Real-Time Interactive Visualization of Deformations of Singularities - - PowerPoint PPT Presentation

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Real-Time Interactive Visualization of Deformations of Singularities

Oliver Labs

Universit¨ at des Saarlandes, Germany E-Mail: Labs@Math.Uni-Sb.de, mail@OliverLabs.net

• Sept. 3, 2006, ICMS ’06 at Castro Urdiales, Spain

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularitie

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Surf and Surfaces (I)

Definition

A real algebraic surface of degree d in ❘3 is (the zero-set of) a polynomial in three variables of degree d: xd + a1xd−1y + a2xd−1z + a3xd−1 + ... + ak.

◮ SURF was one of the first (or the first?) tools to visualize

algebraic surfaces which are specified by their defining polynomial.

◮ To my knowledge, it is still the quickest one. And it can

draw curves on surfaces. Drawback: very non-intuitive usage; problems with lower-dimensional parts; limited number of surfaces, . . .

◮ Other tools: POVRAY (raytracing), LSMP (triangulation).

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Surf and Surfaces (I)

Definition

A real algebraic surface of degree d in ❘3 is (the zero-set of) a polynomial in three variables of degree d: xd + a1xd−1y + a2xd−1z + a3xd−1 + ... + ak.

◮ SURF was one of the first (or the first?) tools to visualize

algebraic surfaces which are specified by their defining polynomial.

◮ To my knowledge, it is still the quickest one. And it can

draw curves on surfaces. Drawback: very non-intuitive usage; problems with lower-dimensional parts; limited number of surfaces, . . .

◮ Other tools: POVRAY (raytracing), LSMP (triangulation).

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Surf and Surfaces (I)

Definition

A real algebraic surface of degree d in ❘3 is (the zero-set of) a polynomial in three variables of degree d: xd + a1xd−1y + a2xd−1z + a3xd−1 + ... + ak.

◮ SURF was one of the first (or the first?) tools to visualize

algebraic surfaces which are specified by their defining polynomial.

◮ To my knowledge, it is still the quickest one. And it can

draw curves on surfaces. Drawback: very non-intuitive usage; problems with lower-dimensional parts; limited number of surfaces, . . .

◮ Other tools: POVRAY (raytracing), LSMP (triangulation).

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Surf and Surfaces (I)

Definition

A real algebraic surface of degree d in ❘3 is (the zero-set of) a polynomial in three variables of degree d: xd + a1xd−1y + a2xd−1z + a3xd−1 + ... + ak.

◮ SURF was one of the first (or the first?) tools to visualize

algebraic surfaces which are specified by their defining polynomial.

◮ To my knowledge, it is still the quickest one. And it can

draw curves on surfaces. Drawback: very non-intuitive usage; problems with lower-dimensional parts; limited number of surfaces, . . .

◮ Other tools: POVRAY (raytracing), LSMP (triangulation).

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Surf and Surfaces (II)

◮ SURF: main draw-back: the non-intuitive usage and

rotation.

◮ SURFEX: an extension of SURF which allows the users to

produce nice pictures and movies easily.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Surf and Surfaces (II)

◮ SURF: main draw-back: the non-intuitive usage and

rotation.

◮ SURFEX: an extension of SURF which allows the users to

produce nice pictures and movies easily. A surface of degree 7 with 99 real singularities.

• Constructed using computer alge-

bra (SINGULAR), see my article on arxiv.org: math.AG/0409348 (2004)

• Visualized using SURF via SUR-

FEX.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Surf and Surfaces

SURFEX in action

Singularities Constructing Cubic Surfaces Projective Plane Curves Some Visualization Issues Plans for the Future Conclusion

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularitie

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Singularities

Definition

A singularity of a surface S in ❘3 is a point p ∈ S, s.t. all the partial derivatives vanish:

∂S ∂x (p) = ∂S ∂y (p) = ∂S ∂z (p) = 0.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Singularities

(a): node (A−

1 )

x2 − y2 + z2 = 0

Definition

A singularity of a surface S in ❘3 is a point p ∈ S, s.t. all the partial derivatives vanish:

∂S ∂x (p) = ∂S ∂y (p) = ∂S ∂z (p) = 0.

