planes.Graph of with x and y complexAlgebraic closure22(1)yxx=−22 3 212 11123yyxxxx−= −−3212 2 2 1 223yyxx xx=− − +Some comments on why the graph of the systemis a surface.12 12Letting and , then solving for and in terms of and ,xs xt y y s t==1211 22we would essentially have , , ( , ) and ( , )xsxtyyst y yst=== =12 1 2These are parametric equations for a surface in , , , spacexx yy1212(a nice mapping of a 2-D , plane into 4-D , , , space.)st x x y y12for ( , ) and ( , ) which can be pieced together to get the whole graph.yst ystThe situation is a little more complicated in that the algebra leads to several solutionsGraph of with x and y complexAlgebraic closure22(1)yxx=−22 3 212 11123yyxxxx−= −−3212 2 2 1 223yyxx xx=− − +22for 0, 0xy==11(the , - plane)xy22( 1) becomesyxx=−21for 0, 0xy==22211(1)yxx=− −12(the , - plane)xy22( 1) becomesyxx=−211( )[( ) 1]xx=−−−22111(1)yxx=−1x2y1x1yGraph of with x and y complex22(1)yxx=−22 2Recall, the graph of ( 1) in is equivalent to the graph of the systemyxx=− C223 21211123212 2 2 1 2323yyxxxxyyxx xx⎧−=−−⎨=− − +⎩Now lets look at the intersection of4in .R2the graph with the 3-space 0.x=1x2y1y22312111220yyxxyy⎧−=−⎨=⎩The system of equations becomes21so 0 or 0yy==11 12and the intersection (a curve) lies in only the , - plane or the , - plane.xy xy2for 0,y =22111(1)yxx=−1for 0,y =22211(1)yxx=−−1x2y1x1yGraph of with x and y complex22(1)yxx=−1x1y1x2y211 12The intersection of the graph with the 3-space 0