The curve of intersection of a torus with a plane is called toric section. Even if both surfaces are rather simple to define and are described by rather simple equations, the toric section has a rather complicated equation and can assume rather interesting shapes.
In this article I’ll discuss some properties of this curve, investigate its differences with the most renowned conic section, show how to build its general quartic equation, how a toric section can also be generated by intersecting a cylinder with a cone and finally describe how it is possible to represent it in the 3D Graphics view of Geogebra…
This simulation could help students to playfully discover and experiment the properties of the parabolic motion and to understand the concept of flux of a vector field (ball velocity) through a surface (the one delimited by the basketball ring).
I made it also to explore the limits of what can be done with Geogebra in building a complex simulation and I’m rather impressed with the width of its opportunities, especially through the scripting and with its (rather hidden) potential.
Note: given complexity of the simulation it’s advisable to download the .ggb file and run it locally through the free Geogebra Classic desktop program. The web app may be rather slow and jerky.
The geogebra file can be downloaded in this page of the Geogebra material portal (link).
The Geogebra Classic desktop program can be downloaded at this page.
Update June 2017
A new version of the simulation with rebounds, allowing rimshots and bankshots, has been published here at the Geogebra material portal.
Three charged particles, two positive (blue) and one negative (red) are released from rest at the vertices of an isosceles triangle (equilateral in the initial setting).
It’s assumed that the particles have the same charge (but for the sign), the same mass (inertia) and that only the electric force acts on them.
The system dynamics will be just driven by Coulombian attractive/repulsive forces.
Anyway, given the strong symmetries in the initial conditions and given the conservation of energy and momentum, the system can be reduced to just a couple of differential equations, since the position/velocity of one of the blue particles is enough to set the positions/velocities of the other two. Here’s a video of the resulting “Electric dance”:
Every time I try to get some deeper insight about Newton’s gravitational law I stumble upon the geometrical properties of the ellipse.
After many years of these strange and challenging encounters I really think that the ellipse is a rich and wonderful trove of geometrical nuances and subtleties…
On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of . It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and a false alarm rate estimated to be less than 1 event per 203 000 years, equivalent to a significance greater than 5.1σ. The source lies at a luminosity distance of Mpc corresponding to a redshift . In the source frame, the initial black hole masses are M⊙ and M⊙, and the final black hole mass is M⊙, with M⊙ radiated in gravitational waves. All uncertainties define 90% credible intervals. These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.
Well, actually that is not a complete Apollonian gasket, but it can give the idea.
To produce a full gasket the code should be longer than that allowed by a twitter length, but I think that a basic one could be done in about 500 characters or less.
A new interactive simulation created with the software Wolfram Mathematica, reproducing an anticipatory model for pedestrian interactions, is now available at this page of the CDF simulation section.
Here’s the demo video posted on youtube:
The model is based on the paper:
A universal power law governing pedestrian interactions by Ioannis Karamouzas, Brian Skinner, and Stephen J. Guy
published on 2 December 2014 in Physical Review Letters
The article, together with some other interesting related material, is also available in this page of the Applied Motion Lab, University of Minnesota.
The main point of the model is that the interaction force between two pedestrian is not based on their distance (as it’d happen for, say, electrons) but rather on their time-to-collision, which is defined as “the duration of time for which two pedestrians could continue walking at their current velocities before colliding”.
So, in this model you won’t see nearby pedestrians repel each other if their trajectories are not such to produce a collision in the next few seconds. That makes possible for pedestrians to walk side by side as it happens in the real world.
On the other hand two pedestrians about to collide will try to change their motion (in velocity direction and/or speed).
See the CDF simulationspage for further details.