Example

◮ nodes (A1) ◮ cusps (A2)

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Singularities

(a): node (A−

1 )

x2 − y2 + z2 = 0

(b): node (solitary, A+

1 )

x2 + y2 + z2 = 0

Definition

A singularity of a surface S in ❘3 is a point p ∈ S, s.t. all the partial derivatives vanish:

∂S ∂x (p) = ∂S ∂y (p) = ∂S ∂z (p) = 0.

Example

◮ nodes (A1) ◮ cusps (A2)

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Singularities

(a): node (A−

1 )

x2 − y2 + z2 = 0

(b): node (solitary, A+

1 )

x2 + y2 + z2 = 0

(c): cusp (A−

2 )

x3 − y2 + z2 = 0

Definition

A singularity of a surface S in ❘3 is a point p ∈ S, s.t. all the partial derivatives vanish:

∂S ∂x (p) = ∂S ∂y (p) = ∂S ∂z (p) = 0.

Example

◮ nodes (A1) ◮ cusps (A2)

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Singularities

(a): node (A−

1 )

x2 − y2 + z2 = 0

(b): node (solitary, A+

1 )

x2 + y2 + z2 = 0

(c): cusp (A−

2 )

x3 − y2 + z2 = 0

(d): cusp (A+

2

x3 + y 2 + z2 = 0

Definition

A singularity of a surface S in ❘3 is a point p ∈ S, s.t. all the partial derivatives vanish:

∂S ∂x (p) = ∂S ∂y (p) = ∂S ∂z (p) = 0.

Example

◮ nodes (A1) ◮ cusps (A2)

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Simple Singularities on Cubic Surfaces

Name Old Name Normal Form Coxeter Diagram µ❘ ν A−

2k

B2k+1 x2k+1 + y2 − z2 , 2k k = 1, 2 A+

2k

B2k+1 x2k+1 + y2 + z2 k − 1 k = 1 A−

2k−1 B2k

x2k + y 2 − z2 , 2k − 1 k = 2, 3 A+

2k−1 B2k

x2k − y 2 − z2 1 k − 1 k = 2 A−

1

C2 x2 + y2 − z2 1 A

1

C2 x2 + y2 + z2 1 D−

4

U6 x2y − y 3 − z2 4 D+

4

U6 x2y + y 3 + z2 2 1 D−

5

U7 x2y + y 4 − z2 5 E−

6

U8 x3 + y4 − z2 6 Multiplicity: degree of the lowest order term. Tangent Cone: homogeneous part of the lowest order. The old names come from the tangent cone:

conical or bi-/uni-planar . Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Simple Singularities on Cubic Surfaces

Name Old Name Normal Form Coxeter Diagram µ❘ ν A−

2k

B2k+1 x2k+1 + y2 − z2 , 2k k = 1, 2 A+

2k

B2k+1 x2k+1 + y2 + z2 k − 1 k = 1 A−

2k−1 B2k

x2k + y 2 − z2 , 2k − 1 k = 2, 3 A+

2k−1 B2k

x2k − y 2 − z2 1 k − 1 k = 2 A−

1

C2 x2 + y2 − z2 1 A

1

C2 x2 + y2 + z2 1 D−

4

U6 x2y − y 3 − z2 4 D+

4

U6 x2y + y 3 + z2 2 1 D−

5

U7 x2y + y 4 − z2 5 E−

6

U8 x3 + y4 − z2 6 Multiplicity: degree of the lowest order term. Tangent Cone: homogeneous part of the lowest order. The old names come from the tangent cone:

conical or bi-/uni-planar . Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Simple Singularities on Cubic Surfaces

