Throughout the whole calendar we tried to show animations which explain some algebraic or geometric concept or proof.
This last one is relatively useless. It just shows a nice surface which looks like a product of planes when seen from far away...
Showing posts with label Oliver Labs. Show all posts
Showing posts with label Oliver Labs. Show all posts
Friday, 5 August 2011
Polar Curves
The polar curve PC,p from a point p to a plane curve C of degree d is a plane curve of degree d-1 which intersects the original curve C exactly in those points ti for which the line through p and ti is a tangent to C through p.
In our example, you see a one parameter family of projective cubic plane curves (black) in the spherical view, together with its polar (white) from the green point, and the (red) tangents from the point to the black curve.
When a tangent at a flex point of the (black) curve passes through the green point, or when an ordinary double point develops, one can see that two of the intersection points of the polar (white) with the curve (black) coincide. The same thus also holds for two red tangents from the green point to the black curve. Moreover, when a cusp singularity develops, four of the intersection points and thus four red tangents coincide.
The point in which the two white great circles intersect, i.e. the singular point of the polar, is a flex point of the (black) curve. The (red) tangent at that point passes through the green point throughout the whole animation.
This film was made by Oliver Labs using Singular and surfex.
In our example, you see a one parameter family of projective cubic plane curves (black) in the spherical view, together with its polar (white) from the green point, and the (red) tangents from the point to the black curve.
When a tangent at a flex point of the (black) curve passes through the green point, or when an ordinary double point develops, one can see that two of the intersection points of the polar (white) with the curve (black) coincide. The same thus also holds for two red tangents from the green point to the black curve. Moreover, when a cusp singularity develops, four of the intersection points and thus four red tangents coincide.
The point in which the two white great circles intersect, i.e. the singular point of the polar, is a flex point of the (black) curve. The (red) tangent at that point passes through the green point throughout the whole animation.
This film was made by Oliver Labs using Singular and surfex.
Blowup in a Point
Today, we present the standard picture which appears in any algebraic geometry text book in the form of an animation: the blowup of the affine plane in the origin. Over each point of the plane there is a unique point in the blowup except for the origin where we have a whole line, called the exceptional line (green). The points on the exceptional line correspond to tangent directions of the affine plane in the origin. Since each lines through the origin passes it in a different direction, the corresponding lines on the blowup do not intersect.
This also allows us to find a smooth curve (blue) on the blowup that lies over the singular blue curve in the plane. The singular point of the plane curve has two preimages since the curve passes through the origin in two different directions (white).
If C is any singular curve lying on a smooth surface it is a classical theorem, that one can find a smooth curve D mapping to C, by iterating this process.
This film was made by Hans-Christian v. Bothmer and Oliver Labs using surfex.
This also allows us to find a smooth curve (blue) on the blowup that lies over the singular blue curve in the plane. The singular point of the plane curve has two preimages since the curve passes through the origin in two different directions (white).
If C is any singular curve lying on a smooth surface it is a classical theorem, that one can find a smooth curve D mapping to C, by iterating this process.
This film was made by Hans-Christian v. Bothmer and Oliver Labs using surfex.
Dual Curves
At the left, you see a projective cubic plane curve (black), together with its tangents (white). At the right, you see its dual curve (red) which consists of those points in the dual projective plane which correspond to the tangent at a given point of the original curve, together with the tangent at that point (green).
At the node (the double point) of the black curve, one seees that there is not a unique tangent at that point. In the dual curve, this is reflected by the fact that the tangent at the corresponding point touches it in two points.
Similarly, once the point on the original curve in which we draw the tangent passes the flex point, the tangent changes its direction. This is reflected on the dual curve by the fact that the curve changes its direction, i.e. it has a cusp singularity.
This film was made by Oliver Labs using surfex.
At the node (the double point) of the black curve, one seees that there is not a unique tangent at that point. In the dual curve, this is reflected by the fact that the tangent at the corresponding point touches it in two points.
Similarly, once the point on the original curve in which we draw the tangent passes the flex point, the tangent changes its direction. This is reflected on the dual curve by the fact that the curve changes its direction, i.e. it has a cusp singularity.
This film was made by Oliver Labs using surfex.
