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Peter Liepa
Visual math, mainly geometry -- euclidean, hyperbolic, projective, conformal, computational.
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Peter Liepa Jan 5
Replying to @octonion
Russets, although you’ll probably never find them at your Kroger. Maybe they don’t store well, or have sufficient eye appeal, but they have never been mainstream like the industrial varieties shown and discussed in this thread. Grown in Ontario.
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Peter Liepa Dec 25
Replying to @RoyWiggins
Nice! I’m surprised that the transformed circles still look like circles. Did you do anything special, or did it just work out that way?
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Peter Liepa Dec 7
Replying to @CodePen
Live version and code at . (Doesn't work well on iPhone Safari, but what else is new.) 3/3
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Peter Liepa Dec 7
Replying to @peterliepa
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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Peter Liepa Dec 7
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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Peter Liepa Dec 4
Replying to @geogebra
of the Hesse Transfer Principle in Richter-Gebert's Perspectives on Projective Geometry. 2/2
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Peter Liepa Dec 4
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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Peter Liepa Nov 29
Replying to @74WTungsteno
"[The] centre of the parabola is its only point on the line at infinity, which is [in] the direction of the axis of the parabola. The parabola is tangent to the line at infinity at its centre." (From )
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Peter Liepa Oct 29
Replying to @keenanisalive
Restricted to the unit circle, hyperbolic transformations are equivalent to projective transformations, and can be expressed as 3x3 matrices. Thus all Pythagorean triples (viewed as vectors) can be generated by applying 3x3 matrices. This leads to .
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Peter Liepa Sep 24
Replying to @keenanisalive
... perhaps by Regge himself (a physicist who worked with simplicial approximations of spacetime). I think it was from the 1950/60's. Can't remember whether the formula was 3D or nD. 2/2
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Peter Liepa Sep 24
Replying to @keenanisalive
Can you email this 15 years back in time to me when I was struggling with this stuff? The 3d case is mentioned in Tu et al (2006 eqn 20). I remember seeing it in the physics literature going as far back as a typescript on Regge calculus ... 1/2
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Peter Liepa Sep 23
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.
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Peter Liepa Sep 18
Replying to @peterliepa
Conformal mappings for the given pictures are normalized Jacobi sn, inverse stereoscopic, cos+i sin, and a frequency modulated version of cos+i sin. These were rendered in online Mathematica and have some artifacts. 3/3
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Peter Liepa Sep 18
Replying to @peterliepa
To compute these neighborhoods, find a conformal mapping that takes a real interval (or the entire real axis) to the circle, and then map the image of a grid containing the domain. (This of course can be generalized with maps to any curve) 2/3
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Peter Liepa Sep 18
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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Peter Liepa Sep 17
Replying to @ICERM
If that link doesn't work try
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Peter Liepa Aug 9
I feel this way about math (especially geometry): "Writing songs is a great thing. It's like a jigsaw puzzle and a kaleidoscope put together" -
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Peter Liepa Aug 5
On my list of 500 mathy things to do is to tesselate the flat version into a triangular mesh, glue the corresponding boundaries, and then try curvature flow to get the final genus 3 surface, e.g. as in by et al.
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Peter Liepa Aug 1
Replying to @peterliepa
[2/2]: A Riemann sphere tessellated by a packing of hyperbolic tessellations. Induced by a Kleinian reflection group whose limit set is a Sierpinski circle packing. Other visualizations by Chéritat are at and
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Peter Liepa Aug 1
[1/2]: Planet Chéritat - . Webgl implementation of "a Kleinian reflection group with a Sierpinski limit set" devised by Arnaud Chéritat ( )
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