Name Old Name Normal Form Coxeter Diagram µ❘ ν A−

2k

B2k+1 x2k+1 + y2 − z2 , 2k k = 1, 2 A+

2k

B2k+1 x2k+1 + y2 + z2 k − 1 k = 1 A−

2k−1 B2k

x2k + y 2 − z2 , 2k − 1 k = 2, 3 A+

2k−1 B2k

x2k − y 2 − z2 1 k − 1 k = 2 A−

1

C2 x2 + y2 − z2 1 A

1

C2 x2 + y2 + z2 1 D−

4

U6 x2y − y 3 − z2 4 D+

4

U6 x2y + y 3 + z2 2 1 D−

5

U7 x2y + y 4 − z2 5 E−

6

U8 x3 + y4 − z2 6 Multiplicity: degree of the lowest order term. Tangent Cone: homogeneous part of the lowest order. The old names come from the tangent cone:

conical or bi-/uni-planar . Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Simple Singularities on Cubic Surfaces

Name Old Name Normal Form Coxeter Diagram µ❘ ν A−

2k

B2k+1 x2k+1 + y2 − z2 , 2k k = 1, 2 A+

2k

B2k+1 x2k+1 + y2 + z2 k − 1 k = 1 A−

2k−1 B2k

x2k + y 2 − z2 , 2k − 1 k = 2, 3 A+

2k−1 B2k

x2k − y 2 − z2 1 k − 1 k = 2 A−

1

C2 x2 + y2 − z2 1 A

1

C2 x2 + y2 + z2 1 D−

4

U6 x2y − y 3 − z2 4 D+

4

U6 x2y + y 3 + z2 2 1 D−

5

U7 x2y + y 4 − z2 5 E−

6

U8 x3 + y4 − z2 6 Multiplicity: degree of the lowest order term. Tangent Cone: homogeneous part of the lowest order. The old names come from the tangent cone:

conical or bi-/uni-planar . Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Classification of Singularities using SINGULAR

CLASSIFY.LIB can classify many types of isolated hypersurface

singularities. If we want to find out all the types of all isolated singularities of a given surface, we have to:

◮ compute all coordinates of the isolated singularities (via

(zero-dimensional) primary decomposition),

◮ translate each singularity to the origin

◮ switch to a local ring ◮ user CLASSIFY.LIB to classify the singularity

◮ make a summary of the results

Example

classify: a cubic surface classify: a nonic surfex: a nonic Oliver Labs Real-Time Interactive Visualization of Deformations of Singularitie

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Classification of Singularities using SINGULAR

CLASSIFY.LIB can classify many types of isolated hypersurface

singularities. If we want to find out all the types of all isolated singularities of a given surface, we have to:

◮ compute all coordinates of the isolated singularities (via

(zero-dimensional) primary decomposition),

◮ translate each singularity to the origin

◮ switch to a local ring ◮ user CLASSIFY.LIB to classify the singularity

◮ make a summary of the results

Example

classify: a cubic surface classify: a nonic surfex: a nonic Oliver Labs Real-Time Interactive Visualization of Deformations of Singularitie

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Classification of Singularities using SINGULAR

CLASSIFY.LIB can classify many types of isolated hypersurface

singularities. If we want to find out all the types of all isolated singularities of a given surface, we have to:

◮ compute all coordinates of the isolated singularities (via

(zero-dimensional) primary decomposition),

◮ translate each singularity to the origin

◮ switch to a local ring ◮ user CLASSIFY.LIB to classify the singularity

◮ make a summary of the results

Example

classify: a cubic surface classify: a nonic surfex: a nonic Oliver Labs Real-Time Interactive Visualization of Deformations of Singularitie

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Classification of Singularities using SINGULAR

CLASSIFY.LIB can classify many types of isolated hypersurface

singularities. If we want to find out all the types of all isolated singularities of a given surface, we have to:

