Peter Liepa

@peterliepa

Visual math, mainly geometry -- euclidean, hyperbolic, projective, conformal, computational.

Toronto
Joined May 2009

Media

  1. 7 Dec 2019

    The gradient used is the gradient of the angle ACB for two fixed points A and B. That's an unusual gradient, with a funky formula, but the circular trajectories follow from the fact that inscribed angles in a circle subtended by the same chord (AB here) are equal. 2/3

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  2. 7 Dec 2019

    A gradient flow where the trajectories form orthogonal sets of coaxial circles. Some liberties taken in this visualization, because particles flow in both directions along flow lines and level sets. (short thread)

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  3. 4 Dec 2019

    Projective geometry is endlessly fascinating. The six points of a complete quadrilateral (blue) are centrally projected to a conic. The images of opposite points define 3 lines (green) that are concurrent (as indicated by the red arrow). Based on an illustration 1/2

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  4. 23 Sep 2019

    Just for fun, here's a "conformal neighborhood" of the curve. The small gap in the upper left is where |t|>100. I'm impressed that you guys could come up with both implicit and parametric formulae for the curve. I didn't know the latter was even possible.

  5. 18 Sep 2019

    Conformal neighborhoods of circles and subcircles. I.e. square grids that follow the curve. Square size depends on arc speed of underlying parameterization. Grid on one side of the curve is Schwarz reflection of grid on the other side, which here is inversion in the circle. 1/3

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  6. 1 Aug 2019

    [1/2]: Planet Chéritat - . Webgl implementation of "a Kleinian reflection group with a Sierpinski limit set" devised by Arnaud Chéritat ( )

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  7. 24 Jun 2019

    Oddly satisfying contrail doodler. (requires a mouse).

  8. 13 Jun 2019

    From the archives, vintage 2010. A hyperbolic tiling mapped to the exterior of a filled Julia set. Frames from a zoom sequence -- wide shot, mid-zoom, closeup.

  9. 7 Jun 2019
    Replying to

    This is a version showing particle trails. Perhaps not exactly what you were requesting, but it gives an idea of how particles move approximately along geodesics.

  10. 5 Jun 2019

    Hyperbolic billiards. A close-up.

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  11. 5 Jun 2019

    Hyperbolic billiards. Elastic collisions with clumping. There are really only three balls - black, green, blue.

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  12. 23 May 2019

    I'm working on a hyperbolic doodler -- an online paint program for triangle group tessellations. Here are some test images. Wabi-sabi meets the hyperbolic plane meets webgl.

  13. 8 May 2019

    Conformal morph of rectangle to the unit disk. Via Jacobi sn(). Created using and

  14. 8 May 2019

    Conformal mapping of rectangles of varying aspect ratio (w/h) to the unit disk. Via Jacobi sn(). Created using and

  15. 23 Apr 2019

    A sphere with Jacobi's ellipsoidal parameterization. A single parameter controls the location of the umbilics and stands in for the eccentricity of the general triaxial ellipsoid. At both extremes this becomes a conventional spherical parameterization.

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  16. 17 Apr 2019

    Elliptic curve y² = 4x³ - ax + 1 and its conformal neighborhood for a∈[2,4]. The curve transitions from one component to two at a=3. It is parameterized here with the elliptic functions Weierstrass ℘,℘'. The half period rectangle is shown in light red in the background

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  17. 8 Apr 2019

    Schwarzian reflection of a grid in an ellipse. The original grid is green, the reflection is red. When the ellipse is squashed to a line, the grid is mirror reflected. When the ellipse is a circle, the ellipse is inverted in the circle.

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  18. 8 Apr 2019

    A conformal neighbourhood of elliptic curve y² = 4x³ - 4x + 1.

  19. 9 Nov 2018
  20. 25 Jul 2018

    Carpet fractal: . These behave like Julia sets of rational quadratics, but are rectangular. The white shapes coming in from the border remind me of Trix rabbits.

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