How to find a conformal map of the unit disk on a given simply-connected domain

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By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of circles and radii around the center of the disk, like on this image taken from this site:



enter image description here



I am interested in finding such transformations for the simply-connected domains having natural origin: oak and maple leaves:



enter image description hereenter image description here



Is it possible to find and draw corresponding conformal maps?



Maybe there are some online instruments (like Wolframalpha or Maple) for doing such tasks.



The purpose of this activity is to obtain an attractive image for the cover of a textbook on univalent maps of the unit disk.






cv.complex-variables complex-geometry conformal-geometry







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  • For these complicated shapes one has to use special software. As far as I know, it is not free. You have to shop for it.
    – Alexandre Eremenko
    36 mins ago














up vote
3
down vote

favorite












By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of circles and radii around the center of the disk, like on this image taken from this site:



enter image description here



I am interested in finding such transformations for the simply-connected domains having natural origin: oak and maple leaves:



enter image description hereenter image description here



Is it possible to find and draw corresponding conformal maps?



Maybe there are some online instruments (like Wolframalpha or Maple) for doing such tasks.



The purpose of this activity is to obtain an attractive image for the cover of a textbook on univalent maps of the unit disk.






cv.complex-variables complex-geometry conformal-geometry







share|cite|improve this question





















  • For these complicated shapes one has to use special software. As far as I know, it is not free. You have to shop for it.
    – Alexandre Eremenko
    36 mins ago












up vote
3
down vote

favorite









up vote
3
down vote

favorite











By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of circles and radii around the center of the disk, like on this image taken from this site:



enter image description here



I am interested in finding such transformations for the simply-connected domains having natural origin: oak and maple leaves:



enter image description hereenter image description here



Is it possible to find and draw corresponding conformal maps?



Maybe there are some online instruments (like Wolframalpha or Maple) for doing such tasks.



The purpose of this activity is to obtain an attractive image for the cover of a textbook on univalent maps of the unit disk.






cv.complex-variables complex-geometry conformal-geometry







share|cite|improve this question













By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of circles and radii around the center of the disk, like on this image taken from this site:



enter image description here



I am interested in finding such transformations for the simply-connected domains having natural origin: oak and maple leaves:



enter image description hereenter image description here



Is it possible to find and draw corresponding conformal maps?



Maybe there are some online instruments (like Wolframalpha or Maple) for doing such tasks.



The purpose of this activity is to obtain an attractive image for the cover of a textbook on univalent maps of the unit disk.






cv.complex-variables complex-geometry conformal-geometry




cv.complex-variables complex-geometry conformal-geometry






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asked 1 hour ago









Taras Banakh

14.6k12984




14.6k12984











  • For these complicated shapes one has to use special software. As far as I know, it is not free. You have to shop for it.
    – Alexandre Eremenko
    36 mins ago
















  • For these complicated shapes one has to use special software. As far as I know, it is not free. You have to shop for it.
    – Alexandre Eremenko
    36 mins ago















For these complicated shapes one has to use special software. As far as I know, it is not free. You have to shop for it.
– Alexandre Eremenko
36 mins ago




For these complicated shapes one has to use special software. As far as I know, it is not free. You have to shop for it.
– Alexandre Eremenko
36 mins ago










3 Answers
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2
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Here is the Statue of David, conformally mapped.These guys wrote the software: http://gsl.lab.asu.edu/doc/surfacecm.html. They are the Geometry Systems Lab at Arizona State University.






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  • Thank you for the link, but it is too complicated (I mean 3D). What I need is just a planar conformal map (more precisely, the image of such a map).
    – Taras Banakh
    54 mins ago

















up vote
2
down vote













The geometry processing group at Carnegie Mellon University recently developed an algorithm called Boundary First Flattening that allows you to efficiently and interactively compute conformal parameterizations of triangle meshes. You can download the software here: https://geometrycollective.github.io/boundary-first-flattening/



It is incredibly powerful and easy to use. Unlike almost all previous algorithms for conformal parameterization it allows for significant control over the boundary shape.






share|cite|improve this answer








New contributor




yousuf soliman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

















  • Thank you for the link. If you know how to use it, could you please produce a conformal map of the unit disk onto the maple or oak leaf? Or it will take too much time?
    – Taras Banakh
    4 mins ago


















up vote
2
down vote













You may want to look at Don Marshall's Zipper algorithm:
https://sites.math.washington.edu/~marshall/zipper.html






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    3 Answers
    3






    active

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    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    2
    down vote













    Here is the Statue of David, conformally mapped.These guys wrote the software: http://gsl.lab.asu.edu/doc/surfacecm.html. They are the Geometry Systems Lab at Arizona State University.






    share|cite|improve this answer




















    • Thank you for the link, but it is too complicated (I mean 3D). What I need is just a planar conformal map (more precisely, the image of such a map).
      – Taras Banakh
      54 mins ago














    up vote
    2
    down vote













    Here is the Statue of David, conformally mapped.These guys wrote the software: http://gsl.lab.asu.edu/doc/surfacecm.html. They are the Geometry Systems Lab at Arizona State University.






    share|cite|improve this answer




















    • Thank you for the link, but it is too complicated (I mean 3D). What I need is just a planar conformal map (more precisely, the image of such a map).
      – Taras Banakh
      54 mins ago












    up vote
    2
    down vote










    up vote
    2
    down vote









    Here is the Statue of David, conformally mapped.These guys wrote the software: http://gsl.lab.asu.edu/doc/surfacecm.html. They are the Geometry Systems Lab at Arizona State University.






