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: x d + a 1 x d − 1 y + a 2 x d − 1 z + a 3 x d − 1 + ... + a k . ◮ 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: P OV R AY (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: x d + a 1 x d − 1 y + a 2 x d − 1 z + a 3 x d − 1 + ... + a k . ◮ 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: P OV R AY (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: x d + a 1 x d − 1 y + a 2 x d − 1 z + a 3 x d − 1 + ... + a k . ◮ 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: P OV R AY (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: x d + a 1 x d − 1 y + a 2 x d − 1 z + a 3 x d − 1 + ... + a k . ◮ 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: P OV R AY (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 (S INGULAR ), 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 Singularities Some Visualization Issues Constructing Cubic Surfaces Plans for the Future Projective Plane Curves Conclusion 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 Singularities Some Visualization Issues Constructing Cubic Surfaces Plans for the Future Projective Plane Curves Conclusion 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 Singularities Some Visualization Issues Constructing Cubic Surfaces Plans for the Future Projective Plane Curves Conclusion Singularities (a): node ( A − 1 ) Definition x 2 − y 2 + z 2 = 0 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 (A 1 ) ◮ cusps (A 2 ) Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces SURFEX in action Singularities Some Visualization Issues Constructing Cubic Surfaces Plans for the Future Projective Plane Curves Conclusion Singularities (a): node ( A − 1 ) Definition x 2 − y 2 + z 2 = 0 A singularity of a surface S in ❘ 3 is a point p ∈ S, s.t. all (b): node (solitary, A + 1 ) the partial derivatives vanish: x 2 + y 2 + z 2 = 0 ∂ S ∂ x ( p ) = ∂ S ∂ y ( p ) = ∂ S ∂ z ( p ) = 0 . Example ◮ nodes (A 1 ) ◮ cusps (A 2 ) Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces SURFEX in action Singularities Some Visualization Issues Constructing Cubic Surfaces Plans for the Future Projective Plane Curves Conclusion Singularities (a): node ( A − 1 ) Definition x 2 − y 2 + z 2 = 0 A singularity of a surface S in ❘ 3 is a point p ∈ S, s.t. all (b): node (solitary, A + 1 ) the partial derivatives vanish: x 2 + y 2 + z 2 = 0 ∂ x ( p ) = ∂ S ∂ S ∂ y ( p ) = ∂ S ∂ z ( p ) = 0 . (c): cusp ( A − 2 ) Example x 3 − y 2 + z 2 = 0 ◮ nodes (A 1 ) ◮ cusps (A 2 ) Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces SURFEX in action Singularities Some Visualization Issues Constructing Cubic Surfaces Plans for the Future Projective Plane Curves Conclusion Singularities (a): node ( A − 1 ) Definition x 2 − y 2 + z 2 = 0 A singularity of a surface S in ❘ 3 is a point p ∈ S, s.t. all (b): node (solitary, A + 1 ) the partial derivatives vanish: x 2 + y 2 + z 2 = 0 ∂ x ( p ) = ∂ S ∂ S ∂ y ( p ) = ∂ S ∂ z ( p ) = 0 . (c): cusp ( A − 2 ) Example x 3 − y 2 + z 2 = 0 ◮ nodes (A 1 ) (d): cusp ( A + ◮ cusps (A 2 ) 2 x 3 + y 2 + z 2 = 0 Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities

Surf and Surfaces SURFEX in action Singularities Some Visualization Issues Constructing Cubic Surfaces Plans for the Future Projective Plane Curves Conclusion Simple Singularities on Cubic Surfaces Name Old Name Normal Form Coxeter Diagram µ ❘ ν x 2 k + 1 + y 2 − z 2 A − B 2 k + 1 , 2 k 0 k = 1 , 2 2 k x 2 k + 1 + y 2 + z 2 A + B 2 k + 1 0 k − 1 k = 1 2 k x 2 k + y 2 − z 2 A − 2 k − 1 B 2 k , 2 k − 1 0 k = 2 , 3 x 2 k − y 2 − z 2 A + 2 k − 1 B 2 k 1 k − 1 k = 2 x 2 + y 2 − z 2 A − C 2 1 0 1 x 2 + y 2 + z 2 A � C 2 1 0 1 x 2 y − y 3 − z 2 D − U 6 4 0 4 x 2 y + y 3 + z 2 D + U 6 2 1 4 x 2 y + y 4 − z 2 D − U 7 5 0 5 x 3 + y 4 − z 2 E − U 8 6 0 6 Multiplicity : degree of the lowest order term. Tangent Cone : homogeneous part of the lowest order. conical or bi-/uni-planar . The old names come from the tangent cone: Oliver Labs Real-Time Interactive Visualization of Deformations of Singularities