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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. 2/3pic.twitter.com/0Dj5OdGsm6
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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)pic.twitter.com/bz7eEInUwv
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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 1/2pic.twitter.com/vCKDcZofzh
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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.pic.twitter.com/r0pH4tRMSL
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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. 1/3pic.twitter.com/psAPCLRBgA
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[1/2]: Planet Chéritat - https://codepen.io/brainjam/full/jjQawN …. Webgl implementation of "a Kleinian reflection group with a Sierpinski limit set" devised by Arnaud Chéritat ( https://www.math.univ-toulouse.fr/~cheritat/GalIII/galery.html …)
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Oddly satisfying contrail doodler. (requires a mouse). https://codepen.io/brainjam/full/zQQYvN …pic.twitter.com/gGjERxepS5
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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.pic.twitter.com/UD15Cfupa9
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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.pic.twitter.com/sCD3JIznYR
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Hyperbolic billiards. Elastic collisions with clumping. There are really only three balls - black, green, blue.pic.twitter.com/MtHJ0ApRiQ
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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.pic.twitter.com/GIkaBPey5K
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Conformal morph of rectangle to the unit disk. Via Jacobi sn(). Created using https://www.wolframcloud.com/ and
@ezgif_compic.twitter.com/E0hAW1EOUE -
Conformal mapping of rectangles of varying aspect ratio (w/h) to the unit disk. Via Jacobi sn(). Created using https://www.wolframcloud.com/ and
@ezgif_compic.twitter.com/QjHrkNiMew -
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.pic.twitter.com/V7DQafxk4A
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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 backgroundpic.twitter.com/H5XoDaB9eG
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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.pic.twitter.com/J5M1GcGrMS
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A conformal neighbourhood of elliptic curve y² = 4x³ - 4x + 1.pic.twitter.com/eM54BHzhJs
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Carpet fractal: https://codepen.io/brainjam/full/RBgNZb/ …. These behave like Julia sets of rational quadratics, but are rectangular. The white shapes coming in from the border remind me of Trix rabbits.pic.twitter.com/an9Xb1WSXE
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