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Robert Eisele

Computer Science & Machine Learning

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TRACKBALL ROTATION USING QUATERNIONS

July 14th, 2021.

The Arcball or Trackball is an elegant and intuitive input method to rotate and
manipulate a three-dimensional scene with the mouse. The idea behind was
proposed by Ken Shoemake [Shoemake] and is quite easy to understand and to
implement which led to the fact that today it is actually the primary mechanism
that people use to rotate objects in 3D computer environments.

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INVERSE KINEMATICS OF A STEWART PLATFORM

February 24th, 2019.

In industry and science often hydraulic Stewart Platforms are deployed for their
robustness, fast acceleration and retention force, but only few publications
focus on a cheap design using electric motors. In recent years many hobbyists
built Stewart Platforms, but either they are not documented well or they are
built to have a proof of concept without much theory behind. With this article I
want to close this gap, derive the inverse kinematics of the Stewart Platform
using cheap servo motors and provide a software library to visualize and build
own platforms.

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DRAWING AN UPRIGHT STAR POLYGON

January 1st, 2019.

Drawing a regular star with nnn edges that stands upright on the ground should
be easy with a rotation by Δα=1n\Delta\alpha=\frac{1}{n}Δα=n1 around the center
of the star. For each of the 2n2n2n edges we toggle between the inner and the
outer radius like so:

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LIGHTSOUT SOLUTION USING LINEAR ALGEBRA

July 30th, 2018.

LightsOut is an electronic single player game manufactured by Tiger Toys in
1995. In the original version, 25 lights are arranged in a 5 by 5 lattice. Each
of these lights can either be on or off and its state can be inverted with a
button related to that light. Additionally, all four horizontally and vertically
adjacent lights - forming a cross - get inverted as well when a single light
gets pressed. The goal is to turn all lights out, starting with a random pattern
of lights, although we will see that not every pattern has a solution.

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HOW TO PLOT A COVARIANCE ERROR ELLIPSE

April 10th, 2018.

A typical way to visualize two-dimensional gaussian distributed data is plotting
a confidence ellipse. Lets assume we have data D∼N(μ,Σ)D\sim\mathcal{N}(\mu,
\Sigma)D∼N(μ,Σ) and want to plot an ellipse representing the confidence ppp by
calculating the radii of the ellipse, its center and rotation. The following
plot shows randomly drawn data and the ellipses for p∈{0.9,0.95,0.99}p\in\{0.9,
0.95, 0.99\}p∈{0.9,0.95,0.99}:

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CREATE A CIRCLE OUT OF THREE POINTS

February 18th, 2018.

A circle is a set of infinite points in two dimensions. However, a circle is
clearly described by three of such points. The opposite direction is also
correct as long as not all three points are on a straight line.

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MAKE A FAIR COIN FROM A BIASED COIN

January 3rd, 2018.

If we have a biased coin, i.e. a coin that comes up heads with a probability not
equal to 12\frac{1}{2}21 , how can we simulate a fair coin? The naive way would
be throwing the coin 100 times and if the coin came up heads 60 times, the bias
would be 0.6. We could then conduct an experiment to cope with this bias.

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MATHEMATICS OF A ROLLER CHAIN ANIMATION

December 9th, 2017.

Given two circular sprockets C1(M1,r1)C_1(M_1, r_1)C1 (M1 ,r1 ) and
C2(M2,r2)C_2(M_2, r_2)C2 (M2 ,r2 ) with their midpoint and radius, we want to
find a way to put a chain around them and animate the links like this:

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DETERMINE IF A NUMBER IS A PERFECT SQUARE

August 22nd, 2017.

Checking if a number is a perfect square is a trivial task when a square root
function is available. Calculating the floored square root squared should result
in the original number:

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OPERATE A SYMA S107G REMOTE CONTROL HELICOPTER WITH AN ARDUINO

July 14th, 2017.

Some time ago I saw people reverse engineering the infrared protocol of the Syma
S107G toy helicopter and I wanted to get my hands on it as well. It is a coaxial
helicopter, which means that it doesn't require a torque cancelling tail rotor.
Yaw is controlled by using the speed difference to a second rotor underneath the
main rotor, which spins in the opposite direction. It's really astonishing what
good quality you get for just 20 bucks. It crashed quite often and it still
flies like new.

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