◮ compute all coordinates of the isolated singularities (via

(zero-dimensional) primary decomposition),

◮ translate each singularity to the origin

◮ switch to a local ring ◮ user CLASSIFY.LIB to classify the singularity

◮ make a summary of the results

Example

classify: a cubic surface classify: a nonic surfex: a nonic Oliver Labs Real-Time Interactive Visualization of Deformations of Singularitie

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Classification of Singularities using SINGULAR

CLASSIFY.LIB can classify many types of isolated hypersurface

singularities. If we want to find out all the types of all isolated singularities of a given surface, we have to:

◮ compute all coordinates of the isolated singularities (via

(zero-dimensional) primary decomposition),

◮ translate each singularity to the origin

◮ switch to a local ring ◮ user CLASSIFY.LIB to classify the singularity

◮ make a summary of the results

Example

classify: a cubic surface classify: a nonic surfex: a nonic Oliver Labs Real-Time Interactive Visualization of Deformations of Singularitie

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Classification of Singularities using SINGULAR

CLASSIFY.LIB can classify many types of isolated hypersurface

singularities. If we want to find out all the types of all isolated singularities of a given surface, we have to:

◮ compute all coordinates of the isolated singularities (via

(zero-dimensional) primary decomposition),

◮ translate each singularity to the origin

◮ switch to a local ring ◮ user CLASSIFY.LIB to classify the singularity

◮ make a summary of the results

Example

classify: a cubic surface classify: a nonic surfex: a nonic Oliver Labs Real-Time Interactive Visualization of Deformations of Singularitie

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Classification of Singularities using SINGULAR

CLASSIFY.LIB can classify many types of isolated hypersurface

singularities. If we want to find out all the types of all isolated singularities of a given surface, we have to:

◮ compute all coordinates of the isolated singularities (via

(zero-dimensional) primary decomposition),

◮ translate each singularity to the origin

◮ switch to a local ring ◮ user CLASSIFY.LIB to classify the singularity

◮ make a summary of the results

Example

classify: a cubic surface classify: a nonic surfex: a nonic Oliver Labs Real-Time Interactive Visualization of Deformations of Singularitie

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Classification of Singularities using SINGULAR

CLASSIFY.LIB can classify many types of isolated hypersurface

singularities. If we want to find out all the types of all isolated singularities of a given surface, we have to:

◮ compute all coordinates of the isolated singularities (via

(zero-dimensional) primary decomposition),

◮ translate each singularity to the origin

◮ switch to a local ring ◮ user CLASSIFY.LIB to classify the singularity

◮ make a summary of the results

Example

classify: a cubic surface classify: a nonic surfex: a nonic Oliver Labs Real-Time Interactive Visualization of Deformations of Singularitie

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Cubic Surfaces

Classically studied, there are exactly 45 different topological types of cubic surfaces with only simple singularities:

◮ cubic surface homepage:

www.cubics.algebraicsurface.net :

visualizations of all 45 topological types, background information.

◮ xcsprg: real cubic surfaces

which arise as the blowup of the plane in six distinct real points.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Cubic Surfaces

Classically studied, there are exactly 45 different topological types of cubic surfaces with only simple singularities:

◮ cubic surface homepage:

www.cubics.algebraicsurface.net :

visualizations of all 45 topological types, background information.

◮ xcsprg: real cubic surfaces

which arise as the blowup of the plane in six distinct real points.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Cubic Surfaces

Classically studied, there are exactly 45 different topological types of cubic surfaces with only simple singularities:

◮ cubic surface homepage:

www.cubics.algebraicsurface.net :

visualizations of all 45 topological types, background information.