A Monoid with Many Singularities
A monoid surface M is a surface of degree d with only isolated singularities and which has a singularity of multiplicity d-1. If we place this singularity at the origin O, then M has an affine equation of the form: M = f_d + f_{d-1}, where the f_i are homogeneous polynomials of degree i.
It is clear that monoid surfaces are very special. E.g., it is easy to see that any monoid surface is rational, which makes it potentially useful for applications in geometric design.
Moreover, the very difficult question on the maximum number μ(d) of singularities on a surface of degree d which was mentioned in door no. 06 can be answered completely for monoid surfaces: μ(d)=d*(d-1)/2+1. This result was known to Rohn around 1900 at least for the case of quartics. In general, we first found it in a paper of P.H. Johansen, M. Loberg, R. Piene (2006, to appear in the Compass II Proceedings) on (real) monoid hypersurfaces.
The proof for one implication is not difficult and can be found in the paper mentioned above. First, one notes that a point (p0:p1:p2:p3) other than O is singular if and only if the plane curves fd and fd-1 intersect with multiplicity at least 2 at (p1:p2:p3). By Bezout's theorem, this can only happen at d*(d-1)/2 points.
Conversely, the authors of the paper give the construction which our film illustrates to show that this number is attained. In the film, f_5 is the black curve which is a deformation of a regular pentagon, and f_4 is the product of the two cirles. In the film one can see that the 10 singularities occur when the deformed pentagon touches the two circles in 10 points.
This film was made by Oliver Labs using surfex.
It is clear that monoid surfaces are very special. E.g., it is easy to see that any monoid surface is rational, which makes it potentially useful for applications in geometric design.
Moreover, the very difficult question on the maximum number μ(d) of singularities on a surface of degree d which was mentioned in door no. 06 can be answered completely for monoid surfaces: μ(d)=d*(d-1)/2+1. This result was known to Rohn around 1900 at least for the case of quartics. In general, we first found it in a paper of P.H. Johansen, M. Loberg, R. Piene (2006, to appear in the Compass II Proceedings) on (real) monoid hypersurfaces.
The proof for one implication is not difficult and can be found in the paper mentioned above. First, one notes that a point (p0:p1:p2:p3) other than O is singular if and only if the plane curves fd and fd-1 intersect with multiplicity at least 2 at (p1:p2:p3). By Bezout's theorem, this can only happen at d*(d-1)/2 points.
Conversely, the authors of the paper give the construction which our film illustrates to show that this number is attained. In the film, f_5 is the black curve which is a deformation of a regular pentagon, and f_4 is the product of the two cirles. In the film one can see that the 10 singularities occur when the deformed pentagon touches the two circles in 10 points.
This film was made by Oliver Labs using surfex.
The Swallowtail
Consider the univariate polynomial p(x)=x^4+ax^2+bx+c with parameters a,b,c (the cyan-colored curve in the plane which is located in the upper part of our film). It has a double root where both p(x) and its derivative p'(x) vanish. To determine for which a, b, c this happens, we eliminate the variable x from these two equations and obtain the so-called discriminant of p.
Many articles have been written about discriminants; here, we focus on the example p(x) above which is well-known and even appears in many text books. The discriminant s(a,b,c) which is a polynomial of degree 5 in the variables a, b, c, is called the swallowtail. This is the pink surface in our film.
Clearly, p(x) has a double root for those a, b, c for which s(a,b,c) vanishes. But s(a,b,c) holds much more information than that!
Depending on the position of the point (a,b,c) w.r.t. the swallowtail, we can see how many real roots the corresponding polynomial p(x) has and which multiplicity they have.
Of course, the point (0,0,0) corresponds to the case for which the polynomial p(x) has a 4-tuple root 0. On the swallowtail, this is the most singular point. For smooth points (a,b,c) on the swallowtail, the corresponding polynomial p(x) has exactly one double root and either 0 or 2 simple roots depending on the location of the smooth point. Where the left and the right smooth part intersect transversally (i.e. in the lower middle part of our film), this happens twice and p(x) has two distinct double roots. On the real isolated half-parabola, which can be seen in the part of the film where the swallowtail rotates, these two distinct double roots are both complex (and thus complex conjugate).
For points inside the triangular-shaped part of the swallowtail (i.e. in the center part of our film) p(x) has exactly four distinct real roots. On a smooth point of the part of the swallowtail which is the boundary of this triangular-shaped region p(x) has four real roots, two of which coincide. On the cuspidal edge of the triangular-shaped region, this happens twice but not symmetrically which means that p(x) has one tripled root and one simple root.