    share|cite|improve this answer












    Here is the Statue of David, conformally mapped.These guys wrote the software: http://gsl.lab.asu.edu/doc/surfacecm.html. They are the Geometry Systems Lab at Arizona State University.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 58 mins ago









    Ben McKay

    13.7k22758




    13.7k22758











    • Thank you for the link, but it is too complicated (I mean 3D). What I need is just a planar conformal map (more precisely, the image of such a map).
      – Taras Banakh
      54 mins ago
















    • Thank you for the link, but it is too complicated (I mean 3D). What I need is just a planar conformal map (more precisely, the image of such a map).
      – Taras Banakh
      54 mins ago















    Thank you for the link, but it is too complicated (I mean 3D). What I need is just a planar conformal map (more precisely, the image of such a map).
    – Taras Banakh
    54 mins ago




    Thank you for the link, but it is too complicated (I mean 3D). What I need is just a planar conformal map (more precisely, the image of such a map).
    – Taras Banakh
    54 mins ago










    up vote
    2
    down vote













    The geometry processing group at Carnegie Mellon University recently developed an algorithm called Boundary First Flattening that allows you to efficiently and interactively compute conformal parameterizations of triangle meshes. You can download the software here: https://geometrycollective.github.io/boundary-first-flattening/



    It is incredibly powerful and easy to use. Unlike almost all previous algorithms for conformal parameterization it allows for significant control over the boundary shape.






    share|cite|improve this answer








    New contributor




    yousuf soliman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.

















    • Thank you for the link. If you know how to use it, could you please produce a conformal map of the unit disk onto the maple or oak leaf? Or it will take too much time?
      – Taras Banakh
      4 mins ago















    up vote
    2
    down vote













    The geometry processing group at Carnegie Mellon University recently developed an algorithm called Boundary First Flattening that allows you to efficiently and interactively compute conformal parameterizations of triangle meshes. You can download the software here: https://geometrycollective.github.io/boundary-first-flattening/



    It is incredibly powerful and easy to use. Unlike almost all previous algorithms for conformal parameterization it allows for significant control over the boundary shape.






    share|cite|improve this answer








    New contributor




    yousuf soliman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.

















    • Thank you for the link. If you know how to use it, could you please produce a conformal map of the unit disk onto the maple or oak leaf? Or it will take too much time?
      – Taras Banakh
      4 mins ago













    up vote
    2
    down vote










    up vote
    2
    down vote









    The geometry processing group at Carnegie Mellon University recently developed an algorithm called Boundary First Flattening that allows you to efficiently and interactively compute conformal parameterizations of triangle meshes. You can download the software here: https://geometrycollective.github.io/boundary-first-flattening/



    It is incredibly powerful and easy to use. Unlike almost all previous algorithms for conformal parameterization it allows for significant control over the boundary shape.






    share|cite|improve this answer








    New contributor




    yousuf soliman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.









    The geometry processing group at Carnegie Mellon University recently developed an algorithm called Boundary First Flattening that allows you to efficiently and interactively compute conformal parameterizations of triangle meshes. You can download the software here: https://geometrycollective.github.io/boundary-first-flattening/



    It is incredibly powerful and easy to use. Unlike almost all previous algorithms for conformal parameterization it allows for significant control over the boundary shape.







    share|cite|improve this answer








    New contributor




    yousuf soliman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.









    share|cite|improve this answer



    share|cite|improve this answer






    New contributor




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    answered 54 mins ago









    yousuf soliman

    12113




    12113




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    yousuf soliman is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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    • Thank you for the link. If you know how to use it, could you please produce a conformal map of the unit disk onto the maple or oak leaf? Or it will take too much time?
      – Taras Banakh
      4 mins ago

















    • Thank you for the link. If you know how to use it, could you please produce a conformal map of the unit disk onto the maple or oak leaf? Or it will take too much time?
      – Taras Banakh
      4 mins ago
















    Thank you for the link. If you know how to use it, could you please produce a conformal map of the unit disk onto the maple or oak leaf? Or it will take too much time?
    – Taras Banakh
    4 mins ago





    Thank you for the link. If you know how to use it, could you please produce a conformal map of the unit disk onto the maple or oak leaf? Or it will take too much time?
    – Taras Banakh
    4 mins ago











    up vote
    2
    down vote













    You may want to look at Don Marshall's Zipper algorithm:
    https://sites.math.washington.edu/~marshall/zipper.html






    share|cite|improve this answer








    New contributor




    user130757 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.





















      up vote
      2
      down vote













      You may want to look at Don Marshall's Zipper algorithm:
      https://sites.math.washington.edu/~marshall/zipper.html






      share|cite|improve this answer








      New contributor




      user130757 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.



















        up vote
        2
        down vote










        up vote
        2
        down vote









        You may want to look at Don Marshall's Zipper algorithm:
        https://sites.math.washington.edu/~marshall/zipper.html






        share|cite|improve this answer








        New contributor




        user130757 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.









        You may want to look at Don Marshall's Zipper algorithm:
        https://sites.math.washington.edu/~marshall/zipper.html







        share|cite|improve this answer








        New contributor




        user130757 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
        Check out our Code of Conduct.









        share|cite|improve this answer



        share|cite|improve this answer






        New contributor




        user130757 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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        answered 21 mins ago









        user130757

        211




        211




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        user130757 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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