◮ xcsprg: real cubic surfaces

which arise as the blowup of the plane in six distinct real points.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Constructing one Nice Example for Every Topological Type of Cubic Surfaces

There are at least three methods of construction:

◮ Via known projective equations. ◮ By direct Construction. ◮ By deforming known examples.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Constructing one Nice Example for Every Topological Type of Cubic Surfaces

There are at least three methods of construction:

◮ Via known projective equations. ◮ By direct Construction. ◮ By deforming known examples.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Constructing one Nice Example for Every Topological Type of Cubic Surfaces

There are at least three methods of construction:

◮ Via known projective equations. ◮ By direct Construction. ◮ By deforming known examples.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Constructing one Nice Example for Every Topological Type of Cubic Surfaces

There are at least three methods of construction:

◮ Via known projective equations. ◮ By direct Construction. ◮ By deforming known examples.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Via known projective equations

For the projective case, Schl¨ afli already gave equations in 1863, see also Cayley (1869).

Example (1A5 and 1A1)

f = wxz + y2z + x3 = 0. The choice w = z − 1 gives our affine equation

KM44 . Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

By direct Construction

. . . using symmetry and local equations of singularities.

Example (3A2)

Locally, an A2-singularity has the form: x2 − y2 + z3 = 0. x2 − y2 are two intersecting lines in the x, y-plane. To get three A2-singularities, we can thus take the union of three real lines which do not meet in one common point, e.g.: pl := (x − 1)(y − 1)(x + y). Then, pl + z3 has three A−

2 -singularities:

+ z3 =

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

By Deforming known Examples

This method was essentially already known to Klein in the 19th century: We can add g ∈ (x, y, z)2 · ( ∂f

∂x , ∂f ∂y , ∂f ∂z ) to the equation f of a

singularity at the origin without changing its topological type.

Example (A2-singularity)

f = x2 − y2 + z3. Thus, ( ∂f

∂x , ∂f ∂y , ∂f ∂z ) = (x, y, z2).

Thus, f + z2 · x also defines an A2-singularity at the origin

→ . → → Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion Singularities Constructing Cubic Surfaces Projective Plane Curves

Projective Plane Curves

Using the feature of drawing intersection curves of surfaces, we can also visualize projective plane curves on a sphere.

◮ projective conics, ◮ projective fermat cubic, ◮ a cusp singularity.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Non-Reduced Varieties

LSMP

SURF or SURFEX SURFEX via SURFEX.LIB

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Non-Reduced Varieties

LSMP

SURF or SURFEX SURFEX via SURFEX.LIB

SURFEX.LIB can first compute the radical of a variety to

circumvent such problems: poly nonred = (xˆ2+yˆ2-zˆ2-3ˆ2)ˆ2; plotRotated(nonred, list(x,y,z), 3);

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Real Lower-Dimensional Parts (I)

LSMP

SURF or SURFEX SURFEX via SURFEX.LIB

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Real Lower-Dimensional Parts (I)

LSMP

SURF or SURFEX SURFEX via SURFEX.LIB

SURFEX.LIB can first compute compute the singular locus which

then enables SURFEX to draw the real lower-dimensional part of surfaces.

real one dimensional.sin

In simple cases, LSMP can do this without computer algebra, but it already fails with the swallowtail.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Real Lower-Dimensional Parts (II)

SURF or SURFEX SURFEX via SURFEX.LIB

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Real Lower-Dimensional Parts (II)

SURF or SURFEX SURFEX via SURFEX.LIB

SURFEX.LIB can first compute compute the singular locus which

then enables SURFEX to draw the real lower-dimensional part of surfaces.

cubic A1p.sin

In simple cases, LSMP can do this without computer algebra.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Plans for the Future

◮ Original Goal: SPICY, space and plane interactive

constructive and algebraic geometry

Dandelin , ◮ idea taken from: the cubic surface program xcsprg (a

project that I developed during the writing of my diploma thesis, advisor: D. van Straten).

◮ too much work, so:

◮ implement an independant surface visualizer SURFEX

(already done!)