This film was made by Oliver Labs using surfex. We thank Frank-Olaf Schreyer for useful comments.
Many articles have been written about discriminants; here, we focus on the example p(x) above which is well-known and even appears in many text books. The discriminant s(a,b,c) which is a polynomial of degree 5 in the variables a, b, c, is called the swallowtail. This is the pink surface in our film.
Clearly, p(x) has a double root for those a, b, c for which s(a,b,c) vanishes. But s(a,b,c) holds much more information than that!
Depending on the position of the point (a,b,c) w.r.t. the swallowtail, we can see how many real roots the corresponding polynomial p(x) has and which multiplicity they have.
Of course, the point (0,0,0) corresponds to the case for which the polynomial p(x) has a 4-tuple root 0. On the swallowtail, this is the most singular point. For smooth points (a,b,c) on the swallowtail, the corresponding polynomial p(x) has exactly one double root and either 0 or 2 simple roots depending on the location of the smooth point. Where the left and the right smooth part intersect transversally (i.e. in the lower middle part of our film), this happens twice and p(x) has two distinct double roots. On the real isolated half-parabola, which can be seen in the part of the film where the swallowtail rotates, these two distinct double roots are both complex (and thus complex conjugate).
For points inside the triangular-shaped part of the swallowtail (i.e. in the center part of our film) p(x) has exactly four distinct real roots. On a smooth point of the part of the swallowtail which is the boundary of this triangular-shaped region p(x) has four real roots, two of which coincide. On the cuspidal edge of the triangular-shaped region, this happens twice but not symmetrically which means that p(x) has one tripled root and one simple root.
This film was made by Oliver Labs using surfex. We thank Frank-Olaf Schreyer for useful comments.
Lines on Singular Cubics
A smooth (complex) cubic surface contains exactly 27 distinct lines. On a real cubic surface, all these 27 lines can be real. When singularities occur, some of the lines fall upon each other. E.g., on a cubic surface with three cusp singularities, 3 times 9 fall together, so that a total number of 3 distinct lines remains.
The film shows what happens when deforming this three-cuspidal surface in four different ways. The color of the lines reflects their multiplicities: orange is 9, magenta is 6, green is 3, red is 2.
More such movies can be found on our website http://www.cubics.algebraicsurface.net/
The film shows what happens when deforming this three-cuspidal surface in four different ways. The color of the lines reflects their multiplicities: orange is 9, magenta is 6, green is 3, red is 2.
More such movies can be found on our website http://www.cubics.algebraicsurface.net/
Pinkham's Deformations
The movie shows the projection into three-space of deformations of the cone over the rational normal curve of degree four, which is embedded in five-space. This singularity can be deformed in two essentially distinct ways, as was discovered by H. Pinkham. The projection and the properties of the two projected deformations were discovered by R. Pellikaan and T. de Jong.
This film was made by Duco van Straten and Oliver Labs using surfex.
This film was made by Duco van Straten and Oliver Labs using surfex.
Vanishing Cycles
The movie shows a generic deformation of the A4-Singularity.
One can clearly see the chain consisting of four cycles that
is contracted in the singularity: the vanishing cycles.
Note that these four cycles are the real part of a chain
of four two-spheres in the complex surface.
This film was made by Duco van Straten and Oliver Labs using surfex.
One can clearly see the chain consisting of four cycles that
is contracted in the singularity: the vanishing cycles.
Note that these four cycles are the real part of a chain
of four two-spheres in the complex surface.
This film was made by Duco van Straten and Oliver Labs using surfex.
Cubic Surfaces as Blowup
A classical result in algebraic geometry, probably first shown by A. Clebsch in the 19th century, says that any smooth complex cubic surface arises as the blowup (see no. 09 of our advent calendar) of the plane in six distinct points. A. Clebsch also already explained that the cubic surface admits a singularity if three of the points are on a common line or if all six are on a common conic; otherwise the six points are called points in general position. Our film starts with the smooth cubic surface corresponding to the six points in pentagon-symmetric position shown in the leftmost image below (the so-called Clebsch Cubic) and ends with the one with 4 singularities (the so-called Cayley Cubic).
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