◮ connect it to existing constructive geometry tools, such as

Cinderella, etc.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Plans for the Future

◮ Original Goal: SPICY, space and plane interactive

constructive and algebraic geometry

Dandelin , ◮ idea taken from: the cubic surface program xcsprg (a

project that I developed during the writing of my diploma thesis, advisor: D. van Straten).

◮ too much work, so:

◮ implement an independant surface visualizer SURFEX

(already done!)

◮ connect it to existing constructive geometry tools, such as

Cinderella, etc.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Plans for the Future

◮ Original Goal: SPICY, space and plane interactive

constructive and algebraic geometry

Dandelin , ◮ idea taken from: the cubic surface program xcsprg (a

project that I developed during the writing of my diploma thesis, advisor: D. van Straten).

◮ too much work, so:

◮ implement an independant surface visualizer SURFEX

(already done!)

◮ connect it to existing constructive geometry tools, such as

Cinderella, etc.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Plans for the Future

◮ Original Goal: SPICY, space and plane interactive

constructive and algebraic geometry

Dandelin , ◮ idea taken from: the cubic surface program xcsprg (a

project that I developed during the writing of my diploma thesis, advisor: D. van Straten).

◮ too much work, so:

◮ implement an independant surface visualizer SURFEX

(already done!)

◮ connect it to existing constructive geometry tools, such as

Cinderella, etc.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Plans for the Future

◮ Original Goal: SPICY, space and plane interactive

constructive and algebraic geometry

Dandelin , ◮ idea taken from: the cubic surface program xcsprg (a

project that I developed during the writing of my diploma thesis, advisor: D. van Straten).

◮ too much work, so:

◮ implement an independant surface visualizer SURFEX

(already done!)

◮ connect it to existing constructive geometry tools, such as

Cinderella, etc.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Conclusion

◮ The cubic surface program xcsprg. ◮ The visualization software surfex as a tool for visualizing

real algebraic surfaces, including deformations of these.

◮ Nice real affine equations for all topological types of cubic

surfaces with isolated singularities can be produced easily using surfex.

◮ Future:

◮ connect SURFEX to constructive geometry tools in the style

• f SPICY,

◮ use computer algebra as a real-time-back-end for SURFEX /

constructive geometry software to compute additional data.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Conclusion

◮ The cubic surface program xcsprg. ◮ The visualization software surfex as a tool for visualizing

real algebraic surfaces, including deformations of these.

◮ Nice real affine equations for all topological types of cubic

surfaces with isolated singularities can be produced easily using surfex.

◮ Future:

◮ connect SURFEX to constructive geometry tools in the style

• f SPICY,

◮ use computer algebra as a real-time-back-end for SURFEX /

constructive geometry software to compute additional data.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Conclusion

◮ The cubic surface program xcsprg. ◮ The visualization software surfex as a tool for visualizing

real algebraic surfaces, including deformations of these.

◮ Nice real affine equations for all topological types of cubic

surfaces with isolated singularities can be produced easily using surfex.

◮ Future:

◮ connect SURFEX to constructive geometry tools in the style

• f SPICY,

◮ use computer algebra as a real-time-back-end for SURFEX /

constructive geometry software to compute additional data.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Conclusion

◮ The cubic surface program xcsprg. ◮ The visualization software surfex as a tool for visualizing

real algebraic surfaces, including deformations of these.

◮ Nice real affine equations for all topological types of cubic

surfaces with isolated singularities can be produced easily using surfex.

◮ Future:

◮ connect SURFEX to constructive geometry tools in the style

• f SPICY,

◮ use computer algebra as a real-time-back-end for SURFEX /

constructive geometry software to compute additional data.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces

SURFEX in action

Some Visualization Issues Plans for the Future Conclusion

Thank you

Thank you for your attention. Oliver Labs www.surfex.AlgebraicSurface.net www.OliverLabs.net www.AlgebraicSurface.net www.CubicSurface.net Unfortunately, the domains above are currently not available; it will hopefully take me only one or two weeks to make them

• nline again.

Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities